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In Chinese belief, the asterism consisting of Altair, β Aquilae and γ Aquilae is known as Hé Gǔ (; lit. "river drum"). The Chinese name for Altair is thus Hé Gǔ èr (; lit. "river drum two", meaning the "second star of the drum at the river"). However, Altair is better known by its other names: Qiān Niú Xīng ( / ) or Niú Láng Xīng (), translated as the cowherd star. These names are an allusion to a love story, The Cowherd and the Weaver Girl, in which Niulang (represented by Altair) and his two children (represented by β Aquilae and γ Aquilae) are separated from respectively their wife and mother Zhinu (represented by Vega) by the Milky Way. They are only permitted to meet once a year, when magpies form a bridge to allow them to cross the Milky Way.
The people of Micronesia called Altair Mai-lapa, meaning "big/old breadfruit", while the Māori people called this star Poutu-te-rangi, meaning "pillar of heaven".
In Western astrology, the star was ill-omened, portending danger from reptiles.
This star is one of the asterisms used by Bugis sailors for navigation, called bintoéng timoro, meaning "eastern star".
A group of Japanese scientists sent a radio signal to Altair in 1983 with the hopes of contacting extraterrestrial life.
NASA announced Altair as the name of the Lunar Surface Access Module (LSAM) on December 13, 2007. The Russian-made Beriev Be-200 Altair seaplane is also named after the star.
Visual companions
The bright primary star has the multiple star designation WDS 19508+0852A and has several faint visual companion stars, WDS 19508+0852B, C, D, E, F and G. All are much more distant than Altair and not physically associated. | Altair | Wikipedia | 428 | 3078 | https://en.wikipedia.org/wiki/Altair | Physical sciences | Notable stars | Astronomy |
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.
The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.
There are three kinds of asymptotes: horizontal, vertical and oblique. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to
More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.
Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph. The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.
Introduction | Asymptote | Wikipedia | 401 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see Line). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
Consider the graph of the function shown in this section. The coordinates of the points on the curve are of the form where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of , .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large becomes, its reciprocal is never 0, so the curve never actually touches the x-axis. Similarly, as the values of become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of , 100, 1,000, 10,000 ..., become larger and larger. So the curve extends further and further upward as it comes closer and closer to the y-axis. Thus, both the x and y-axis are asymptotes of the curve. These ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below. | Asymptote | Wikipedia | 413 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
Asymptotes of functions
The asymptotes most commonly encountered in the study of calculus are of curves of the form . These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x-axis. Vertical asymptotes are vertical lines (perpendicular to the x-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞.
Vertical asymptotes
The line x = a is a vertical asymptote of the graph of the function if at least one of the following statements is true:
where is the limit as x approaches the value a from the left (from lesser values), and is the limit as x approaches a from the right.
For example, if ƒ(x) = x/(x–1), the numerator approaches 1 and the denominator approaches 0 as x approaches 1. So
and the curve has a vertical asymptote x = 1.
The function ƒ(x) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote. For example, for the function
has a limit of +∞ as , ƒ(x) has the vertical asymptote , even though ƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or a vertical line in general) in more than one point. Moreover, if a function is continuous at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
at .
This function has a vertical asymptote at because
and
. | Asymptote | Wikipedia | 508 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
The derivative of is the function
.
For the sequence of points
for
that approaches both from the left and from the right, the values are constantly . Therefore, both one-sided limits of at can be neither nor . Hence doesn't have a vertical asymptote at .
Horizontal asymptotes
Horizontal asymptotes are horizontal lines that the graph of the function approaches as . The horizontal line y = c is a horizontal asymptote of the function y = ƒ(x) if
or .
In the first case, ƒ(x) has y = c as asymptote when x tends to , and in the second ƒ(x) has y = c as an asymptote as x tends to .
For example, the arctangent function satisfies
and
So the line is a horizontal asymptote for the arctangent when x tends to , and is a horizontal asymptote for the arctangent when x tends to .
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function has a horizontal asymptote at y = 0 when x tends both to and because, respectively,
Other common functions that have one or two horizontal asymptotes include (that has an hyperbola as it graph), the Gaussian function the error function, and the logistic function.
Oblique asymptotes
When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote. A function ƒ(x) is asymptotic to the straight line (m ≠ 0) if
In the first case the line is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line is an oblique asymptote of ƒ(x) when x tends to −∞.
An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits
Elementary methods for identifying asymptotes
The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).
General computation of oblique asymptotes for functions | Asymptote | Wikipedia | 506 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
The oblique asymptote, for the function f(x), will be given by the equation y = mx + n. The value for m is computed first and is given by
where a is either or depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
Having m then the value for n can be computed by
where a should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist. Otherwise is the oblique asymptote of ƒ(x) as x tends to a.
For example, the function has
and then
so that is the asymptote of ƒ(x) when x tends to +∞.
The function has
and then
, which does not exist.
So does not have an asymptote when x tends to +∞.
Asymptotes for rational functions
A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x = 0, and x = 1, but not at x = 2.
Oblique asymptotes of rational functions
When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function | Asymptote | Wikipedia | 474 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
shown to the right. As the value of x increases, f approaches the asymptote y = x. This is because the other term, 1/(x+1), approaches 0.
If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as x increases, but the quotient will not be linear, and the function does not have an oblique asymptote.
Transformations of known functions
If a known function has an asymptote (such as y=0 for f(x)=ex), then the translations of it also have an asymptote.
If x=a is a vertical asymptote of f(x), then x=a+h is a vertical asymptote of f(x-h)
If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k
If a known function has an asymptote, then the scaling of the function also have an asymptote.
If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x)
For example, f(x)=ex-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.
General definition
Let be a parametric plane curve, in coordinates A(t) = (x(t),y(t)). Suppose that the curve tends to infinity, that is:
A line ℓ is an asymptote of A if the distance from the point A(t) to ℓ tends to zero as t → b. From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote. | Asymptote | Wikipedia | 432 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
For example, the upper right branch of the curve y = 1/x can be defined parametrically as x = t, y = 1/t (where t > 0). First, x → ∞ as t → ∞ and the distance from the curve to the x-axis is 1/t which approaches 0 as t → ∞. Therefore, the x-axis is an asymptote of the curve. Also, y → ∞ as t → 0 from the right, and the distance between the curve and the y-axis is t which approaches 0 as t → 0. So the y-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is then the distance from the point A(t) = (x(t),y(t)) to the line is given by
if γ(t) is a change of parameterization then the distance becomes
which tends to zero simultaneously as the previous expression.
An important case is when the curve is the graph of a real function (a function of one real variable and returning real values). The graph of the function y = ƒ(x) is the set of points of the plane with coordinates (x,ƒ(x)). For this, a parameterization is
This parameterization is to be considered over the open intervals (a,b), where a can be −∞ and b can be +∞.
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is x = c, for some real number c. The non-vertical case has equation , where m and are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
Curvilinear asymptotes | Asymptote | Wikipedia | 468 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
Let be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred to as an asymptote of A, when there is no risk of confusion with linear asymptotes.
For example, the function
has a curvilinear asymptote , which is known as a parabolic asymptote because it is a parabola rather than a straight line.
Asymptotes and curve sketching
Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity. In order to get better approximations of the curve, curvilinear asymptotes have also been used although the term asymptotic curve seems to be preferred.
Algebraic curves
The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity. For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, although they also make sense when defined in this way for curves over an arbitrary field.
A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
A plane algebraic curve is defined by an equation of the form P(x,y) = 0 where P is a polynomial of degree n
where Pk is homogeneous of degree k. Vanishing of the linear factors of the highest degree term Pn defines the asymptotes of the curve: setting , if , then the line | Asymptote | Wikipedia | 475 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
is an asymptote if and are not both zero. If and , there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a , even when it does not have any parabola that is a curvilinear asymptote. If the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
Over the complex numbers, Pn splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals, Pn splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve has no real points outside the square , but its highest order term gives the linear factor x with multiplicity 4, leading to the unique asymptote x=0.
Asymptotic cone
The hyperbola
has the two asymptotes
The equation for the union of these two lines is
Similarly, the hyperboloid
is said to have the asymptotic cone
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
More generally, consider a surface that has an implicit equation
where the are homogeneous polynomials of degree and . Then the equation defines a cone which is centered at the origin. It is called an asymptotic cone, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity. | Asymptote | Wikipedia | 382 | 3107 | https://en.wikipedia.org/wiki/Asymptote | Mathematics | Mathematical analysis | null |
Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers.
Another distinction is based on the numeral system employed to perform calculations. Decimal arithmetic is the most common. It uses the basic numerals from 0 to 9 and their combinations to express numbers. Binary arithmetic, by contrast, is used by most computers and represents numbers as combinations of the basic numerals 0 and 1. Computer arithmetic deals with the specificities of the implementation of binary arithmetic on computers. Some arithmetic systems operate on mathematical objects other than numbers, such as interval arithmetic and matrix arithmetic.
Arithmetic operations form the basis of many branches of mathematics, such as algebra, calculus, and statistics. They play a similar role in the sciences, like physics and economics. Arithmetic is present in many aspects of daily life, for example, to calculate change while shopping or to manage personal finances. It is one of the earliest forms of mathematics education that students encounter. Its cognitive and conceptual foundations are studied by psychology and philosophy.
The practice of arithmetic is at least thousands and possibly tens of thousands of years old. Ancient civilizations like the Egyptians and the Sumerians invented numeral systems to solve practical arithmetic problems in about 3000 BCE. Starting in the 7th and 6th centuries BCE, the ancient Greeks initiated a more abstract study of numbers and introduced the method of rigorous mathematical proofs. The ancient Indians developed the concept of zero and the decimal system, which Arab mathematicians further refined and spread to the Western world during the medieval period. The first mechanical calculators were invented in the 17th century. The 18th and 19th centuries saw the development of modern number theory and the formulation of axiomatic foundations of arithmetic. In the 20th century, the emergence of electronic calculators and computers revolutionized the accuracy and speed with which arithmetic calculations could be performed. | Arithmetic | Wikipedia | 448 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Definition, etymology, and related fields
Arithmetic is the fundamental branch of mathematics that studies numbers and their operations. In particular, it deals with numerical calculations using the arithmetic operations of addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and logarithm. The term arithmetic has its root in the Latin term which derives from the Ancient Greek words (arithmos), meaning , and (arithmetike tekhne), meaning .
There are disagreements about its precise definition. According to a narrow characterization, arithmetic deals only with natural numbers. However, the more common view is to include operations on integers, rational numbers, real numbers, and sometimes also complex numbers in its scope. Some definitions restrict arithmetic to the field of numerical calculations. When understood in a wider sense, it also includes the study of how the concept of numbers developed, the analysis of properties of and relations between numbers, and the examination of the axiomatic structure of arithmetic operations.
Arithmetic is closely related to number theory and some authors use the terms as synonyms. However, in a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality. Traditionally, it is known as higher arithmetic.
Numbers
Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different kinds of numbers and different numeral systems to represent them.
Kinds
The main kinds of numbers employed in arithmetic are natural numbers, whole numbers, integers, rational numbers, and real numbers. The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as . The symbol of the natural numbers is . The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as and have the symbol . Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers. The set of integers encompasses both positive and negative whole numbers. It has the symbol and can be expressed as . | Arithmetic | Wikipedia | 458 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Based on how natural and whole numbers are used, they can be distinguished into cardinal and ordinal numbers. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?".
A number is rational if it can be represented as the ratio of two integers. For instance, the rational number is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are and . The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is . Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For instance, 0.3 is equal to , and 25.12 is equal to . Every rational number corresponds to a finite or a repeating decimal.
Irrational numbers are numbers that cannot be expressed through the ratio of two integers. They are often required to describe geometric magnitudes. For example, if a right triangle has legs of the length 1 then the length of its hypotenuse is given by the irrational number . is another irrational number and describes the ratio of a circle's circumference to its diameter. The decimal representation of an irrational number is infinite without repeating decimals. The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is . Even wider classes of numbers include complex numbers and quaternions.
Numeral systems
A numeral is a symbol to represent a number and numeral systems are representational frameworks. They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number. Numeral systems are either positional or non-positional. All early numeral systems were non-positional. For non-positional numeral systems, the value of a digit does not depend on its position in the numeral. | Arithmetic | Wikipedia | 465 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
The simplest non-positional system is the unary numeral system. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers. Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks.
Egyptian hieroglyphics had a more complex non-positional numeral system. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For instance, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000.
A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplicand of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the Hindu–Arabic numeral system, the radix is 10. This means that the first digit is multiplied by , the next digit is multiplied by , and so on. For example, the decimal numeral 532 stands for . Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.
Another positional numeral system used extensively in computer arithmetic is the binary system, which has a radix of 2. This means that the first digit is multiplied by , the next digit by , and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for . In computing, each digit in the binary notation corresponds to one bit. The earliest positional system was developed by ancient Babylonians and had a radix of 60.
Operations | Arithmetic | Wikipedia | 495 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are functions that have numbers both as input and output. The most important operations in arithmetic are addition, subtraction, multiplication, and division. Further operations include exponentiation, extraction of roots, and logarithm. If these operations are performed on variables rather than numbers, they are sometimes referred to as algebraic operations.
Two important concepts in relation to arithmetic operations are identity elements and inverse elements. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the additive inverse of the number 6 is -6 since their sum is 0.
There are not only inverse elements but also inverse operations. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in . Defined more formally, the operation "" is an inverse of the operation "" if it fulfills the following condition: if and only if .
Commutativity and associativity are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, is the same as . Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since is the same as .
Addition and subtraction
Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is . Examples are and . The term summation is used if several additions are performed in a row. Counting is a type of repeated addition in which the number 1 is continuously added. | Arithmetic | Wikipedia | 472 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is . Examples are and . Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance . This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.
The additive identity element is 0 and the additive inverse of a number is the negative of that number. For instance, and . Addition is both commutative and associative.
Multiplication and division
Multiplication is an arithmetic operation in which two numbers, called the multiplier and the multiplicand, are combined into a single number called the product. The symbols of multiplication are , , and *. Examples are and . If the multiplicand is a natural number then multiplication is the same as repeated addition, as in .
Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the quotient. The symbols of division are and . Examples are and . Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its reciprocal. The reciprocal of a number is 1 divided by that number. For instance, .
The multiplicative identity element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, and . Multiplication is both commutative and associative.
Exponentiation and logarithm
Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in superscript right after the base. Examples are and ^. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in . | Arithmetic | Wikipedia | 472 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of and the cube root of a number is the same as raising the number to the power of . Examples are and .
Logarithm is the inverse of exponentiation. The logarithm of a number to the base is the exponent to which must be raised to produce . For instance, since , the logarithm base 10 of 1000 is 3. The logarithm of to base is denoted as , or without parentheses, , or even without the explicit base, , when the base can be understood from context. So, the previous example can be written .
Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in . However, exponentiation does not have a general identity element since 1 is not the neutral element for the base. Exponentiation and logarithm are neither commutative nor associative.
Types
Different types of arithmetic systems are discussed in the academic literature. They differ from each other based on what type of number they operate on, what numeral system they use to represent them, and whether they operate on mathematical objects other than numbers.
Integer arithmetic
Integer arithmetic is the branch of arithmetic that deals with the manipulation of positive and negative whole numbers. Simple one-digit operations can be performed by following or memorizing a table that presents the results of all possible combinations, like an addition table or a multiplication table. Other common methods are verbal counting and finger-counting. | Arithmetic | Wikipedia | 352 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
For operations on numbers with more than one digit, different techniques can be employed to calculate the result by using several one-digit operations in a row. For example, in the method addition with carries, the two numbers are written one above the other. Starting from the rightmost digit, each pair of digits is added together. The rightmost digit of the sum is written below them. If the sum is a two-digit number then the leftmost digit, called the "carry", is added to the next pair of digits to the left. This process is repeated until all digits have been added. Other methods used for integer additions are the number line method, the partial sum method, and the compensation method. A similar technique is utilized for subtraction: it also starts with the rightmost digit and uses a "borrow" or a negative carry for the column on the left if the result of the one-digit subtraction is negative.
A basic technique of integer multiplication employs repeated addition. For example, the product of can be calculated as . A common technique for multiplication with larger numbers is called long multiplication. This method starts by writing the multiplier above the multiplicand. The calculation begins by multiplying the multiplier only with the rightmost digit of the multiplicand and writing the result below, starting in the rightmost column. The same is done for each digit of the multiplicand and the result in each case is shifted one position to the left. As a final step, all the individual products are added to arrive at the total product of the two multi-digit numbers. Other techniques used for multiplication are the grid method and the lattice method. Computer science is interested in multiplication algorithms with a low computational complexity to be able to efficiently multiply very large integers, such as the Karatsuba algorithm, the Schönhage–Strassen algorithm, and the Toom–Cook algorithm. A common technique used for division is called long division. Other methods include short division and chunking. | Arithmetic | Wikipedia | 413 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. One way to ensure that the result is an integer is to round the result to a whole number. However, this method leads to inaccuracies as the original value is altered. Another method is to perform the division only partially and retain the remainder. For example, 7 divided by 2 is 3 with a remainder of 1. These difficulties are avoided by rational number arithmetic, which allows for the exact representation of fractions.
A simple method to calculate exponentiation is by repeated multiplication. For instance, the exponentiation of can be calculated as . A more efficient technique used for large exponents is exponentiation by squaring. It breaks down the calculation into a number of squaring operations. For example, the exponentiation can be written as . By taking advantage of repeated squaring operations, only 7 individual operations are needed rather than the 64 operations required for regular repeated multiplication. Methods to calculate logarithms include the Taylor series and continued fractions. Integer arithmetic is not closed under logarithm and under exponentiation with negative exponents, meaning that the result of these operations is not always an integer.
Number theory | Arithmetic | Wikipedia | 279 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Number theory studies the structure and properties of integers as well as the relations and laws between them. Some of the main branches of modern number theory include elementary number theory, analytic number theory, algebraic number theory, and geometric number theory. Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include divisibility, factorization, and primality. Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like how prime numbers are distributed and the claim that every even number is a sum of two prime numbers. Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of fields and rings, as in algebraic number fields like the ring of integers. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane. Further branches of number theory are probabilistic number theory, which employs methods from probability theory, combinatorial number theory, which relies on the field of combinatorics, computational number theory, which approaches number-theoretic problems with computational methods, and applied number theory, which examines the application of number theory to fields like physics, biology, and cryptography.
Influential theorems in number theory include the fundamental theorem of arithmetic, Euclid's theorem, and Fermat's Last Theorem. According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the number 18 is not a prime number and can be represented as , all of which are prime numbers. The number 19, by contrast, is a prime number that has no other prime factorization. Euclid's theorem states that there are infinitely many prime numbers. Fermat's Last Theorem is the statement that no positive integer values exist for , , and that solve the equation if is greater than . | Arithmetic | Wikipedia | 398 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Rational number arithmetic
Rational number arithmetic is the branch of arithmetic that deals with the manipulation of numbers that can be expressed as a ratio of two integers. Most arithmetic operations on rational numbers can be calculated by performing a series of integer arithmetic operations on the numerators and the denominators of the involved numbers. If two rational numbers have the same denominator then they can be added by adding their numerators and keeping the common denominator. For example, . A similar procedure is used for subtraction. If the two numbers do not have the same denominator then they must be transformed to find a common denominator. This can be achieved by scaling the first number with the denominator of the second number while scaling the second number with the denominator of the first number. For instance, .
Two rational numbers are multiplied by multiplying their numerators and their denominators respectively, as in . Dividing one rational number by another can be achieved by multiplying the first number with the reciprocal of the second number. This means that the numerator and the denominator of the second number change position. For example, . Unlike integer arithmetic, rational number arithmetic is closed under division as long as the divisor is not 0.
Both integer arithmetic and rational number arithmetic are not closed under exponentiation and logarithm. One way to calculate exponentiation with a fractional exponent is to perform two separate calculations: one exponentiation using the numerator of the exponent followed by drawing the nth root of the result based on the denominator of the exponent. For example, . The first operation can be completed using methods like repeated multiplication or exponentiation by squaring. One way to get an approximate result for the second operation is to employ Newton's method, which uses a series of steps to gradually refine an initial guess until it reaches the desired level of accuracy. The Taylor series or the continued fraction method can be utilized to calculate logarithms. | Arithmetic | Wikipedia | 417 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
The decimal fraction notation is a special way of representing rational numbers whose denominator is a power of 10. For instance, the rational numbers , , and are written as 0.1, 3.71, and 0.0044 in the decimal fraction notation. Modified versions of integer calculation methods like addition with carry and long multiplication can be applied to calculations with decimal fractions. Not all rational numbers have a finite representation in the decimal notation. For example, the rational number corresponds to 0.333... with an infinite number of 3s. The shortened notation for this type of repeating decimal is 0.. Every repeating decimal expresses a rational number.
Real number arithmetic
Real number arithmetic is the branch of arithmetic that deals with the manipulation of both rational and irrational numbers. Irrational numbers are numbers that cannot be expressed through fractions or repeated decimals, like the root of 2 and . Unlike rational number arithmetic, real number arithmetic is closed under exponentiation as long as it uses a positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1.
Irrational numbers involve an infinite non-repeating series of decimal digits. Because of this, there is often no simple and accurate way to express the results of arithmetic operations like or In cases where absolute precision is not required, the problem of calculating arithmetic operations on real numbers is usually addressed by truncation or rounding. For truncation, a certain number of leftmost digits are kept and remaining digits are discarded or replaced by zeros. For example, the number has an infinite number of digits starting with 3.14159.... If this number is truncated to 4 decimal places, the result is 3.141. Rounding is a similar process in which the last preserved digit is increased by one if the next digit is 5 or greater but remains the same if the next digit is less than 5, so that the rounded number is the best approximation of a given precision for the original number. For instance, if the number is rounded to 4 decimal places, the result is 3.142 because the following digit is a 5, so 3.142 is closer to than 3.141. These methods allow computers to efficiently perform approximate calculations on real numbers.
Approximations and errors | Arithmetic | Wikipedia | 468 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
In science and engineering, numbers represent estimates of physical quantities derived from measurement or modeling. Unlike mathematically exact numbers such as or scientifically relevant numerical data are inherently inexact, involving some measurement uncertainty. One basic way to express the degree of certainty about each number's value and avoid false precision is to round each measurement to a certain number of digits, called significant digits, which are implied to be accurate. For example, a person's height measured with a tape measure might only be precisely known to the nearest centimeter, so should be presented as 1.62 meters rather than 1.6217 meters. If converted to imperial units, this quantity should be rounded to 64 inches or 63.8 inches rather than 63.7795 inches, to clearly convey the precision of the measurement. When a number is written using ordinary decimal notation, leading zeros are not significant, and trailing zeros of numbers not written with a decimal point are implicitly considered to be non-significant. For example, the numbers 0.056 and 1200 each have only 2 significant digits, but the number 40.00 has 4 significant digits. Representing uncertainty using only significant digits is a relatively crude method, with some unintuitive subtleties; explicitly keeping track of an estimate or upper bound of the approximation error is a more sophisticated approach. In the example, the person's height might be represented as meters or .
In performing calculations with uncertain quantities, the uncertainty should be propagated to calculated quantities. When adding or subtracting two or more quantities, add the absolute uncertainties of each summand together to obtain the absolute uncertainty of the sum. When multiplying or dividing two or more quantities, add the relative uncertainties of each factor together to obtain the relative uncertainty of the product. When representing uncertainty by significant digits, uncertainty can be coarsely propagated by rounding the result of adding or subtracting two or more quantities to the leftmost last significant decimal place among the summands, and by rounding the result of multiplying or dividing two or more quantities to the least number of significant digits among the factors. (See .) | Arithmetic | Wikipedia | 436 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
More sophisticated methods of dealing with uncertain values include interval arithmetic and affine arithmetic. Interval arithmetic describes operations on intervals. Intervals can be used to represent a range of values if one does not know the precise magnitude, for example, because of measurement errors. Interval arithmetic includes operations like addition and multiplication on intervals, as in and . It is closely related to affine arithmetic, which aims to give more precise results by performing calculations on affine forms rather than intervals. An affine form is a number together with error terms that describe how the number may deviate from the actual magnitude.
The precision of numerical quantities can be expressed uniformly using normalized scientific notation, which is also convenient for concisely representing numbers which are much larger or smaller than 1. Using scientific notation, a number is decomposed into the product of a number between 1 and 10, called the significand, and 10 raised to some integer power, called the exponent. The significand consists of the significant digits of the number, and is written as a leading digit 1–9 followed by a decimal point and a sequence of digits 0–9. For example, the normalized scientific notation of the number 8276000 is with significand 8.276 and exponent 6, and the normalized scientific notation of the number 0.00735 is with significand 7.35 and exponent −3. Unlike ordinary decimal notation, where trailing zeros of large numbers are implicitly considered to be non-significant, in scientific notation every digit in the significand is considered significant, and adding trailing zeros indicates higher precision. For example, while the number 1200 implicitly has only 2 significant digits, the number explicitly has 3. | Arithmetic | Wikipedia | 349 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
A common method employed by computers to approximate real number arithmetic is called floating-point arithmetic. It represents real numbers similar to the scientific notation through three numbers: a significand, a base, and an exponent. The precision of the significand is limited by the number of bits allocated to represent it. If an arithmetic operation results in a number that requires more bits than are available, the computer rounds the result to the closest representable number. This leads to rounding errors. A consequence of this behavior is that certain laws of arithmetic are violated by floating-point arithmetic. For example, floating-point addition is not associative since the rounding errors introduced can depend on the order of the additions. This means that the result of is sometimes different from the result of The most common technical standard used for floating-point arithmetic is called IEEE 754. Among other things, it determines how numbers are represented, how arithmetic operations and rounding are performed, and how errors and exceptions are handled. In cases where computation speed is not a limiting factor, it is possible to use arbitrary-precision arithmetic, for which the precision of calculations is only restricted by the computer's memory.
Tool use
Forms of arithmetic can also be distinguished by the tools employed to perform calculations and include many approaches besides the regular use of pen and paper. Mental arithmetic relies exclusively on the mind without external tools. Instead, it utilizes visualization, memorization, and certain calculation techniques to solve arithmetic problems. One such technique is the compensation method, which consists in altering the numbers to make the calculation easier and then adjusting the result afterward. For example, instead of calculating , one calculates which is easier because it uses a round number. In the next step, one adds to the result to compensate for the earlier adjustment. Mental arithmetic is often taught in primary education to train the numerical abilities of the students.
The human body can also be employed as an arithmetic tool. The use of hands in finger counting is often introduced to young children to teach them numbers and simple calculations. In its most basic form, the number of extended fingers corresponds to the represented quantity and arithmetic operations like addition and subtraction are performed by extending or retracting fingers. This system is limited to small numbers compared to more advanced systems which employ different approaches to represent larger quantities. The human voice is used as an arithmetic aid in verbal counting. | Arithmetic | Wikipedia | 478 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Tally marks are a simple system based on external tools other than the body. This system relies on mark making, such as strokes drawn on a surface or notches carved into a wooden stick, to keep track of quantities. Some forms of tally marks arrange the strokes in groups of five to make them easier to read.
The abacus is a more advanced tool to represent numbers and perform calculations. An abacus usually consists of a series of rods, each holding several beads. Each bead represents a quantity, which is counted if the bead is moved from one end of a rod to the other. Calculations happen by manipulating the positions of beads until the final bead pattern reveals the result. Related aids include counting boards, which use tokens whose value depends on the area on the board in which they are placed, and counting rods, which are arranged in horizontal and vertical patterns to represent different numbers.
Sectors and slide rules are more refined calculating instruments that rely on geometric relationships between different scales to perform both basic and advanced arithmetic operations. Printed tables were particularly relevant as an aid to look up the results of operations like logarithm and trigonometric functions.
Mechanical calculators automate manual calculation processes. They present the user with some form of input device to enter numbers by turning dials or pressing keys. They include an internal mechanism usually consisting of gears, levers, and wheels to perform calculations and display the results. For electronic calculators and computers, this procedure is further refined by replacing the mechanical components with electronic circuits like microprocessors that combine and transform electric signals to perform calculations.
Others
There are many other types of arithmetic. Modular arithmetic operates on a finite set of numbers. If an operation would result in a number outside this finite set then the number is adjusted back into the set, similar to how the hands of clocks start at the beginning again after having completed one cycle. The number at which this adjustment happens is called the modulus. For example, a regular clock has a modulus of 12. In the case of adding 4 to 9, this means that the result is not 13 but 1. The same principle applies also to other operations, such as subtraction, multiplication, and division.
Some forms of arithmetic deal with operations performed on mathematical objects other than numbers. Interval arithmetic describes operations on intervals. Vector arithmetic and matrix arithmetic describe arithmetic operations on vectors and matrices, like vector addition and matrix multiplication. | Arithmetic | Wikipedia | 490 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Arithmetic systems can be classified based on the numeral system they rely on. For instance, decimal arithmetic describes arithmetic operations in the decimal system. Other examples are binary arithmetic, octal arithmetic, and hexadecimal arithmetic.
Compound unit arithmetic describes arithmetic operations performed on magnitudes with compound units. It involves additional operations to govern the transformation between single unit and compound unit quantities. For example, the operation of reduction is used to transform the compound quantity 1 h 90 min into the single unit quantity 150 min.
Non-Diophantine arithmetics are arithmetic systems that violate traditional arithmetic intuitions and include equations like and . They can be employed to represent some real-world situations in modern physics and everyday life. For instance, the equation can be used to describe the observation that if one raindrop is added to another raindrop then they do not remain two separate entities but become one.
Axiomatic foundations
Axiomatic foundations of arithmetic try to provide a small set of laws, called axioms, from which all fundamental properties of and operations on numbers can be derived. They constitute logically consistent and systematic frameworks that can be used to formulate mathematical proofs in a rigorous manner. Two well-known approaches are the Dedekind–Peano axioms and set-theoretic constructions.
The Dedekind–Peano axioms provide an axiomatization of the arithmetic of natural numbers. Their basic principles were first formulated by Richard Dedekind and later refined by Giuseppe Peano. They rely only on a small number of primitive mathematical concepts, such as 0, natural number, and successor. The Peano axioms determine how these concepts are related to each other. All other arithmetic concepts can then be defined in terms of these primitive concepts.
0 is a natural number.
For every natural number, there is a successor, which is also a natural number.
The successors of two different natural numbers are never identical.
0 is not the successor of a natural number.
If a set contains 0 and every successor then it contains every natural number.
Numbers greater than 0 are expressed by repeated application of the successor function . For example, is and is . Arithmetic operations can be defined as mechanisms that affect how the successor function is applied. For instance, to add to any number is the same as applying the successor function two times to this number. | Arithmetic | Wikipedia | 473 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Various axiomatizations of arithmetic rely on set theory. They cover natural numbers but can also be extended to integers, rational numbers, and real numbers. Each natural number is represented by a unique set. 0 is usually defined as the empty set . Each subsequent number can be defined as the union of the previous number with the set containing the previous number. For example, , , and . Integers can be defined as ordered pairs of natural numbers where the second number is subtracted from the first one. For instance, the pair (9, 0) represents the number 9 while the pair (0, 9) represents the number -9. Rational numbers are defined as pairs of integers where the first number represents the numerator and the second number represents the denominator. For example, the pair (3, 7) represents the rational number . One way to construct the real numbers relies on the concept of Dedekind cuts. According to this approach, each real number is represented by a partition of all rational numbers into two sets, one for all numbers below the represented real number and the other for the rest. Arithmetic operations are defined as functions that perform various set-theoretic transformations on the sets representing the input numbers to arrive at the set representing the result.
History
The earliest forms of arithmetic are sometimes traced back to counting and tally marks used to keep track of quantities. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed. However, a basic sense of numbers may predate these findings and might even have existed before the development of language. | Arithmetic | Wikipedia | 349 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
It was not until the emergence of ancient civilizations that a more complex and structured approach to arithmetic began to evolve, starting around 3000 BCE. This became necessary because of the increased need to keep track of stored items, manage land ownership, and arrange exchanges. All the major ancient civilizations developed non-positional numeral systems to facilitate the representation of numbers. They also had symbols for operations like addition and subtraction and were aware of fractions. Examples are Egyptian hieroglyphics as well as the numeral systems invented in Sumeria, China, and India. The first positional numeral system was developed by the Babylonians starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient. Abacuses have been utilized as hand-operated calculating tools since ancient times as efficient means for performing complex calculations.
Early civilizations primarily used numbers for concrete practical purposes, like commercial activities and tax records, but lacked an abstract concept of number itself. This changed with the ancient Greek mathematicians, who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems. Another novel feature was their use of proofs to establish mathematical truths and validate theories. A further contribution was their distinction of various classes of numbers, such as even numbers, odd numbers, and prime numbers. This included the discovery that numbers for certain geometrical lengths are irrational and therefore cannot be expressed as a fraction. The works of Thales of Miletus and Pythagoras in the 7th and 6th centuries BCE are often regarded as the inception of Greek mathematics. Diophantus was an influential figure in Greek arithmetic in the 3rd century BCE because of his numerous contributions to number theory and his exploration of the application of arithmetic operations to algebraic equations.
The ancient Indians were the first to develop the concept of zero as a number to be used in calculations. The exact rules of its operation were written down by Brahmagupta in around 628 CE. The concept of zero or none existed long before, but it was not considered an object of arithmetic operations. Brahmagupta further provided a detailed discussion of calculations with negative numbers and their application to problems like credit and debt. The concept of negative numbers itself is significantly older and was first explored in Chinese mathematics in the first millennium BCE. | Arithmetic | Wikipedia | 474 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Indian mathematicians also developed the positional decimal system used today, in particular the concept of a zero digit instead of empty or missing positions. For example, a detailed treatment of its operations was provided by Aryabhata around the turn of the 6th century CE. The Indian decimal system was further refined and expanded to non-integers during the Islamic Golden Age by Middle Eastern mathematicians such as Al-Khwarizmi. His work was influential in introducing the decimal numeral system to the Western world, which at that time relied on the Roman numeral system. There, it was popularized by mathematicians like Leonardo Fibonacci, who lived in the 12th and 13th centuries and also developed the Fibonacci sequence. During the Middle Ages and Renaissance, many popular textbooks were published to cover the practical calculations for commerce. The use of abacuses also became widespread in this period. In the 16th century, the mathematician Gerolamo Cardano conceived the concept of complex numbers as a way to solve cubic equations.
The first mechanical calculators were developed in the 17th century and greatly facilitated complex mathematical calculations, such as Blaise Pascal's calculator and Gottfried Wilhelm Leibniz's stepped reckoner. The 17th century also saw the discovery of the logarithm by John Napier.
In the 18th and 19th centuries, mathematicians such as Leonhard Euler and Carl Friedrich Gauss laid the foundations of modern number theory. Another development in this period concerned work on the formalization and foundations of arithmetic, such as Georg Cantor's set theory and the Dedekind–Peano axioms used as an axiomatization of natural-number arithmetic. Computers and electronic calculators were first developed in the 20th century. Their widespread use revolutionized both the accuracy and speed with which even complex arithmetic computations can be calculated.
In various fields
Education
Arithmetic education forms part of primary education. It is one of the first forms of mathematics education that children encounter. Elementary arithmetic aims to give students a basic sense of numbers and to familiarize them with fundamental numerical operations like addition, subtraction, multiplication, and division. It is usually introduced in relation to concrete scenarios, like counting beads, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in early arithmetic education are number lines, addition and multiplication tables, counting blocks, and abacuses. | Arithmetic | Wikipedia | 482 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
Later stages focus on a more abstract understanding and introduce the students to different types of numbers, such as negative numbers, fractions, real numbers, and complex numbers. They further cover more advanced numerical operations, like exponentiation, extraction of roots, and logarithm. They also show how arithmetic operations are employed in other branches of mathematics, such as their application to describe geometrical shapes and the use of variables in algebra. Another aspect is to teach the students the use of algorithms and calculators to solve complex arithmetic problems.
Psychology
The psychology of arithmetic is interested in how humans and animals learn about numbers, represent them, and use them for calculations. It examines how mathematical problems are understood and solved and how arithmetic abilities are related to perception, memory, judgment, and decision making. For example, it investigates how collections of concrete items are first encountered in perception and subsequently associated with numbers. A further field of inquiry concerns the relation between numerical calculations and the use of language to form representations. Psychology also explores the biological origin of arithmetic as an inborn ability. This concerns pre-verbal and pre-symbolic cognitive processes implementing arithmetic-like operations required to successfully represent the world and perform tasks like spatial navigation.
One of the concepts studied by psychology is numeracy, which is the capability to comprehend numerical concepts, apply them to concrete situations, and reason with them. It includes a fundamental number sense as well as being able to estimate and compare quantities. It further encompasses the abilities to symbolically represent numbers in numbering systems, interpret numerical data, and evaluate arithmetic calculations. Numeracy is a key skill in many academic fields. A lack of numeracy can inhibit academic success and lead to bad economic decisions in everyday life, for example, by misunderstanding mortgage plans and insurance policies.
Philosophy
The philosophy of arithmetic studies the fundamental concepts and principles underlying numbers and arithmetic operations. It explores the nature and ontological status of numbers, the relation of arithmetic to language and logic, and how it is possible to acquire arithmetic knowledge.
According to Platonism, numbers have mind-independent existence: they exist as abstract objects outside spacetime and without causal powers. This view is rejected by intuitionists, who claim that mathematical objects are mental constructions. Further theories are logicism, which holds that mathematical truths are reducible to logical truths, and formalism, which states that mathematical principles are rules of how symbols are manipulated without claiming that they correspond to entities outside the rule-governed activity. | Arithmetic | Wikipedia | 495 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
The traditionally dominant view in the epistemology of arithmetic is that arithmetic truths are knowable a priori. This means that they can be known by thinking alone without the need to rely on sensory experience. Some proponents of this view state that arithmetic knowledge is innate while others claim that there is some form of rational intuition through which mathematical truths can be apprehended. A more recent alternative view was suggested by naturalist philosophers like Willard Van Orman Quine, who argue that mathematical principles are high-level generalizations that are ultimately grounded in the sensory world as described by the empirical sciences.
Others
Arithmetic is relevant to many fields. In daily life, it is required to calculate change when shopping, manage personal finances, and adjust a cooking recipe for a different number of servings. Businesses use arithmetic to calculate profits and losses and analyze market trends. In the field of engineering, it is used to measure quantities, calculate loads and forces, and design structures. Cryptography relies on arithmetic operations to protect sensitive information by encrypting data and messages.
Arithmetic is intimately connected to many branches of mathematics that depend on numerical operations. Algebra relies on arithmetic principles to solve equations using variables. These principles also play a key role in calculus in its attempt to determine rates of change and areas under curves. Geometry uses arithmetic operations to measure the properties of shapes while statistics utilizes them to analyze numerical data. Due to the relevance of arithmetic operations throughout mathematics, the influence of arithmetic extends to most sciences such as physics, computer science, and economics. These operations are used in calculations, problem-solving, data analysis, and algorithms, making them integral to scientific research, technological development, and economic modeling. | Arithmetic | Wikipedia | 336 | 3118 | https://en.wikipedia.org/wiki/Arithmetic | Mathematics | Mathematics | null |
In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.
An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.
Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum.
Multiplicative and additive functions
An arithmetic function a is
completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n;
completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n;
Two whole numbers m and n are called coprime if their greatest common divisor is 1, that is, if there is no prime number that divides both of them.
Then an arithmetic function a is
additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n;
multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n.
Notation
In this article, and mean that the sum or product is over all prime numbers:
and
Similarly, and mean that the sum or product is over all prime powers with strictly positive exponent (so is not included):
The notations and mean that the sum or product is over all positive divisors of n, including 1 and n. For example, if , then
The notations can be combined: and mean that the sum or product is over all prime divisors of n. For example, if n = 18, then
and similarly and mean that the sum or product is over all prime powers dividing n. For example, if n = 24, then | Arithmetic function | Wikipedia | 458 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
Ω(n), ω(n), νp(n) – prime power decomposition
The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.)
It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define the p-adic valuation νp(n) to be the exponent of the highest power of the prime p that divides n. That is, if p is one of the pi then νp(n) = ai, otherwise it is zero. Then
In terms of the above the prime omega functions ω and Ω are defined by
To avoid repetition, formulas for the functions listed in this article are, whenever possible, given in terms of n and the corresponding pi, ai, ω, and Ω.
Multiplicative functions
σk(n), τ(n), d(n) – divisor sums
σk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number.
σ1(n), the sum of the (positive) divisors of n, is usually denoted by σ(n).
Since a positive number to the zero power is one, σ0(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or τ(n) (for the German Teiler = divisors).
Setting k = 0 in the second product gives
φ(n) – Euler totient function
φ(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n.
Jk(n) – Jordan totient function
Jk(n), the Jordan totient function, is the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. It is a generalization of Euler's totient, .
μ(n) – Möbius function
μ(n), the Möbius function, is important because of the Möbius inversion formula. See , below. | Arithmetic function | Wikipedia | 509 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
This implies that μ(1) = 1. (Because Ω(1) = ω(1) = 0.)
τ(n) – Ramanujan tau function
τ(n), the Ramanujan tau function, is defined by its generating function identity:
Although it is hard to say exactly what "arithmetical property of n" it "expresses", (τ(n) is (2π)−12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function) it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain σk(n) and rk(n) functions (because these are also coefficients in the expansion of modular forms).
cq(n) – Ramanujan's sum
cq(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity:
Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixed value of n it is multiplicative in q:
If q and r are coprime, then
ψ(n) – Dedekind psi function
The Dedekind psi function, used in the theory of modular functions, is defined by the formula
Completely multiplicative functions
λ(n) – Liouville function
λ(n), the Liouville function, is defined by
χ(n) – characters
All Dirichlet characters χ(n) are completely multiplicative. Two characters have special notations:
The principal character (mod n) is denoted by χ0(a) (or χ1(a)). It is defined as
The quadratic character (mod n) is denoted by the Jacobi symbol for odd n (it is not defined for even n):
In this formula is the Legendre symbol, defined for all integers a and all odd primes p by
Following the normal convention for the empty product,
Additive functions
ω(n) – distinct prime divisors
ω(n), defined above as the number of distinct primes dividing n, is additive (see Prime omega function).
Completely additive functions
Ω(n) – prime divisors
Ω(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive (see Prime omega function). | Arithmetic function | Wikipedia | 499 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
νp(n) – p-adic valuation of an integer n
For a fixed prime p, νp(n), defined above as the exponent of the largest power of p dividing n, is completely additive.
Logarithmic derivative
, where is the arithmetic derivative.
Neither multiplicative nor additive
π(x), Π(x), ϑ(x), ψ(x) – prime-counting functions
These important functions (which are not arithmetic functions) are defined for non-negative real arguments, and are used in the various statements and proofs of the prime number theorem. They are summation functions (see the main section just below) of arithmetic functions which are neither multiplicative nor additive.
π(x), the prime-counting function, is the number of primes not exceeding x. It is the summation function of the characteristic function of the prime numbers.
A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, etc. It is the summation function of the arithmetic function which takes the value 1/k on integers which are the kth power of some prime number, and the value 0 on other integers.
ϑ(x) and ψ(x), the Chebyshev functions, are defined as sums of the natural logarithms of the primes not exceeding x.
The second Chebyshev function ψ(x) is the summation function of the von Mangoldt function just below.
Λ(n) – von Mangoldt function
Λ(n), the von Mangoldt function, is 0 unless the argument n is a prime power , in which case it is the natural logarithm of the prime p:
p(n) – partition function
p(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different:
λ(n) – Carmichael function
λ(n), the Carmichael function, is the smallest positive number such that for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n. | Arithmetic function | Wikipedia | 472 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
For powers of odd primes and for 2 and 4, λ(n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n:
and for general n it is the least common multiple of λ of each of the prime power factors of n:
h(n) – class number
h(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples.
rk(n) – sum of k squares
rk(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different.
D(n) – Arithmetic derivative
Using the Heaviside notation for the derivative, the arithmetic derivative D(n) is a function such that
if n prime, and
(the product rule)
Summation functions
Given an arithmetic function a(n), its summation function A(x) is defined by
A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0.
Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:
Individual values of arithmetic functions may fluctuate wildly – as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x.
A classical example of this phenomenon is given by the divisor summatory function, the summation function of d(n), the number of divisors of n:
An average order of an arithmetic function is some simpler or better-understood function which has the same summation function asymptotically, and hence takes the same values "on average". We say that g is an average order of f if | Arithmetic function | Wikipedia | 501 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
as x tends to infinity. The example above shows that d(n) has the average order log(n).
Dirichlet convolution
Given an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges):
Fa(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is ζ(s) the Riemann zeta function.
The generating function of the Möbius function is the inverse of the zeta function:
Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The product Fa(s)Fb(s) can be computed as follows:
It is a straightforward exercise to show that if c(n) is defined by
then
This function c is called the Dirichlet convolution of a and b, and is denoted by .
A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function:
Multiplying by the inverse of the zeta function gives the Möbius inversion formula:
If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative.
Relations among the functions
There are a great many formulas connecting arithmetical functions with each other and with the functions of analysis, especially powers, roots, and the exponential and log functions. The page divisor sum identities contains many more generalized and related examples of identities involving arithmetic functions.
Here are a few examples:
Dirichlet convolutions
where λ is the Liouville function.
Möbius inversion
Möbius inversion
Möbius inversion
Möbius inversion
Möbius inversion
where λ is the Liouville function.
Möbius inversion
Sums of squares
For all (Lagrange's four-square theorem).
where the Kronecker symbol has the values
There is a formula for r3 in the section on class numbers below.
where .
where
Define the function as | Arithmetic function | Wikipedia | 454 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
That is, if n is odd, is the sum of the kth powers of the divisors of n, that is, and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n.
Adopt the convention that Ramanujan's if x is not an integer.
Divisor sum convolutions
Here "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series:
The sequence is called the convolution or the Cauchy product of the sequences an and bn.
These formulas may be proved analytically (see Eisenstein series) or by elementary methods.
where τ(n) is Ramanujan's function.
Since σk(n) (for natural number k) and τ(n) are integers, the above formulas can be used to prove congruences for the functions. See Ramanujan tau function for some examples.
Extend the domain of the partition function by setting
This recurrence can be used to compute p(n).
Class number related
Peter Gustav Lejeune Dirichlet discovered formulas that relate the class number h of quadratic number fields to the Jacobi symbol.
An integer D is called a fundamental discriminant if it is the discriminant of a quadratic number field. This is equivalent to D ≠ 1 and either a) D is squarefree and D ≡ 1 (mod 4) or b) D ≡ 0 (mod 4), D/4 is squarefree, and D/4 ≡ 2 or 3 (mod 4).
Extend the Jacobi symbol to accept even numbers in the "denominator" by defining the Kronecker symbol:
Then if D < −4 is a fundamental discriminant
There is also a formula relating r3 and h. Again, let D be a fundamental discriminant, D < −4. Then
Prime-count related
Let be the nth harmonic number. Then
is true for every natural number n if and only if the Riemann hypothesis is true.
The Riemann hypothesis is also equivalent to the statement that, for all n > 5040,
(where γ is the Euler–Mascheroni constant). This is Robin's theorem.
Menon's identity
In 1965 P Kesava Menon proved | Arithmetic function | Wikipedia | 508 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
This has been generalized by a number of mathematicians. For example,
B. Sury
N. Rao where a1, a2, ..., as are integers, gcd(a1, a2, ..., as, n) = 1.
László Fejes Tóth where m1 and m2 are odd, m = lcm(m1, m2).
In fact, if f is any arithmetical function
where stands for Dirichlet convolution.
Miscellaneous
Let m and n be distinct, odd, and positive. Then the Jacobi symbol satisfies the law of quadratic reciprocity:
Let D(n) be the arithmetic derivative. Then the logarithmic derivative See Arithmetic derivative for details.
Let λ(n) be Liouville's function. Then
and
Let λ(n) be Carmichael's function. Then
Further,
See Multiplicative group of integers modulo n and Primitive root modulo n.
Note that
Compare this with
where τ(n) is Ramanujan's function.
First 100 values of some arithmetic functions | Arithmetic function | Wikipedia | 219 | 3170 | https://en.wikipedia.org/wiki/Arithmetic%20function | Mathematics | Functions: General | null |
Agar ( or ), or agar-agar, is a jelly-like substance consisting of polysaccharides obtained from the cell walls of some species of red algae, primarily from “ogonori” and “tengusa”. As found in nature, agar is a mixture of two components, the linear polysaccharide agarose and a heterogeneous mixture of smaller molecules called agaropectin. It forms the supporting structure in the cell walls of certain species of algae and is released on boiling. These algae are known as agarophytes, belonging to the Rhodophyta (red algae) phylum. The processing of food-grade agar removes the agaropectin, and the commercial product is essentially pure agarose.
Agar has been used as an ingredient in desserts throughout Asia and also as a solid substrate to contain culture media for microbiological work. Agar can be used as a laxative; an appetite suppressant; a vegan substitute for gelatin; a thickener for soups; in fruit preserves, ice cream, and other desserts; as a clarifying agent in brewing; and for sizing paper and fabrics.
Etymology
The word agar comes from agar-agar, the Malay name for red algae (Gigartina, Eucheuma, Gracilaria) from which the jelly is produced. It is also known as Kanten () (from the phrase kan-zarashi tokoroten () or "cold-exposed agar"), Japanese isinglass, China grass, Ceylon moss or Jaffna moss. Gracilaria edulis or its synonym G. lichenoides is specifically referred to as agal-agal or Ceylon agar.
History | Agar | Wikipedia | 374 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Macroalgae have been used widely as food by coastal cultures, especially in Southeast Asia. In the Philippines, Gracilaria, known as gulaman (also guraman, gar-garao, or gulaman dagat, among other names) in Tagalog, have been harvested and used as food for centuries, eaten both fresh or sun-dried and turned into jellies. The earliest historical attestation is from the Vocabulario de la lengua tagala (1754) by the Jesuit priests Juan de Noceda and Pedro de Sanlucar, where golaman or gulaman was defined as "una yerva, de que se haze conserva a modo de Halea, naze en la mar" ("a herb, from which a jam-like preserve is made, grows in the sea"), with an additional entry for guinolaman to refer to food made with the jelly.
Carrageenan, derived from gusô (Eucheuma spp.), which also congeals into a gel-like texture is also used similarly among the Visayan peoples and have been recorded in the even earlier Diccionario De La Lengua Bisaya, Hiligueina y Haraia de la isla de Panay y Sugbu y para las demas islas (c.1637) of the Augustinian missionary Alonso de Méntrida . In the book, Méntrida describes gusô as being cooked until it melts, and then allowed to congeal into a sour dish.
In Ambon Island in the Maluku Islands of Indonesia, agar is extracted from Graciliaria and eaten as a type of pickle or a sauce. Jelly seaweeds were also favoured and foraged by Malay communities living on the coasts of the Riau Archipelago and Singapore in Southeast Asia for centuries. 19th century records indicate that dried Graciliaria were one of the bulk exports of British Malaya to China. Poultices made from agar were also used for swollen knee joints and sores in Johore and Singapore.
The application of agar as a food additive in Japan is alleged to have been discovered in 1658 by Mino Tarōzaemon (), an innkeeper in current Fushimi-ku, Kyoto who, according to legend, was said to have discarded surplus seaweed soup (Tokoroten) and noticed that it gelled later after a winter night's freezing. | Agar | Wikipedia | 510 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Agar was first subjected to chemical analysis in 1859 by the French chemist Anselme Payen, who had obtained agar from the marine algae Gelidium corneum.
Beginning in the late 19th century, agar began to be used as a solid medium for growing various microbes. Agar was first described for use in microbiology in 1882 by the German microbiologist Walther Hesse, an assistant working in Robert Koch's laboratory, on the suggestion of his wife Fanny Hesse. Agar quickly supplanted gelatin as the base of microbiological media, due to its higher melting temperature, allowing microbes to be grown at higher temperatures without the media liquefying.
With its newfound use in microbiology, agar production quickly increased. This production centered on Japan, which produced most of the world's agar until World War II. However, with the outbreak of World War II, many nations were forced to establish domestic agar industries in order to continue microbiological research. Around the time of World War II, approximately 2,500 tons of agar were produced annually. By the mid-1970s, production worldwide had increased dramatically to approximately 10,000 tons each year. Since then, production of agar has fluctuated due to unstable and sometimes over-utilized seaweed populations.
Chemical composition
Agar consists of a mixture of two polysaccharides: agarose and agaropectin, with agarose making up about 70% of the mixture, while agaropectin makes about 30% of it. Agarose is a linear polymer, made up of repeating units of agarobiose, a disaccharide made up of D-galactose and 3,6-anhydro-L-galactopyranose. Agaropectin is a heterogeneous mixture of smaller molecules that occur in lesser amounts, and is made up of alternating units of D-galactose and L-galactose heavily modified with acidic side-groups, such as sulfate, glucuronate, and pyruvate. | Agar | Wikipedia | 432 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Physical properties
Agar exhibits a phenomenon known as hysteresis whereby, when mixed with water, it solidifies and forms a gel below about , which is called the gel point, and melts above , which is the melting point. Hysteresis is the property of having a difference between the gel point and melting point temperatures. This property lends a suitable balance between easy melting and good gel stability at relatively high temperatures. Since many scientific applications require incubation at temperatures close to human body temperature (37 °C), agar is more appropriate than other solidifying agents that melt at this temperature, such as gelatin.
Uses
Culinary
Agar-agar is a natural vegetable gelatin counterpart. It is white and semi-translucent when sold in packages as washed and dried strips or in powdered form. It can be used to make jellies, puddings, and custards. When making jelly, it is boiled in water until the solids dissolve. Sweetener, flavoring, coloring, fruits and or vegetables are then added, and the liquid is poured into molds to be served as desserts and vegetable aspics or incorporated with other desserts such as a layer of jelly in a cake.
Agar-agar is approximately 80% dietary fiber, so it can serve as an intestinal regulator. Its bulking quality has been behind fad diets in Asia, for example the kanten (the Japanese word for agar-agar) diet. Once ingested, kanten triples in size and absorbs water. This results in the consumers feeling fuller. | Agar | Wikipedia | 325 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Asian culinary
One use of agar in Japanese cuisine is in anmitsu, a dessert made of small cubes of agar jelly and served in a bowl with various fruits or other ingredients. It is also the main ingredient in mizu yōkan, another popular Japanese food. In Philippine cuisine, it is used to make the jelly bars in the various gulaman refreshments like sago't gulaman, samalamig, or desserts such as buko pandan, agar flan, halo-halo, fruit cocktail jelly, and the black and red gulaman used in various fruit salads. In Vietnamese cuisine, jellies made of flavored layers of agar-agar, called thạch, are a popular dessert, and are often made in ornate molds for special occasions. In Indian cuisine, agar is used for making desserts. In Burmese cuisine, a sweet jelly known as kyauk kyaw is made from agar. Agar jelly is widely used in Taiwanese bubble tea.
Other culinary
It can be used as addition to (or as a replacement for) pectin in jelly, jam, or marmalade, as a substitute to gelatin for its superior gelling properties, and as a strengthening ingredient in souffles and custards. Another use of agar-agar is in a Russian dish ptich'ye moloko (bird's milk), a rich jellified custard (or soft meringue) used as a cake filling or chocolate-glazed as individual sweets.
Agar-agar may also be used as the gelling agent in gel clarification, a culinary technique used to clarify stocks, sauces, and other liquids. Mexico has traditional candies made out of Agar gelatin, most of them in colorful, half-circle shapes that resemble a melon or watermelon fruit slice, and commonly covered with sugar. They are known in Spanish as Dulce de Agar (Agar sweets)
Agar-agar is an allowed nonorganic/nonsynthetic additive used as a thickener, gelling agent, texturizer, moisturizer, emulsifier, flavor enhancer, and absorbent in certified organic foods.
Microbiology
Agar plate | Agar | Wikipedia | 473 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
An agar plate or Petri dish is used to provide a growth medium using a mix of agar and other nutrients in which microorganisms, including bacteria and fungi, can be cultured and observed under the microscope. Agar is indigestible for many organisms so that microbial growth does not affect the gel used and it remains stable. Agar is typically sold commercially as a powder that can be mixed with water and prepared similarly to gelatin before use as a growth medium. Nutrients are typically added to meet the nutritional needs of the microbes organism, the formulations of which may be "undefined" where the precise composition is unknown, or "defined" where the exact chemical composition is known. Agar is often dispensed using a sterile media dispenser.
Different algae produce various types of agar. Each agar has unique properties that suit different purposes. Because of the agarose component, the agar solidifies. When heated, agarose has the potential to melt and then solidify. Because of this property, they are referred to as "physical gels". In contrast, polyacrylamide polymerization is an irreversible process, and the resulting products are known as chemical gels.
There are a variety of different types of agar that support the growth of different microorganisms. A nutrient agar may be permissive, allowing for the cultivation of any non-fastidious microorganisms; a commonly-used nutrient agar for bacteria is the Luria Bertani (LB) agar which contains lysogeny broth, a nutrient-rich medium used for bacterial growth. Additionally, 2216 Marine Broth (MB) agar, with high salt content, is optimized for growing heterotrophic marine bacteria like those of the Vibrio genus, while Terrific Broth (TB) agar is used to non-selectively culture high yields of the bacterium E. coli. More generally, enriched media is an agar variety that is infused with the necessary nutrients required by fastidious organisms to grow. Despite the large diversity of agar mediums, yeast extract is a common ingredient across all varieties as it is a macronutrient that provides a nitrogen source for all bacterial cell types. | Agar | Wikipedia | 471 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Other fastidious organisms may require the addition of different biological fluids such as horse or sheep blood, serum, egg yolk, and so on. Agar plates can also be selective, and can be used to promote the growth of bacteria of interest while inhibiting others. A variety of chemicals may be added to create an environment favourable for specific types of bacteria or bacteria with certain properties, but not conducive for growth of others. For example, antibiotics may be added in cloning experiments whereby bacteria with antibiotic-resistant plasmid are selected. In addition to antibiotic treated agar, other selective and indicator agar plates include TCBS agar and MacConkey agar. Thiosulfate citrate bile salts sucrose (TCBS) agar is used to differentiate Vibrio species based on their sucrose metabolism, since only some will metabolize the sucrose in the plate and change its pH. Indicator dyes included in the gel will display a visual change of the pH by changing the gel color from green to yellow. MacConkey agar contains bile salts and crystal violet to selectively grow gram-negative bacteria and differentiate between species using pH-indicator dyes that demonstrate lactose metabolism properties.
Motility assays
As a gel, an agar or agarose medium is porous and therefore can be used to measure microorganism motility and mobility. The gel's porosity is directly related to the concentration of agarose in the medium, so various levels of effective viscosity (from the cell's "point of view") can be selected, depending on the experimental objectives.
A common identification assay involves culturing a sample of the organism deep within a block of nutrient agar. Cells will attempt to grow within the gel structure. Motile species will be able to migrate, albeit slowly, throughout the gel, and infiltration rates can then be visualized, whereas non-motile species will show growth only along the now-empty path introduced by the invasive initial sample deposition. | Agar | Wikipedia | 423 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Another setup commonly used for measuring chemotaxis and chemokinesis utilizes the under-agarose cell migration assay, whereby a layer of agarose gel is placed between a cell population and a chemoattractant. As a concentration gradient develops from the diffusion of the chemoattractant into the gel, various cell populations requiring different stimulation levels to migrate can then be visualized over time using microphotography as they tunnel upward through the gel against gravity along the gradient.
Plant biology
Research grade agar is used extensively in plant biology as it is optionally supplemented with a nutrient and/or vitamin mixture that allows for seedling germination in Petri dishes under sterile conditions (given that the seeds are sterilized as well). Nutrient and/or vitamin supplementation for Arabidopsis thaliana is standard across most experimental conditions. Murashige & Skoog (MS) nutrient mix and Gamborg's B5 vitamin mix in general are used. A 1.0% agar/0.44% MS+vitamin dH2O solution is suitable for growth media between normal growth temps.
When using agar, within any growth medium, it is important to know that the solidification of the agar is pH-dependent. The optimal range for solidification is between 5.4 and 5.7. Usually, the application of potassium hydroxide is needed to increase the pH to this range. A general guideline is about 600 μl 0.1M KOH per 250 ml GM. This entire mixture can be sterilized using the liquid cycle of an autoclave.
This medium nicely lends itself to the application of specific concentrations of phytohormones etc. to induce specific growth patterns in that one can easily prepare a solution containing the desired amount of hormone, add it to the known volume of GM, and autoclave to both sterilize and evaporate off any solvent that may have been used to dissolve the often-polar hormones. This hormone/GM solution can be spread across the surface of Petri dishes sown with germinated and/or etiolated seedlings.
Experiments with the moss Physcomitrella patens, however, have shown that choice of the gelling agent – agar or Gelrite – does influence phytohormone sensitivity of the plant cell culture. | Agar | Wikipedia | 488 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Other uses
Agar is used:
As an impression material in dentistry.
As a medium to precisely orient the tissue specimen and secure it by agar pre-embedding (especially useful for small endoscopy biopsy specimens) for histopathology processing
To make salt bridges and gel plugs for use in electrochemistry.
In formicariums as a transparent substitute for sand and a source of nutrition.
As a natural ingredient in forming modeling clay for young children to play with.
As an allowed biofertilizer component in organic farming.
As a substrate for precipitin reactions in immunology.
At different times as a substitute for gelatin in photographic emulsions, arrowroot in preparing silver paper and as a substitute for fish glue in resist etching.
As an MRI elastic gel phantom to mimic tissue mechanical properties in Magnetic Resonance Elastography
Gelidium agar is used primarily for bacteriological plates. Gracilaria agar is used mainly in food applications.
In 2016, AMAM, a Japanese company, developed a prototype for Agar-based commercial packaging system called Agar Plasticity, intended as a replacement for oil-based plastic packaging. | Agar | Wikipedia | 242 | 3262 | https://en.wikipedia.org/wiki/Agar | Biology and health sciences | Carbohydrates | Biology |
Acid rain is rain or any other form of precipitation that is unusually acidic, meaning that it has elevated levels of hydrogen ions (low pH). Most water, including drinking water, has a neutral pH that exists between 6.5 and 8.5, but acid rain has a pH level lower than this and ranges from 4–5 on average. The more acidic the acid rain is, the lower its pH is. Acid rain can have harmful effects on plants, aquatic animals, and infrastructure. Acid rain is caused by emissions of sulfur dioxide and nitrogen oxide, which react with the water molecules in the atmosphere to produce acids.
Acid rain has been shown to have adverse impacts on forests, freshwaters, soils, microbes, insects and aquatic life-forms. In ecosystems, persistent acid rain reduces tree bark durability, leaving flora more susceptible to environmental stressors such as drought, heat/cold and pest infestation. Acid rain is also capable of detrimenting soil composition by stripping it of nutrients such as calcium and magnesium which play a role in plant growth and maintaining healthy soil. In terms of human infrastructure, acid rain also causes paint to peel, corrosion of steel structures such as bridges, and weathering of stone buildings and statues as well as having impacts on human health.
Some governments, including those in Europe and North America, have made efforts since the 1970s to reduce the release of sulfur dioxide and nitrogen oxide into the atmosphere through air pollution regulations. These efforts have had positive results due to the widespread research on acid rain starting in the 1960s and the publicized information on its harmful effects. The main source of sulfur and nitrogen compounds that result in acid rain are anthropogenic, but nitrogen oxides can also be produced naturally by lightning strikes and sulfur dioxide is produced by volcanic eruptions.
Definition
"Acid rain" is rain with a pH less than 5. "Clean" or unpolluted rain has a pH greater than 5 but still less than pH = 7 owing to the acidity caused by carbon dioxide acid according to the following reactions:
A variety of natural and human-made sources contribute to the acidity. For example nitric acid produced by electric discharge in the atmosphere such as lightning. The usual anthropogenic sources are sulfur dioxide and nitrogen oxide. They react with water (as does carbon dioxide) to give solutions with pH < 5. Occasional pH readings in rain and fog water of well below 2.4 have been reported in industrialized areas.
History | Acid rain | Wikipedia | 498 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Acid rain was first systematically studied in Europe in the 1960s and in the United States and Canada in the following decade.
In Europe
The corrosive effect of polluted, acidic city air on limestone and marble was noted in the 17th century by John Evelyn, who remarked upon the poor condition of the Arundel marbles.
Since the Industrial Revolution, emissions of sulfur dioxide and nitrogen oxides into the atmosphere have increased. In 1852, Robert Angus Smith was the first to show the relationship between acid rain and atmospheric pollution in Manchester, England. Smith coined the term "acid rain" in 1872.
In the late 1960s, scientists began widely observing and studying the phenomenon. At first, the main focus in this research lay on local effects of acid rain. Waldemar Christofer Brøgger was the first to acknowledge long-distance transportation of pollutants crossing borders from the United Kingdom to Norway – a problem systematically studied by Brynjulf Ottar in the 1970s. Ottar's work was strongly influenced by Swedish soil scientist Svante Odén, who had drawn widespread attention to Europe's acid rain problem in popular newspapers and wrote a landmark paper on the subject in 1968.
In the United States
The earliest report about acid rain in the United States came from chemical evidence gathered from Hubbard Brook Valley; public awareness of acid rain in the US increased in the 1970s after The New York Times reported on these findings.
In 1972, a group of scientists, including Gene Likens, discovered the rain that was deposited at White Mountains of New Hampshire was acidic. The pH of the sample was measured to be 4.03 at Hubbard Brook. The Hubbard Brook Ecosystem Study followed up with a series of research studies that analyzed the environmental effects of acid rain. The alumina from soils neutralized acid rain that mixed with stream water at Hubbard Brook. The result of this research indicated that the chemical reaction between acid rain and aluminium leads to an increasing rate of soil weathering. Experimental research examined the effects of increased acidity in streams on ecological species. In 1980, scientists modified the acidity of Norris Brook, New Hampshire, and observed the change in species' behaviors. There was a decrease in species diversity, an increase in community dominants, and a reduction in the food web complexity. | Acid rain | Wikipedia | 456 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
In 1980, the US Congress passed an Acid Deposition Act. This Act established an 18-year assessment and research program under the direction of the National Acidic Precipitation Assessment Program (NAPAP). NAPAP enlarged a network of monitoring sites to determine how acidic precipitation was, seeking to determine long-term trends, and established a network for dry deposition. Using a statistically based sampling design, NAPAP quantified the effects of acid rain on a regional basis by targeting research and surveys to identify and quantify the impact of acid precipitation on freshwater and terrestrial ecosystems. NAPAP also assessed the effects of acid rain on historical buildings, monuments, and building materials. It also funded extensive studies on atmospheric processes and potential control programs.
From the start, policy advocates from all sides attempted to influence NAPAP activities to support their particular policy advocacy efforts, or to disparage those of their opponents. For the US Government's scientific enterprise, a significant impact of NAPAP were lessons learned in the assessment process and in environmental research management to a relatively large group of scientists, program managers, and the public.
In 1981, the National Academy of Sciences was looking into research about the controversial issues regarding acid rain. President Ronald Reagan dismissed the issues of acid rain until his personal visit to Canada and confirmed that the Canadian border suffered from the drifting pollution from smokestacks originating in the US Midwest. Reagan honored the agreement to Canadian Prime Minister Pierre Trudeau's enforcement of anti-pollution regulation. In 1982, Reagan commissioned William Nierenberg to serve on the National Science Board. Nierenberg selected scientists including Gene Likens to serve on a panel to draft a report on acid rain. In 1983, the panel of scientists came up with a draft report, which concluded that acid rain is a real problem and solutions should be sought. White House Office of Science and Technology Policy reviewed the draft report and sent Fred Singer's suggestions of the report, which cast doubt on the cause of acid rain. The panelists revealed rejections against Singer's positions and submitted the report to Nierenberg in April. In May 1983, the House of Representatives voted against legislation controlling sulfur emissions. There was a debate about whether Nierenberg delayed the release of the report. Nierenberg denied the saying about his suppression of the report and stated that it was withheld after the House's vote because it was not ready to be published. | Acid rain | Wikipedia | 483 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
In 1991, the US National Acid Precipitation Assessment Program (NAPAP) provided its first assessment of acid rain in the United States. It reported that 5% of New England Lakes were acidic, with sulfates being the most common problem. They noted that 2% of the lakes could no longer support Brook Trout, and 6% of the lakes were unsuitable for the survival of many minnow species. Subsequent Reports to Congress have documented chemical changes in soil and freshwater ecosystems, nitrogen saturation, soil nutrient decreases, episodic acidification, regional haze, and damage to historical monuments.
Meanwhile, in 1990, the US Congress passed a series of amendments to the Clean Air Act. Title IV of these amendments established a cap and trade system designed to control emissions of sulfur dioxide and nitrogen oxides. Both these emissions proved to cause a significant problem for U.S. citizens and their access to healthy, clean air. Title IV called for a total reduction of about 10 million tons of SO2 emissions from power plants, close to a 50% reduction. It was implemented in two phases. Phase I began in 1995 and limited sulfur dioxide emissions from 110 of the largest power plants to 8.7 million tons of sulfur dioxide. One power plant in New England (Merrimack) was in Phase I. Four other plants (Newington, Mount Tom, Brayton Point, and Salem Harbor) were added under other program provisions. Phase II began in 2000 and affects most of the power plants in the country.
During the 1990s, research continued. On March 10, 2005, the EPA issued the Clean Air Interstate Rule (CAIR). This rule provides states with a solution to the problem of power plant pollution that drifts from one state to another. CAIR will permanently cap emissions of SO2 and NOx in the eastern United States. When fully implemented, CAIR will reduce SO2 emissions in 28 eastern states and the District of Columbia by over 70% and NOx emissions by over 60% from 2003 levels.
Overall, the program's cap and trade program has been successful in achieving its goals. Since the 1990s, SO2 emissions have dropped 40%, and according to the Pacific Research Institute, acid rain levels have dropped 65% since 1976. Conventional regulation was used in the European Union, which saw a decrease of over 70% in SO2 emissions during the same period.
In 2007, total SO2 emissions were 8.9 million tons, achieving the program's long-term goal ahead of the 2010 statutory deadline. | Acid rain | Wikipedia | 507 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
In 2007 the EPA estimated that by 2010, the overall costs of complying with the program for businesses and consumers would be $1 billion to $2 billion a year, only one-fourth of what was initially predicted. Forbes says: "In 2010, by which time the cap and trade system had been augmented by the George W. Bush administration's Clean Air Interstate Rule, SO2 emissions had fallen to 5.1 million tons."
The term citizen science can be traced back as far as January 1989 to a campaign by the Audubon Society to measure acid rain. Scientist Muki Haklay cites in a policy report for the Wilson Center entitled 'Citizen Science and Policy: A European Perspective' a first use of the term 'citizen science' by R. Kerson in the magazine MIT Technology Review from January 1989. Quoting from the Wilson Center report: "The new form of engagement in science received the name "citizen science". The first recorded example of using the term is from 1989, describing how 225 volunteers across the US collected rain samples to assist the Audubon Society in an acid-rain awareness-raising campaign. The volunteers collected samples, checked for acidity, and reported to the organization. The information was then used to demonstrate the full extent of the phenomenon."
In Canada
Canadian Harold Harvey was among the first to research a "dead" lake. In 1971, he and R. J. Beamish published a report, "Acidification of the La Cloche Mountain Lakes", documenting the gradual deterioration of fish stocks in 60 lakes in Killarney Park in Ontario, which they had been studying systematically since 1966. | Acid rain | Wikipedia | 330 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
In the 1970s and 80s, acid rain was a major topic of research at the Experimental Lakes Area (ELA) in Northwestern Ontario, Canada. Researchers added sulfuric acid to whole lakes in controlled ecosystem experiments to simulate the effects of acid rain. Because its remote conditions allowed for whole-ecosystem experiments, research at the ELA showed that the effect of acid rain on fish populations started at concentrations much lower than those observed in laboratory experiments. In the context of a food web, fish populations crashed earlier than when acid rain had direct toxic effects to the fish because the acidity led to crashes in prey populations (e.g. mysids). As experimental acid inputs were reduced, fish populations and lake ecosystems recovered at least partially, although invertebrate populations have still not completely returned to the baseline conditions. This research showed both that acidification was linked to declining fish populations and that the effects could be reversed if sulfuric acid emissions decreased, and influenced policy in Canada and the United States.
In 1985, seven Canadian provinces (all except British Columbia, Alberta, and Saskatchewan) and the federal government signed the Eastern Canada Acid Rain Program. The provinces agreed to limit their combined sulfur dioxide emissions to 2.3 million tonnes by 1994. The Canada-US Air Quality Agreement was signed in 1991. In 1998, all federal, provincial, and territorial Ministers of Energy and Environment signed The Canada-Wide Acid Rain Strategy for Post-2000, which was designed to protect lakes that are more sensitive than those protected by earlier policies.
In India
Increased risk might be posed by the expected rise in total sulphur emissions from 4,400 kilotonnes (kt) in 1990 to 6,500 kt in 2000, 10,900 kt in 2010 and 18,500 in 2020.
Emissions of chemicals leading to acidification
The most important gas which leads to acidification is sulfur dioxide. Emissions of nitrogen oxides which are oxidized to form nitric acid are of increasing importance due to stricter controls on emissions of sulfur compounds. 70 Tg(S) per year in the form of SO2 comes from fossil fuel combustion and industry, 2.8 Tg(S) from wildfires, and 7–8 Tg(S) per year from volcanoes. | Acid rain | Wikipedia | 457 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Natural phenomena
The principal natural phenomena that contribute acid-producing gases to the atmosphere are emissions from volcanoes. Thus, for example, fumaroles from the Laguna Caliente crater of Poás Volcano create extremely high amounts of acid rain and fog, with acidity as high as a pH of 2, clearing an area of any vegetation and frequently causing irritation to the eyes and lungs of inhabitants in nearby settlements. Acid-producing gasses are also created by biological processes that occur on the land, in wetlands, and in the oceans. The major biological source of sulfur compounds is dimethyl sulfide.
Nitric acid in rainwater is an important source of fixed nitrogen for plant life, and is also produced by electrical activity in the atmosphere such as lightning.
Acidic deposits have been detected in glacial ice thousands of years old in remote parts of the globe.
Human activity
The principal cause of acid rain is sulfur and nitrogen compounds from human sources, such as electricity generation, animal agriculture, factories, and motor vehicles. These also include power plants, which use electric power generators that account for a quarter of nitrogen oxides and two-thirds of sulfur dioxide within the atmosphere. Industrial acid rain is a substantial problem in China and Russia and areas downwind from them. These areas all burn sulfur-containing coal to generate heat and electricity.
The problem of acid rain has not only increased with population and industrial growth, but has become more widespread. The use of tall smokestacks to reduce local pollution has contributed to the spread of acid rain by releasing gases into regional atmospheric circulation; dispersal from these taller stacks causes pollutants to be carried farther, causing widespread ecological damage. Often deposition occurs a considerable distance downwind of the emissions, with mountainous regions tending to receive the greatest deposition (because of their higher rainfall). An example of this effect is the low pH of rain which falls in Scandinavia. Regarding low pH and pH imbalances in correlation to acid rain, low levels, or those under the pH value of 7, are considered acidic. Acid rain falls at a pH value of roughly 4, making it harmful to consume for humans. When these low pH levels fall in specific regions, they not only affect the environment but also human health. With acidic pH levels in humans comes hair loss, low urinary pH, severe mineral imbalances, constipation, and many cases of chronic disorders like Fibromyalgia and Basal Carcinoma. | Acid rain | Wikipedia | 490 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Chemical process
Combustion of fuels and smelting of some ores produce sulfur dioxide and nitric oxides. They are converted into sulfuric acid and nitric acid.
In the gas phase sulfur dioxide is oxidized to sulfuric acid:
Nitrogen dioxide reacts with hydroxyl radicals to form nitric acid:
NO2 + OH· → HNO3
The detailed mechanisms depend on the presence water and traces of iron and manganese. A number of oxidants are capable of these reactions aside from O2, these include ozone, hydrogen peroxide, and oxygen.
Acid deposition
Wet deposition
Wet deposition of acids occurs when any form of precipitation (rain, snow, and so on) removes acids from the atmosphere and delivers it to the Earth's surface. This can result from the deposition of acids produced in the raindrops (see aqueous phase chemistry above) or by the precipitation removing the acids either in clouds or below clouds. Wet removal of both gases and aerosols are both of importance for wet deposition.
Dry deposition
Acid deposition also occurs via dry deposition in the absence of precipitation. This can be responsible for as much as 20 to 60% of total acid deposition. This occurs when particles and gases stick to the ground, plants or other surfaces.
Adverse effects
Acid rain has been shown to have adverse impacts on forests, freshwaters and soils, killing insect and aquatic life-forms as well as causing damage to buildings and having impacts on human health.
Surface waters and aquatic animals
Sulfuric acid and nitric acid have multiple impacts on aquatic ecosystems, including acidification, increased nitrogen and aluminum content, and alteration of biogeochemical processes. Both the lower pH and higher aluminium concentrations in surface water that occur as a result of acid rain can cause damage to fish and other aquatic animals. At pH lower than 5 most fish eggs will not hatch and lower pH can kill adult fish. As lakes and rivers become more acidic, biodiversity is reduced. Acid rain has eliminated insect life and some fish species, including the brook trout in some lakes, streams, and creeks in geographically sensitive areas, such as the Adirondack Mountains of the United States. | Acid rain | Wikipedia | 435 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
However, the extent to which acid rain contributes directly or indirectly via runoff from the catchment to lake and river acidity (i.e., depending on characteristics of the surrounding watershed) is variable. The United States Environmental Protection Agency's (EPA) website states: "Of the lakes and streams surveyed, acid rain caused acidity in 75% of the acidic lakes and about 50% of the acidic streams". Lakes hosted by silicate basement rocks are more acidic than lakes within limestone or other basement rocks with a carbonate composition (i.e. marble) due to buffering effects by carbonate minerals, even with the same amount of acid rain.
Soils
Soil biology and chemistry can be seriously damaged by acid rain. Some microbes are unable to tolerate changes to low pH and are killed. The enzymes of these microbes are denatured (changed in shape so they no longer function) by the acid. The hydronium ions of acid rain also mobilize toxins, such as aluminium, and leach away essential nutrients and minerals such as magnesium.
2 H+ (aq) + Mg2+ (clay) 2 H+ (clay) + Mg2+ (aq)
Soil chemistry can be dramatically changed when base cations, such as calcium and magnesium, are leached by acid rain, thereby affecting sensitive species, such as sugar maple (Acer saccharum).
Soil acidification | Acid rain | Wikipedia | 285 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Impacts of acidic water and soil acidification on plants could be minor or in most cases major. Most minor cases which do not result in fatality of plant life can be attributed to the plants being less susceptible to acidic conditions and/or the acid rain being less potent. However, even in minor cases, the plant will eventually die due to the acidic water lowering the plant's natural pH. Acidic water enters the plant and causes important plant minerals to dissolve and get carried away; which ultimately causes the plant to die of lack of minerals for nutrition. In major cases, which are more extreme, the same process of damage occurs as in minor cases, which is removal of essential minerals, but at a much quicker rate. Likewise, acid rain that falls on soil and on plant leaves causes drying of the waxy leaf cuticle, which ultimately causes rapid water loss from the plant to the outside atmosphere and eventually results in death of the plant. Soil acidification can lead to a decline in soil microbes as a result of a change in pH, which would have an adverse effect on plants due to their dependence on soil microbes to access nutrients. To see if a plant is being affected by soil acidification, one can closely observe the plant leaves. If the leaves are green and look healthy, the soil pH is normal and acceptable for plant life. But if the plant leaves have yellowing between the veins on their leaves, that means the plant is suffering from acidification and is unhealthy. Moreover, a plant suffering from soil acidification cannot photosynthesize; the acid-water-induced process of drying out of the plant can destroy chloroplast organelles. Without being able to photosynthesize, a plant cannot create nutrients for its own survival or oxygen for the survival of aerobic organisms, which affects most species on Earth and ultimately ends the purpose of the plant's existence.
Forests and other vegetation
Adverse effects may be indirectly related to acid rain, like the acid's effects on soil (see above) or high concentration of gaseous precursors to acid rain. High altitude forests are especially vulnerable as they are often surrounded by clouds and fog which are more acidic than rain. | Acid rain | Wikipedia | 445 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Plants are capable of adapting to acid rain. On Jinyun Mountain, Chongqing, plant species were seen adapting to new environmental conditions. The affects on the species ranged from being beneficial to detrimental. With natural rainfall or mild acid rainfall, the biochemical and physiological characteristics of plant seedlings were enhanced. Once the pH increases reaches the threshold of 3.5, the acid rain can no longer be beneficial and begins to have negative affects.
Acid rain can negatively impact photosynthesis in plant leaves, when leaves are exposed to a lower pH, photosynthesis is impacted due to the decline in chlorophyll. Acid rain also has the ability to cause deformation to leaves at a cellular level, examples include; tissue scaring and changes to the stomatal, epidermis and mesophyll cells. Additional impacts of acid rain includes a decline in cuticle thickness present on the leaf surface. Because acid rain damages leaves, this directly impacts a plants ability to have a strong canopy cover, a decline in canopy cover can lead plants to be more vulnerable to diseases.
Dead or dying trees often appear in areas impacted by acid rain. Acid rain causes aluminum to leach from the soil, posing risks to both plant and animal life. Furthermore, it strips the soil of critical minerals and nutrients necessary for tree growth.
At higher altitudes, acidic fog and clouds can deplete nutrients from tree foliage, leading to discolored or dead leaves and needles. This depletion compromises the trees' ability to absorb sunlight, weakening them and diminishing their capacity to endure cold conditions.
Other plants can also be damaged by acid rain, but the effect on food crops is minimized by the application of lime and fertilizers to replace lost nutrients. In cultivated areas, limestone may also be added to increase the ability of the soil to keep the pH stable, but this tactic is largely unusable in the case of wilderness lands. When calcium is leached from the needles of red spruce, these trees become less cold tolerant and exhibit winter injury and even death. Acid rain may also affect crop productivity by necrosis or changes to soil nutrients, which ultimately prevent plants from reaching maturity.
Ocean acidification | Acid rain | Wikipedia | 444 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Acid rain has a much less harmful effect on oceans on a global scale, but it creates an amplified impact in the shallower waters of coastal waters. Acid rain can cause the ocean's pH to fall, known as ocean acidification, making it more difficult for different coastal species to create their exoskeletons that they need to survive. These coastal species link together as part of the ocean's food chain, and without them being a source for other marine life to feed off of, more marine life will die. Coral's limestone skeleton is particularly sensitive to pH decreases, because the calcium carbonate, a core component of the limestone skeleton, dissolves in acidic (low pH) solutions.
In addition to acidification, excess nitrogen inputs from the atmosphere promote increased growth of phytoplankton and other marine plants, which, in turn, may cause more frequent harmful algal blooms and eutrophication (the creation of oxygen-depleted "dead zones") in some parts of the ocean.
Human health effects
Acid rain can negatively impact human health, especially when people breathe in particles released from acid rain. The effects of acid rain on human health are complex and may be seen in several ways, such as respiratory issues for long-term exposure and indirect exposure through contaminated food and water sources.
Nitrogen Dioxide Effects
Exposure to air pollutants associated with acid rain, such as nitrogen dioxide (NO2), may have a negative impact on respiratory health. Water-soluble nitrogen dioxide accumulates in the tiny airways, where it is transformed into nitric and nitrous acids. Pneumonia caused by nitric acids directly damages the epithelial cells lining the airways, resulting in pulmonary edema. Exposure to nitrogen dioxide also reduces the immune response by inhibiting the generation of inflammatory cytokines by alveolar macrophages in response to bacterial infection. In animal studies, the pollutant further reduces respiratory immunity by decreasing mucociliary clearance in the lower respiratory tract, which results in a reduced ability to remove respiratory infections. | Acid rain | Wikipedia | 413 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Sulfur Trioxide Effects
The effects of sulfur trioxide and sulfuric acid are similar because they both produce sulfuric acid when they come into touch with the wet surfaces of your skin or respiratory system. The amount of SO3 breath through the mouth is larger than the amount of SO3 breath through the nose. When humans breathe in sulfur trioxide, small droplets of sulfuric acid will form inside the body and enter the respiratory tract to the lungs depending on the particle size. The effects of SO3 on the respiratory system lead to breathing difficulty in people who have asthma symptoms. Sulfur trioxide also causes very corrosive and irritation on the skin, eye, and gastrointestinal tracts when there is direct exposure to a specific concentration or long-term exposure. Consuming concentrated sulfuric acid has been known to burn the mouth and throat, erode a hole in the stomach, burns when it comes into contact with skin, make your eyes weep if it gets into them, and mortality.
Federal Government's recommendation
Nitrogen Dioxides
A 25 parts per million (ppm) maximum for nitric oxide in working air has been set by the Occupational Safety and Health Administration (OSHA) for an 8-hour workday and a 40-hour workweek. Additionally, OSHA has established a 5-ppm nitrogen dioxide exposure limit for 15 minutes in the workplace.
Sulfur Trioxide
The not-to-exceed limits in the air, water, soil, or food that are recommended by regulations are often based on levels that affect animals before being modified to assist in safeguarding people. Depending on whether they employ different animal studies, have different exposure lengths (e.g., an 8-hour workday versus a 24-hour day), or for other reasons, these not-to-exceed values can vary between federal bodies.
The amount of sulfur dioxide that can be emitted into the atmosphere is capped by the EPA. This reduces the quantity of sulfur dioxide in the air that turns into sulfur trioxide and sulfuric acid. Sulfuric acid concentrations in workroom air are restricted by OSHA to 1 mg/m3. Moreover, NIOSH advises a time-weighted average limit of 1 mg/m3.
When you are aware of NO2 and SO3 exposure, you should talk to your doctor and ask people who are around you, especially children.
Other adverse effects | Acid rain | Wikipedia | 474 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Acid rain can damage buildings, historic monuments, and statues, especially those made of rocks, such as limestone and marble, that contain large amounts of calcium carbonate. Acids in the rain react with the calcium compounds in the stones to create gypsum, which then flakes off.
CaCO3 (s) + H2SO4 (aq) CaSO4 (s) + CO2 (g) + H2O (l)
The effects of this are commonly seen on old gravestones, where acid rain can cause the inscriptions to become completely illegible. Acid rain also increases the corrosion rate of metals, in particular iron, steel, copper and bronze.
Affected areas
Places significantly impacted by acid rain around the globe include most of eastern Europe from Poland northward into Scandinavia, the eastern third of the United States, and southeastern Canada. Other affected areas include the southeastern coast of China and Taiwan.
Prevention methods
Technical solutions
Many coal-firing power stations use flue-gas desulfurization (FGD) to remove sulfur-containing gases from their stack gases. For a typical coal-fired power station, FGD will remove 95% or more of the SO2 in the flue gases. An example of FGD is the wet scrubber which is commonly used. A wet scrubber is basically a reaction tower equipped with a fan that extracts hot smoke stack gases from a power plant into the tower. Lime or limestone in slurry form is also injected into the tower to mix with the stack gases and combine with the sulfur dioxide present. The calcium carbonate of the limestone produces pH-neutral calcium sulfate that is physically removed from the scrubber. That is, the scrubber turns sulfur pollution into industrial sulfates.
In some areas the sulfates are sold to chemical companies as gypsum when the purity of calcium sulfate is high. In others, they are placed in landfill. The effects of acid rain can last for generations, as the effects of pH level change can stimulate the continued leaching of undesirable chemicals into otherwise pristine water sources, killing off vulnerable insect and fish species and blocking efforts to restore native life.
Fluidized bed combustion also reduces the amount of sulfur emitted by power production.
Vehicle emissions control reduces emissions of nitrogen oxides from motor vehicles.
International treaties | Acid rain | Wikipedia | 464 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
International treaties on the long-range transport of atmospheric pollutants have been agreed upon by western countries for some time now. Beginning in 1979, European countries convened in order to ratify general principles discussed during the UNECE Convention. The purpose was to combat Long-Range Transboundary Air Pollution. The 1985 Helsinki Protocol on the Reduction of Sulfur Emissions under the Convention on Long-Range Transboundary Air Pollution furthered the results of the convention. Results of the treaty have already come to fruition, as evidenced by an approximate 40 percent drop in particulate matter in North America. The effectiveness of the Convention in combatting acid rain has inspired further acts of international commitment to prevent the proliferation of particulate matter. Canada and the US signed the Air Quality Agreement in 1991. Most European countries and Canada signed the treaties. Activity of the Long-Range Transboundary Air Pollution Convention remained dormant after 1999, when 27 countries convened to further reduce the effects of acid rain. In 2000, foreign cooperation to prevent acid rain was sparked in Asia for the first time. Ten diplomats from countries ranging throughout the continent convened to discuss ways to prevent acid rain. Following these discussions, the Acid Deposition Monitoring Network in East Asia (EANET) was established in 2001 as an intergovernmental initiative to provide science-based inputs for decision makers and promote international cooperation on acid deposition in East Asia. In 2023, the EANET member countries include Cambodia, China, Indonesia, Japan, Lao PDR, Malaysia, Mongolia, Myanmar, the Philippines, Republic of Korea, Russia, Thailand and Vietnam.
Emissions trading
In this regulatory scheme, every current polluting facility is given or may purchase on an open market an emissions allowance for each unit of a designated pollutant it emits. Operators can then install pollution control equipment, and sell portions of their emissions allowances they no longer need for their own operations, thereby recovering some of the capital cost of their investment in such equipment. The intention is to give operators economic incentives to install pollution controls.
The first emissions trading market was established in the United States by enactment of the Clean Air Act Amendments of 1990. The overall goal of the Acid Rain Program established by the Act is to achieve significant environmental and public health benefits through reductions in emissions of sulfur dioxide (SO2) and nitrogen oxides (NOx), the primary causes of acid rain. To achieve this goal at the lowest cost to society, the program employs both regulatory and market based approaches for controlling air pollution. | Acid rain | Wikipedia | 501 | 3263 | https://en.wikipedia.org/wiki/Acid%20rain | Physical sciences | Precipitation | null |
Brass is an alloy of copper and zinc, in proportions which can be varied to achieve different colours and mechanical, electrical, acoustic and chemical properties, but copper typically has the larger proportion, generally 66% copper and 34% zinc. In use since prehistoric times, it is a substitutional alloy: atoms of the two constituents may replace each other within the same crystal structure.
Brass is similar to bronze, a copper alloy that contains tin instead of zinc. Both bronze and brass may include small proportions of a range of other elements including arsenic, lead, phosphorus, aluminium, manganese and silicon. Historically, the distinction between the two alloys has been less consistent and clear, and increasingly museums use the more general term "copper alloy".
Brass has long been a popular material for its bright gold-like appearance and is still used for drawer pulls and doorknobs. It has also been widely used to make sculpture and utensils because of its low melting point, high workability (both with hand tools and with modern turning and milling machines), durability, and electrical and thermal conductivity. Brasses with higher copper content are softer and more golden in colour; conversely those with less copper and thus more zinc are harder and more silvery in colour.
Brass is still commonly used in applications where corrosion resistance and low friction are required, such as locks, hinges, gears, bearings, ammunition casings, zippers, plumbing, hose couplings, valves, SCUBA regulators, and electrical plugs and sockets. It is used extensively for musical instruments such as horns and bells. The composition of brass makes it a favorable substitute for copper in costume jewelry and fashion jewelry, as it exhibits greater resistance to corrosion. Brass is not as hard as bronze and so is not suitable for most weapons and tools. Nor is it suitable for marine uses, because the zinc reacts with minerals in salt water, leaving porous copper behind; marine brass, with added tin, avoids this, as does bronze.
Brass is often used in situations in which it is important that sparks not be struck, such as in fittings and tools used near flammable or explosive materials.
Properties
Brass is more malleable than bronze or zinc. The relatively low melting point of brass (, depending on composition) and its flow characteristics make it a relatively easy material to cast. By varying the proportions of copper and zinc, the properties of the brass can be changed, allowing hard and soft brasses. The density of brass is . | Brass | Wikipedia | 500 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Today, almost 90% of all brass alloys are recycled. Because brass is not ferromagnetic, ferrous scrap can be separated from it by passing the scrap near a powerful magnet. Brass scrap is melted and recast into billets that are extruded into the desired form and size. The general softness of brass means that it can often be machined without the use of cutting fluid, though there are exceptions to this.
Aluminium makes brass stronger and more corrosion-resistant. Aluminium also causes a highly beneficial hard layer of aluminium oxide (Al2O3) to be formed on the surface that is thin, transparent, and self-healing. Tin has a similar effect and finds its use especially in seawater applications (naval brasses). Combinations of iron, aluminium, silicon, and manganese make brass wear- and tear-resistant. The addition of as little as 1% iron to a brass alloy will result in an alloy with a noticeable magnetic attraction.
Brass will corrode in the presence of moisture, chlorides, acetates, ammonia, and certain acids. This often happens when the copper reacts with sulfur to form a brown and eventually black surface layer of copper sulfide which, if regularly exposed to slightly acidic water such as urban rainwater, can then oxidize in air to form a patina of green-blue copper carbonate. Depending on how the patina layer was formed, it may protect the underlying brass from further damage.
Although copper and zinc have a large difference in electrical potential, the resulting brass alloy does not experience internalized galvanic corrosion because of the absence of a corrosive environment within the mixture. However, if brass is placed in contact with a more noble metal such as silver or gold in such an environment, the brass will corrode galvanically; conversely, if brass is in contact with a less-noble metal such as zinc or iron, the less noble metal will corrode and the brass will be protected. | Brass | Wikipedia | 404 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Lead content
To enhance the machinability of brass, lead is often added in concentrations of about 2%. Since lead has a lower melting point than the other constituents of the brass, it tends to migrate towards the grain boundaries in the form of globules as it cools from casting. The pattern the globules form on the surface of the brass increases the available lead surface area which, in turn, affects the degree of leaching. In addition, cutting operations can smear the lead globules over the surface. These effects can lead to significant lead leaching from brasses of comparatively low lead content.
In October 1999, the California State Attorney General sued 13 key manufacturers and distributors over lead content. In laboratory tests, state researchers found the average brass key, new or old, exceeded the California Proposition 65 limits by an average factor of 19, assuming handling twice a day. In April 2001 manufacturers agreed to reduce lead content to 1.5%, or face a requirement to warn consumers about lead content. Keys plated with other metals are not affected by the settlement, and may continue to use brass alloys with a higher percentage of lead content.
Also in California, lead-free materials must be used for "each component that comes into contact with the wetted surface of pipes and pipe fittings, plumbing fittings and fixtures". On 1 January 2010, the maximum amount of lead in "lead-free brass" in California was reduced from 4% to 0.25% lead.
Corrosion-resistant brass for harsh environments
Dezincification-resistant (DZR or DR) brasses, sometimes referred to as CR (corrosion resistant) brasses, are used where there is a large corrosion risk and where normal brasses do not meet the requirements. Applications with high water temperatures, chlorides present or deviating water qualities (soft water) play a role. DZR-brass is used in water boiler systems. This brass alloy must be produced with great care, with special attention placed on a balanced composition and proper production temperatures and parameters to avoid long-term failures.
An example of DZR brass is the C352 brass, with about 30% zinc, 61–63% copper, 1.7–2.8% lead, and 0.02–0.15% arsenic. The lead and arsenic significantly suppress the zinc loss. | Brass | Wikipedia | 482 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
"Red brasses", a family of alloys with high copper proportion and generally less than 15% zinc, are more resistant to zinc loss. One of the metals called "red brass" is 85% copper, 5% tin, 5% lead, and 5% zinc. Copper alloy C23000, which is also known as "red brass", contains 84–86% copper, 0.05% each iron and lead, with the balance being zinc.
Another such material is gunmetal, from the family of red brasses. Gunmetal alloys contain roughly 88% copper, 8–10% tin, and 2–4% zinc. Lead can be added for ease of machining or for bearing alloys.
"Naval brass", for use in seawater, contains 40% zinc but also 1% tin. The tin addition suppresses zinc-leaching.
The NSF International requires brasses with more than 15% zinc, used in piping and plumbing fittings, to be dezincification-resistant.
Use in musical instruments
The high malleability and workability, relatively good resistance to corrosion, and traditionally attributed acoustic properties of brass, have made it the usual metal of choice for construction of musical instruments whose acoustic resonators consist of long, relatively narrow tubing, often folded or coiled for compactness; silver and its alloys, and even gold, have been used for the same reasons, but brass is the most economical choice. Collectively known as brass instruments, or simply 'the brass', these include the trombone, tuba, trumpet, cornet, flugelhorn, baritone horn, euphonium, tenor horn, and French horn, and many other "horns", many in variously sized families, such as the saxhorns. | Brass | Wikipedia | 360 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Other wind instruments may be constructed of brass or other metals, and indeed most modern student-model flutes and piccolos are made of some variety of brass, usually a cupronickel alloy similar to nickel silver (also known as German silver). Clarinets, especially low clarinets such as the contrabass and subcontrabass, are sometimes made of metal because of limited supplies of the dense, fine-grained tropical hardwoods traditionally preferred for smaller woodwinds. For the same reason, some low clarinets, bassoons and contrabassoons feature a hybrid construction, with long, straight sections of wood, and curved joints, neck, and/or bell of metal. The use of metal also avoids the risks of exposing wooden instruments to changes in temperature or humidity, which can cause sudden cracking. Even though the saxophones and sarrusophones are classified as woodwind instruments, they are normally made of brass for similar reasons, and because their wide, conical bores and thin-walled bodies are more easily and efficiently made by forming sheet metal than by machining wood.
The keywork of most modern woodwinds, including wooden-bodied instruments, is also usually made of an alloy such as nickel silver. Such alloys are stiffer and more durable than the brass used to construct the instrument bodies, but still workable with simple hand tools—a boon to quick repairs. The mouthpieces of both brass instruments and, less commonly, woodwind instruments are often made of brass among other metals as well.
Next to the brass instruments, the most notable use of brass in music is in various percussion instruments, most notably cymbals, gongs, and orchestral (tubular) bells (large "church" bells are normally made of bronze). Small handbells and "jingle bells" are also commonly made of brass.
The harmonica is a free reed aerophone, also often made from brass. In organ pipes of the reed family, brass strips (called tongues) are used as the reeds, which beat against the shallot (or beat "through" the shallot in the case of a "free" reed). Although not part of the brass section, snare drums are also sometimes made of brass. Some parts on electric guitars are also made from brass, especially inertia blocks on tremolo systems for its tonal properties, and for string nuts and saddles for both tonal properties and its low friction.
Germicidal and antimicrobial applications | Brass | Wikipedia | 508 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
The bactericidal properties of brass have been observed for centuries, particularly in marine environments where it prevents biofouling. Depending upon the type and concentration of pathogens and the medium they are in, brass kills these microorganisms within a few minutes to hours of contact.
A large number of independent studies confirm this antimicrobial effect, even against antibiotic-resistant bacteria such as MRSA and VRSA. The mechanisms of antimicrobial action by copper and its alloys, including brass, are a subject of intense and ongoing investigation.
Season cracking
Brass is susceptible to stress corrosion cracking, especially from ammonia or substances containing or releasing ammonia. The problem is sometimes known as season cracking after it was first discovered in brass cartridges used for rifle ammunition during the 1920s in the British Indian Army. The problem was caused by high residual stresses from cold forming of the cases during manufacture, together with chemical attack from traces of ammonia in the atmosphere. The cartridges were stored in stables and the ammonia concentration rose during the hot summer months, thus initiating brittle cracks. The problem was resolved by annealing the cases, and storing the cartridges elsewhere.
Types
Other phases than α, β and γ are ε, a hexagonal intermetallic CuZn3, and η, a solid solution of copper in zinc.
Brass alloys
History
Although forms of brass have been in use since prehistory, its true nature as a copper-zinc alloy was not understood until the post-medieval period because the zinc vapor which reacted with copper to make brass was not recognized as a metal. The King James Bible makes many references to "brass" to translate "nechosheth" (bronze or copper) from Hebrew to English. The earliest brasses may have been natural alloys made by smelting zinc-rich copper ores. By the Roman period brass was being deliberately produced from metallic copper and zinc minerals using the cementation process, the product of which was calamine brass, and variations on this method continued until the mid-19th century. It was eventually replaced by speltering, the direct alloying of copper and zinc metal which was introduced to Europe in the 16th century.
Brass has sometimes historically been referred to as "yellow copper". | Brass | Wikipedia | 450 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Early copper-zinc alloys
In West Asia and the Eastern Mediterranean early copper-zinc alloys are now known in small numbers from a number of 3rd millennium BC sites in the Aegean, Iraq, the United Arab Emirates, Kalmykia, Turkmenistan and Georgia and from 2nd millennium BC sites in western India, Uzbekistan, Iran, Syria, Iraq and Canaan. Isolated examples of copper-zinc alloys are known in China from the 1st century AD, long after bronze was widely used.
The compositions of these early "brass" objects are highly variable and most have zinc contents of between 5% and 15% wt which is lower than in brass produced by cementation. These may be "natural alloys" manufactured by smelting zinc rich copper ores in redox conditions. Many have similar tin contents to contemporary bronze artefacts and it is possible that some copper-zinc alloys were accidental and perhaps not even distinguished from copper. However the large number of copper-zinc alloys now known suggests that at least some were deliberately manufactured and many have zinc contents of more than 12% wt which would have resulted in a distinctive golden colour.
By the 8th–7th century BC Assyrian cuneiform tablets mention the exploitation of the "copper of the mountains" and this may refer to "natural" brass. "Oreikhalkon" (mountain copper), the Ancient Greek translation of this term, was later adapted to the Latin aurichalcum meaning "golden copper" which became the standard term for brass. In the 4th century BC Plato knew orichalkos as rare and nearly as valuable as gold and Pliny describes how aurichalcum had come from Cypriot ore deposits which had been exhausted by the 1st century AD. X-ray fluorescence analysis of 39 orichalcum ingots recovered from a 2,600-year-old shipwreck off Sicily found them to be an alloy made with 75–80% copper, 15–20% zinc and small percentages of nickel, lead and iron.
Roman world | Brass | Wikipedia | 404 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
During the later part of first millennium BC the use of brass spread across a wide geographical area from Britain and Spain in the west to Iran, and India in the east. This seems to have been encouraged by exports and influence from the Middle East and eastern Mediterranean where deliberate production of brass from metallic copper and zinc ores had been introduced. The 4th century BC writer Theopompus, quoted by Strabo, describes how heating earth from Andeira in Turkey produced "droplets of false silver", probably metallic zinc, which could be used to turn copper into oreichalkos. In the 1st century BC the Greek Dioscorides seems to have recognized a link between zinc minerals and brass describing how Cadmia (zinc oxide) was found on the walls of furnaces used to heat either zinc ore or copper and explaining that it can then be used to make brass.
By the first century BC brass was available in sufficient supply to use as coinage in Phrygia and Bithynia, and after the Augustan currency reform of 23 BC it was also used to make Roman dupondii and sestertii. The uniform use of brass for coinage and military equipment across the Roman world may indicate a degree of state involvement in the industry, and brass even seems to have been deliberately boycotted by Jewish communities in Palestine because of its association with Roman authority.
Brass was produced by the cementation process where copper and zinc ore are heated together until zinc vapor is produced which reacts with the copper. There is good archaeological evidence for this process and crucibles used to produce brass by cementation have been found on Roman period sites including Xanten and Nidda in Germany, Lyon in France and at a number of sites in Britain. They vary in size from tiny acorn sized to large amphorae like vessels but all have elevated levels of zinc on the interior and are lidded. They show no signs of slag or metal prills suggesting that zinc minerals were heated to produce zinc vapor which reacted with metallic copper in a solid state reaction. The fabric of these crucibles is porous, probably designed to prevent a buildup of pressure, and many have small holes in the lids which may be designed to release pressure or to add additional zinc minerals near the end of the process. Dioscorides mentioned that zinc minerals were used for both the working and finishing of brass, perhaps suggesting secondary additions. | Brass | Wikipedia | 485 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Brass made during the early Roman period seems to have varied between 20% and 28% wt zinc. The high content of zinc in coinage and brass objects declined after the first century AD and it has been suggested that this reflects zinc loss during recycling and thus an interruption in the production of new brass. However it is now thought this was probably a deliberate change in composition and overall the use of brass increases over this period making up around 40% of all copper alloys used in the Roman world by the 4th century AD.
Medieval period
Little is known about the production of brass during the centuries immediately after the collapse of the Roman Empire. Disruption in the trade of tin for bronze from Western Europe may have contributed to the increasing popularity of brass in the east and by the 6th–7th centuries AD over 90% of copper alloy artefacts from Egypt were made of brass. However other alloys such as low tin bronze were also used and they vary depending on local cultural attitudes, the purpose of the metal and access to zinc, especially between the Islamic and Byzantine world. Conversely the use of true brass seems to have declined in Western Europe during this period in favor of gunmetals and other mixed alloys but by about 1000 brass artefacts are found in Scandinavian graves in Scotland, brass was being used in the manufacture of coins in Northumbria and there is archaeological and historical evidence for the production of calamine brass in Germany and the Low Countries, areas rich in calamine ore. | Brass | Wikipedia | 289 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
These places would remain important centres of brass making throughout the Middle Ages period, especially Dinant. Brass objects are still collectively known as dinanderie in French. The baptismal font at St Bartholomew's Church, Liège in modern Belgium (before 1117) is an outstanding masterpiece of Romanesque brass casting, though also often described as bronze. The metal of the early 12th-century Gloucester Candlestick is unusual even by medieval standards in being a mixture of copper, zinc, tin, lead, nickel, iron, antimony and arsenic with an unusually large amount of silver, ranging from 22.5% in the base to 5.76% in the pan below the candle. The proportions of this mixture may suggest that the candlestick was made from a hoard of old coins, probably Late Roman. Latten is a term for medieval alloys of uncertain and often variable composition often covering decorative borders and similar objects cut from sheet metal, whether of brass or bronze. Especially in Tibetan art, analysis of some objects shows very different compositions from different ends of a large piece. Aquamaniles were typically made in brass in both the European and Islamic worlds.
The cementation process continued to be used but literary sources from both Europe and the Islamic world seem to describe variants of a higher temperature liquid process which took place in open-topped crucibles. Islamic cementation seems to have used zinc oxide known as tutiya or tutty rather than zinc ores for brass-making, resulting in a metal with lower iron impurities. A number of Islamic writers and the 13th century Italian Marco Polo describe how this was obtained by sublimation from zinc ores and condensed onto clay or iron bars, archaeological examples of which have been identified at Kush in Iran. It could then be used for brass making or medicinal purposes. In 10th century Yemen al-Hamdani described how spreading al-iglimiya, probably zinc oxide, onto the surface of molten copper produced tutiya vapor which then reacted with the metal. The 13th century Iranian writer al-Kashani describes a more complex process whereby tutiya was mixed with raisins and gently roasted before being added to the surface of the molten metal. A temporary lid was added at this point presumably to minimize the escape of zinc vapor. | Brass | Wikipedia | 459 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
In Europe a similar liquid process in open-topped crucibles took place which was probably less efficient than the Roman process and the use of the term tutty by Albertus Magnus in the 13th century suggests influence from Islamic technology. The 12th century German monk Theophilus described how preheated crucibles were one sixth filled with powdered calamine and charcoal then topped up with copper and charcoal before being melted, stirred then filled again. The final product was cast, then again melted with calamine. It has been suggested that this second melting may have taken place at a lower temperature to allow more zinc to be absorbed. Albertus Magnus noted that the "power" of both calamine and tutty could evaporate and described how the addition of powdered glass could create a film to bind it to the metal.
German brass making crucibles are known from Dortmund dating to the 10th century AD and from Soest and Schwerte in Westphalia dating to around the 13th century confirm Theophilus' account, as they are open-topped, although ceramic discs from Soest may have served as loose lids which may have been used to reduce zinc evaporation, and have slag on the interior resulting from a liquid process.
Africa
Some of the most famous objects in African art are the lost wax castings of West Africa, mostly from what is now Nigeria, produced first by the Kingdom of Ife and then the Benin Empire. Though normally described as "bronzes", the Benin Bronzes, now mostly in the British Museum and other Western collections, and the large portrait heads such as the Bronze Head from Ife of "heavily leaded zinc-brass" and the Bronze Head of Queen Idia, both also British Museum, are better described as brass, though of variable compositions. Work in brass or bronze continued to be important in Benin art and other West African traditions such as Akan goldweights, where the metal was regarded as a more valuable material than in Europe. | Brass | Wikipedia | 402 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Renaissance and post-medieval Europe
The Renaissance saw important changes to both the theory and practice of brassmaking in Europe. By the 15th century there is evidence for the renewed use of lidded cementation crucibles at Zwickau in Germany. These large crucibles were capable of producing c.20 kg of brass. There are traces of slag and pieces of metal on the interior. Their irregular composition suggests that this was a lower temperature, not entirely liquid, process. The crucible lids had small holes which were blocked with clay plugs near the end of the process presumably to maximize zinc absorption in the final stages. Triangular crucibles were then used to melt the brass for casting.
16th-century technical writers such as Biringuccio, Ercker and Agricola described a variety of cementation brass making techniques and came closer to understanding the true nature of the process noting that copper became heavier as it changed to brass and that it became more golden as additional calamine was added. Zinc metal was also becoming more commonplace. By 1513 metallic zinc ingots from India and China were arriving in London and pellets of zinc condensed in furnace flues at the Rammelsberg in Germany were exploited for cementation brass making from around 1550. | Brass | Wikipedia | 253 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Eventually it was discovered that metallic zinc could be alloyed with copper to make brass, a process known as speltering, and by 1657 the German chemist Johann Glauber had recognized that calamine was "nothing else but unmeltable zinc" and that zinc was a "half ripe metal". However some earlier high zinc, low iron brasses such as the 1530 Wightman brass memorial plaque from England may have been made by alloying copper with zinc and include traces of cadmium similar to those found in some zinc ingots from China.
However, the cementation process was not abandoned, and as late as the early 19th century there are descriptions of solid-state cementation in a domed furnace at around 900–950 °C and lasting up to 10 hours. The European brass industry continued to flourish into the post medieval period buoyed by innovations such as the 16th century introduction of water powered hammers for the production of wares such as pots. By 1559 the Germany city of Aachen alone was capable of producing 300,000 cwt of brass per year. After several false starts during the 16th and 17th centuries the brass industry was also established in England taking advantage of abundant supplies of cheap copper smelted in the new coal fired reverberatory furnace. In 1723 Bristol brass maker Nehemiah Champion patented the use of granulated copper, produced by pouring molten metal into cold water. This increased the surface area of the copper helping it react and zinc contents of up to 33% wt were reported using this new technique. | Brass | Wikipedia | 311 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
In 1738 Nehemiah's son William Champion patented a technique for the first industrial scale distillation of metallic zinc known as distillation per descencum or "the English process". This local zinc was used in speltering and allowed greater control over the zinc content of brass and the production of high-zinc copper alloys which would have been difficult or impossible to produce using cementation, for use in expensive objects such as scientific instruments, clocks, brass buttons and costume jewelry. However Champion continued to use the cheaper calamine cementation method to produce lower-zinc brass and the archaeological remains of bee-hive shaped cementation furnaces have been identified at his works at Warmley. By the mid-to-late 18th century developments in cheaper zinc distillation such as John-Jaques Dony's horizontal furnaces in Belgium and the reduction of tariffs on zinc as well as demand for corrosion-resistant high zinc alloys increased the popularity of speltering and as a result cementation was largely abandoned by the mid-19th century. | Brass | Wikipedia | 212 | 3292 | https://en.wikipedia.org/wiki/Brass | Physical sciences | Specific alloys | null |
Brackish water, sometimes termed brack water, is water occurring in a natural environment that has more salinity than freshwater, but not as much as seawater. It may result from mixing seawater (salt water) and fresh water together, as in estuaries, or it may occur in brackish fossil aquifers. The word comes from the Middle Dutch root brak. Certain human activities can produce brackish water, in particular civil engineering projects such as dikes and the flooding of coastal marshland to produce brackish water pools for freshwater prawn farming. Brackish water is also the primary waste product of the salinity gradient power process. Because brackish water is hostile to the growth of most terrestrial plant species, without appropriate management it can be damaging to the environment (see article on shrimp farms).
Technically, brackish water contains between 0.5 and 30 grams of salt per litre—more often expressed as 0.5 to 30 parts per thousand (‰), which is a specific gravity of between 1.0004 and 1.0226. Thus, brackish covers a range of salinity regimes and is not considered a precisely defined condition. It is characteristic of many brackish surface waters that their salinity can vary considerably over space or time. Water with a salt concentration greater than 30‰ is considered saline.
Brackish water habitats
Estuaries
Brackish water condition commonly occurs when fresh water meets seawater. In fact, the most extensive brackish water habitats worldwide are estuaries, where a river meets the sea. | Brackish water | Wikipedia | 327 | 3336 | https://en.wikipedia.org/wiki/Brackish%20water | Physical sciences | Water: General | Earth science |
The River Thames flowing through London is a classic river estuary. The town of Teddington a few miles west of London marks the boundary between the tidal and non-tidal parts of the Thames, although it is still considered a freshwater river about as far east as Battersea insofar as the average salinity is very low and the fish fauna consists predominantly of freshwater species such as roach, dace, carp, perch, and pike. The Thames Estuary becomes brackish between Battersea and Gravesend, and the diversity of freshwater fish species present is smaller, primarily roach and dace; euryhaline marine species such as flounder, European seabass, mullet, and smelt become much more common. Further east, the salinity increases and the freshwater fish species are completely replaced by euryhaline marine ones, until the river reaches Gravesend, at which point conditions become fully marine and the fish fauna resembles that of the adjacent North Sea and includes both euryhaline and stenohaline marine species. A similar pattern of replacement can be observed with the aquatic plants and invertebrates living in the river.
This type of ecological succession from freshwater to marine ecosystem is typical of river estuaries. River estuaries form important staging points during the migration of anadromous and catadromous fish species, such as salmon, shad and eels, giving them time to form social groups and to adjust to the changes in salinity. Salmon are anadromous, meaning they live in the sea but ascend rivers to spawn; eels are catadromous, living in rivers and streams, but returning to the sea to breed. Besides the species that migrate through estuaries, there are many other fish that use them as "nursery grounds" for spawning or as places young fish can feed and grow before moving elsewhere. Herring and plaice are two commercially important species that use the Thames Estuary for this purpose.
Estuaries are also commonly used as fishing grounds and as places for fish farming or ranching. For example, Atlantic salmon farms are often located in estuaries, although this has caused controversy, because in doing so, fish farmers expose migrating wild fish to large numbers of external parasites such as sea lice that escape from the pens the farmed fish are kept in.
Mangroves | Brackish water | Wikipedia | 467 | 3336 | https://en.wikipedia.org/wiki/Brackish%20water | Physical sciences | Water: General | Earth science |
Another important brackish water habitat is the mangrove swamp or mangal. Many, though not all, mangrove swamps fringe estuaries and lagoons where the salinity changes with each tide. Among the most specialised residents of mangrove forests are mudskippers, fish that forage for food on land, and archer fish, perch-like fish that "spit" at insects and other small animals living in the trees, knocking them into the water where they can be eaten. Like estuaries, mangrove swamps are extremely important breeding grounds for many fish, with species such as snappers, halfbeaks, and tarpon spawning or maturing among them. Besides fish, numerous other animals use mangroves, including such species as the saltwater crocodile, American crocodile, proboscis monkey, diamondback terrapin, and the crab-eating frog, Fejervarya cancrivora (formerly Rana cancrivora). Mangroves represent important nesting sites for numerous birds groups such as herons, storks, spoonbills, ibises, kingfishers, shorebirds and seabirds.
Although often plagued with mosquitoes and other insects that make them unpleasant for humans, mangrove swamps are very important buffer zones between land and sea, and are a natural defense against hurricane and tsunami damage in particular.
The Sundarbans and Bhitarkanika Mangroves are two of the large mangrove forests in the world, both on the coast of the Bay of Bengal.
Brackish seas and lakes
Some seas and lakes are brackish. The Baltic Sea is a brackish sea adjoining the North Sea. Originally the Eridanos river system prior to the Pleistocene, since then it has been flooded by the North Sea but still receives so much freshwater from the adjacent lands that the water is brackish. As seawater is denser, the water in the Baltic is stratified, with seawater at the bottom and freshwater at the top. Limited mixing occurs because of the lack of tides and storms, with the result that the fish fauna at the surface is freshwater in composition while that lower down is more marine. Cod are an example of a species only found in deep water in the Baltic, while pike are confined to the less saline surface waters. | Brackish water | Wikipedia | 458 | 3336 | https://en.wikipedia.org/wiki/Brackish%20water | Physical sciences | Water: General | Earth science |
The Caspian Sea is the world's largest lake and contains brackish water with a salinity about one-third that of normal seawater. The Caspian is famous for its peculiar animal fauna, including one of the few non-marine seals (the Caspian seal) and the great sturgeons, a major source of caviar.
Hudson Bay is a brackish marginal sea of the Arctic Ocean, it remains brackish due its limited connections to the open ocean, very high levels freshwater surface runoff input from the large Hudson Bay drainage basin, and low rate of evaporation due to being completely covered in ice for over half the year.
In the Black Sea the surface water is brackish with an average salinity of about 17–18 parts per thousand compared to 30 to 40 for the oceans. The deep, anoxic water of the Black Sea originates from warm, salty water of the Mediterranean.
Lake Texoma, a reservoir on the border between the U.S. states of Texas and Oklahoma, is a rare example of a brackish lake that is neither part of an endorheic basin nor a direct arm of the ocean, though its salinity is considerably lower than that of the other bodies of water mentioned here. The reservoir was created by the damming of the Red River of the South, which (along with several of its tributaries) receives large amounts of salt from natural seepage from buried deposits in the upstream region. The salinity is high enough that striped bass, a fish normally found only in salt water, has self-sustaining populations in the lake.
Brackish marsh
Other brackish bodies of water
Human uses
Brackish water is being used by humans in many different sectors. It is commonly used as cooling water for power generation and in a variety of ways in the mining, oil, and gas industries. Once desalinated it can also be used for agriculture, livestock, and municipal uses. Brackish water can be treated using reverse osmosis, electrodialysis, and other filtration processes. | Brackish water | Wikipedia | 423 | 3336 | https://en.wikipedia.org/wiki/Brackish%20water | Physical sciences | Water: General | Earth science |
The bit is the most basic unit of information in computing and digital communication. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented as either , but other representations such as true/false, yes/no, on/off, or +/− are also widely used.
The relation between these values and the physical states of the underlying storage or device is a matter of convention, and different assignments may be used even within the same device or program. It may be physically implemented with a two-state device.
A contiguous group of binary digits is commonly called a bit string, a bit vector, or a single-dimensional (or multi-dimensional) bit array.
A group of eight bits is called one byte, but historically the size of the byte is not strictly defined. Frequently, half, full, double and quadruple words consist of a number of bytes which is a low power of two. A string of four bits is usually a nibble.
In information theory, one bit is the information entropy of a random binary variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known. As a unit of information or negentropy, the bit is also known as a shannon, named after Claude E. Shannon. As a measure of the length of a digital string that is encoded as symbols over a 0-1 (binary) alphabet, the bit has been called a binit, but this usage is now rare.
In data compression, the goal is to find a shorter representation for a string, so that it requires fewer bits of storage -- but it must be "compressed" before storage and then (generally) "decompressed" before it is used in a computation. The field of Algorithmic Information Theory is devoted to the study of the "irreducible information content" of a string (i.e. its shortest-possible representation length, in bits), under the assumption that the receiver has minimal a priori knowledge of the method used to compress the string.
The symbol for the binary digit is either "bit", per the IEC 80000-13:2008 standard, or the lowercase character "b", per the IEEE 1541-2002 standard. Use of the latter may create confusion with the capital "B" which is the international standard symbol for the byte.
History | Bit | Wikipedia | 500 | 3364 | https://en.wikipedia.org/wiki/Bit | Physical sciences | Data | null |
Ralph Hartley suggested the use of a logarithmic measure of information in 1928. Claude E. Shannon first used the word "bit" in his seminal 1948 paper "A Mathematical Theory of Communication". He attributed its origin to John W. Tukey, who had written a Bell Labs memo on 9 January 1947 in which he contracted "binary information digit" to simply "bit".
Physical representation
A bit can be stored by a digital device or other physical system that exists in either of two possible distinct states. These may be the two stable states of a flip-flop, two positions of an electrical switch, two distinct voltage or current levels allowed by a circuit, two distinct levels of light intensity, two directions of magnetization or polarization, the orientation of reversible double stranded DNA, etc.
Perhaps the earliest example of a binary storage device was the punched card invented by Basile Bouchon and Jean-Baptiste Falcon (1732), developed by Joseph Marie Jacquard (1804), and later adopted by Semyon Korsakov, Charles Babbage, Herman Hollerith, and early computer manufacturers like IBM. A variant of that idea was the perforated paper tape. In all those systems, the medium (card or tape) conceptually carried an array of hole positions; each position could be either punched through or not, thus carrying one bit of information. The encoding of text by bits was also used in Morse code (1844) and early digital communications machines such as teletypes and stock ticker machines (1870).
The first electrical devices for discrete logic (such as elevator and traffic light control circuits, telephone switches, and Konrad Zuse's computer) represented bits as the states of electrical relays which could be either "open" or "closed". When relays were replaced by vacuum tubes, starting in the 1940s, computer builders experimented with a variety of storage methods, such as pressure pulses traveling down a mercury delay line, charges stored on the inside surface of a cathode-ray tube, or opaque spots printed on glass discs by photolithographic techniques. | Bit | Wikipedia | 429 | 3364 | https://en.wikipedia.org/wiki/Bit | Physical sciences | Data | null |
In the 1950s and 1960s, these methods were largely supplanted by magnetic storage devices such as magnetic-core memory, magnetic tapes, drums, and disks, where a bit was represented by the polarity of magnetization of a certain area of a ferromagnetic film, or by a change in polarity from one direction to the other. The same principle was later used in the magnetic bubble memory developed in the 1980s, and is still found in various magnetic strip items such as metro tickets and some credit cards.
In modern semiconductor memory, such as dynamic random-access memory or a solid-state drive, the two values of a bit are represented by two levels of electric charge stored in a capacitor or a floating-gate MOSFET. In certain types of programmable logic arrays and read-only memory, a bit may be represented by the presence or absence of a conducting path at a certain point of a circuit. In optical discs, a bit is encoded as the presence or absence of a microscopic pit on a reflective surface. In one-dimensional bar codes and two-dimensional QR codes, bits are encoded as lines or squares which may be either black or white.
In modern digital computing, bits are transformed in Boolean logic gates.
Transmission and processing
Bits are transmitted one at a time in serial transmission. By contrast, multiple bits are transmitted simultaneously in a parallel transmission. A serial computer processes information in either a bit-serial or a byte-serial fashion. From the standpoint of data communications, a byte-serial transmission is an 8-way parallel transmission with binary signalling.
In programming languages such as C, a bitwise operation operates on binary strings as though they are vectors of bits, rather than interpreting them as binary numbers.
Data transfer rates are usually measured in decimal SI multiples. For example, a channel capacity may be specified as 8 kbit/s = 8 kb/s = 1 kB/s.
Storage
File sizes are often measured in (binary) IEC multiples of bytes, for example 1 KiB = 1024 bytes = 8192 bits. Confusion may arise in cases where (for historic reasons) filesizes are specified with binary multipliers using the ambiguous prefixes K, M, and G rather than the IEC standard prefixes Ki, Mi, and Gi.
Mass storage devices are usually measured in decimal SI multiples, for example 1 TB = bytes. | Bit | Wikipedia | 488 | 3364 | https://en.wikipedia.org/wiki/Bit | Physical sciences | Data | null |
Confusingly, the storage capacity of a directly-addressable memory device, such as a DRAM chip, or an assemblage of such chips on a memory module, is specified as a binary multiple -- using the ambiguous prefix G rather than the IEC recommended Gi prefix. For example, a DRAM chip that is specified (and advertised) as having "1 GB" of capacity has bytes of capacity. As at 2022, the difference between the popular understanding of a memory system with "8 GB" of capacity, and the SI-correct meaning of "8 GB" was still causing difficulty to software designers.
Unit and symbol
The bit is not defined in the International System of Units (SI). However, the International Electrotechnical Commission issued standard IEC 60027, which specifies that the symbol for binary digit should be 'bit', and this should be used in all multiples, such as 'kbit', for kilobit. However, the lower-case letter 'b' is widely used as well and was recommended by the IEEE 1541 Standard (2002). In contrast, the upper case letter 'B' is the standard and customary symbol for byte.
Multiple bits
Multiple bits may be expressed and represented in several ways. For convenience of representing commonly reoccurring groups of bits in information technology, several units of information have traditionally been used. The most common is the unit byte, coined by Werner Buchholz in June 1956, which historically was used to represent the group of bits used to encode a single character of text (until UTF-8 multibyte encoding took over) in a computer and for this reason it was used as the basic addressable element in many computer architectures. By 1993, the trend in hardware design had converged on the 8-bit byte. However, because of the ambiguity of relying on the underlying hardware design, the unit octet was defined to explicitly denote a sequence of eight bits.
Computers usually manipulate bits in groups of a fixed size, conventionally named "words". Like the byte, the number of bits in a word also varies with the hardware design, and is typically between 8 and 80 bits, or even more in some specialized computers. In the early 21st century, retail personal or server computers have a word size of 32 or 64 bits. | Bit | Wikipedia | 472 | 3364 | https://en.wikipedia.org/wiki/Bit | Physical sciences | Data | null |
The International System of Units defines a series of decimal prefixes for multiples of standardized units which are commonly also used with the bit and the byte. The prefixes kilo (103) through yotta (1024) increment by multiples of one thousand, and the corresponding units are the kilobit (kbit) through the yottabit (Ybit). | Bit | Wikipedia | 78 | 3364 | https://en.wikipedia.org/wiki/Bit | Physical sciences | Data | null |
The byte is a unit of digital information that most commonly consists of eight bits. 1 byte (B) = 8 bits (bit). Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit of memory in many computer architectures. To disambiguate arbitrarily sized bytes from the common 8-bit definition, network protocol documents such as the Internet Protocol () refer to an 8-bit byte as an octet. Those bits in an octet are usually counted with numbering from 0 to 7 or 7 to 0 depending on the bit endianness.
The size of the byte has historically been hardware-dependent and no definitive standards existed that mandated the size. Sizes from 1 to 48 bits have been used. The six-bit character code was an often-used implementation in early encoding systems, and computers using six-bit and nine-bit bytes were common in the 1960s. These systems often had memory words of 12, 18, 24, 30, 36, 48, or 60 bits, corresponding to 2, 3, 4, 5, 6, 8, or 10 six-bit bytes, and persisted, in legacy systems, into the twenty-first century. In this era, bit groupings in the instruction stream were often referred to as syllables or slab, before the term byte became common.
The modern de facto standard of eight bits, as documented in ISO/IEC 2382-1:1993, is a convenient power of two permitting the binary-encoded values 0 through 255 for one byte, as 2 to the power of 8 is 256. The international standard IEC 80000-13 codified this common meaning. Many types of applications use information representable in eight or fewer bits and processor designers commonly optimize for this usage. The popularity of major commercial computing architectures has aided in the ubiquitous acceptance of the 8-bit byte. Modern architectures typically use 32- or 64-bit words, built of four or eight bytes, respectively.
The unit symbol for the byte was designated as the upper-case letter B by the International Electrotechnical Commission (IEC) and Institute of Electrical and Electronics Engineers (IEEE). Internationally, the unit octet explicitly defines a sequence of eight bits, eliminating the potential ambiguity of the term "byte". The symbol for octet, 'o', also conveniently eliminates the ambiguity in the symbol 'B' between byte and bel. | Byte | Wikipedia | 508 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
Etymology and history
The term byte was coined by Werner Buchholz in June 1956, during the early design phase for the IBM Stretch computer, which had addressing to the bit and variable field length (VFL) instructions with a byte size encoded in the instruction. It is a deliberate respelling of bite to avoid accidental mutation to bit.
Another origin of byte for bit groups smaller than a computer's word size, and in particular groups of four bits, is on record by Louis G. Dooley, who claimed he coined the term while working with Jules Schwartz and Dick Beeler on an air defense system called SAGE at MIT Lincoln Laboratory in 1956 or 1957, which was jointly developed by Rand, MIT, and IBM. Later on, Schwartz's language JOVIAL actually used the term, but the author recalled vaguely that it was derived from AN/FSQ-31.
Early computers used a variety of four-bit binary-coded decimal (BCD) representations and the six-bit codes for printable graphic patterns common in the U.S. Army (FIELDATA) and Navy. These representations included alphanumeric characters and special graphical symbols. These sets were expanded in 1963 to seven bits of coding, called the American Standard Code for Information Interchange (ASCII) as the Federal Information Processing Standard, which replaced the incompatible teleprinter codes in use by different branches of the U.S. government and universities during the 1960s. ASCII included the distinction of upper- and lowercase alphabets and a set of control characters to facilitate the transmission of written language as well as printing device functions, such as page advance and line feed, and the physical or logical control of data flow over the transmission media. During the early 1960s, while also active in ASCII standardization, IBM simultaneously introduced in its product line of System/360 the eight-bit Extended Binary Coded Decimal Interchange Code (EBCDIC), an expansion of their six-bit binary-coded decimal (BCDIC) representations used in earlier card punches.
The prominence of the System/360 led to the ubiquitous adoption of the eight-bit storage size, while in detail the EBCDIC and ASCII encoding schemes are different.
In the early 1960s, AT&T introduced digital telephony on long-distance trunk lines. These used the eight-bit μ-law encoding. This large investment promised to reduce transmission costs for eight-bit data. | Byte | Wikipedia | 491 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
In Volume 1 of The Art of Computer Programming (first published in 1968), Donald Knuth uses byte in his hypothetical MIX computer to denote a unit which "contains an unspecified amount of information ... capable of holding at least 64 distinct values ... at most 100 distinct values. On a binary computer a byte must therefore be composed of six bits". He notes that "Since 1975 or so, the word byte has come to mean a sequence of precisely eight binary digits...When we speak of bytes in connection with MIX we shall confine ourselves to the former sense of the word, harking back to the days when bytes were not yet standardized."
The development of eight-bit microprocessors in the 1970s popularized this storage size. Microprocessors such as the Intel 8080, the direct predecessor of the 8086, could also perform a small number of operations on the four-bit pairs in a byte, such as the decimal-add-adjust (DAA) instruction. A four-bit quantity is often called a nibble, also nybble, which is conveniently represented by a single hexadecimal digit.
The term octet unambiguously specifies a size of eight bits. It is used extensively in protocol definitions.
Historically, the term octad or octade was used to denote eight bits as well at least in Western Europe; however, this usage is no longer common. The exact origin of the term is unclear, but it can be found in British, Dutch, and German sources of the 1960s and 1970s, and throughout the documentation of Philips mainframe computers.
Unit symbol
The unit symbol for the byte is specified in IEC 80000-13, IEEE 1541 and the Metric Interchange Format as the upper-case character B.
In the International System of Quantities (ISQ), B is also the symbol of the bel, a unit of logarithmic power ratio named after Alexander Graham Bell, creating a conflict with the IEC specification. However, little danger of confusion exists, because the bel is a rarely used unit. It is used primarily in its decadic fraction, the decibel (dB), for signal strength and sound pressure level measurements, while a unit for one-tenth of a byte, the decibyte, and other fractions, are only used in derived units, such as transmission rates. | Byte | Wikipedia | 485 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
The lowercase letter o for octet is defined as the symbol for octet in IEC 80000-13 and is commonly used in languages such as French and Romanian, and is also combined with metric prefixes for multiples, for example ko and Mo.
Multiple-byte units
More than one system exists to define unit multiples based on the byte. Some systems are based on powers of 10, following the International System of Units (SI), which defines for example the prefix kilo as 1000 (103); other systems are based on powers of two. Nomenclature for these systems has led to confusion. Systems based on powers of 10 use standard SI prefixes (kilo, mega, giga, ...) and their corresponding symbols (k, M, G, ...). Systems based on powers of 2, however, might use binary prefixes (kibi, mebi, gibi, ...) and their corresponding symbols (Ki, Mi, Gi, ...) or they might use the prefixes K, M, and G, creating ambiguity when the prefixes M or G are used.
While the difference between the decimal and binary interpretations is relatively small for the kilobyte (about 2% smaller than the kibibyte), the systems deviate increasingly as units grow larger (the relative deviation grows by 2.4% for each three orders of magnitude). For example, a power-of-10-based terabyte is about 9% smaller than power-of-2-based tebibyte.
Units based on powers of 10
Definition of prefixes using powers of 10—in which 1 kilobyte (symbol kB) is defined to equal 1,000 bytes—is recommended by the International Electrotechnical Commission (IEC). The IEC standard defines eight such multiples, up to 1 yottabyte (YB), equal to 10008 bytes. The additional prefixes ronna- for 10009 and quetta- for 100010 were adopted by the International Bureau of Weights and Measures (BIPM) in 2022. | Byte | Wikipedia | 429 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
This definition is most commonly used for data-rate units in computer networks, internal bus, hard drive and flash media transfer speeds, and for the capacities of most storage media, particularly hard drives, flash-based storage, and DVDs. Operating systems that use this definition include macOS, iOS, Ubuntu, and Debian. It is also consistent with the other uses of the SI prefixes in computing, such as CPU clock speeds or measures of performance.
Units based on powers of 2
A system of units based on powers of 2 in which 1 kibibyte (KiB) is equal to 1,024 (i.e., 210) bytes is defined by international standard IEC 80000-13 and is supported by national and international standards bodies (BIPM, IEC, NIST). The IEC standard defines eight such multiples, up to 1 yobibyte (YiB), equal to 10248 bytes. The natural binary counterparts to ronna- and quetta- were given in a consultation paper of the International Committee for Weights and Measures' Consultative Committee for Units (CCU) as robi- (Ri, 10249) and quebi- (Qi, 102410), but have not yet been adopted by the IEC or ISO.
An alternative system of nomenclature for the same units (referred to here as the customary convention), in which 1 kilobyte (KB) is equal to 1,024 bytes, 1 megabyte (MB) is equal to 10242 bytes and 1 gigabyte (GB) is equal to 10243 bytes is mentioned by a 1990s JEDEC standard. Only the first three multiples (up to GB) are mentioned by the JEDEC standard, which makes no mention of TB and larger. While confusing and incorrect, the customary convention is used by the Microsoft Windows operating system and random-access memory capacity, such as main memory and CPU cache size, and in marketing and billing by telecommunication companies, such as Vodafone, AT&T, Orange and Telstra.
For storage capacity, the customary convention was used by macOS and iOS through Mac OS X 10.5 Leopard and iOS 10, after which they switched to units based on powers of 10.
Parochial units | Byte | Wikipedia | 458 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
Various computer vendors have coined terms for data of various sizes, sometimes with different sizes for the same term even within a single vendor. These terms include double word, half word, long word, quad word, slab, superword and syllable. There are also informal terms. e.g., half byte and nybble for 4 bits, octal K for .
History of the conflicting definitions
Contemporary computer memory has a binary architecture making a definition of memory units based on powers of 2 most practical. The use of the metric prefix kilo for binary multiples arose as a convenience, because is approximately . This definition was popular in early decades of personal computing, with products like the Tandon 5-inch DD floppy format (holding bytes) being advertised as "360 KB", following the -byte convention. It was not universal, however. The Shugart SA-400 5-inch floppy disk held 109,375 bytes unformatted, and was advertised as "110 Kbyte", using the 1000 convention. Likewise, the 8-inch DEC RX01 floppy (1975) held bytes formatted, and was advertised as "256k". Some devices were advertised using a mixture of the two definitions: most notably, floppy disks advertised as "1.44 MB" have an actual capacity of , the equivalent of 1.47 MB or 1.41 MiB.
In 1995, the International Union of Pure and Applied Chemistry's (IUPAC) Interdivisional Committee on Nomenclature and Symbols attempted to resolve this ambiguity by proposing a set of binary prefixes for the powers of 1024, including kibi (kilobinary), mebi (megabinary), and gibi (gigabinary).
In December 1998, the IEC addressed such multiple usages and definitions by adopting the IUPAC's proposed prefixes (kibi, mebi, gibi, etc.) to unambiguously denote powers of 1024. Thus one kibibyte (1 KiB) is 10241 bytes = 1024 bytes, one mebibyte (1 MiB) is 10242 bytes = bytes, and so on.
In 1999, Donald Knuth suggested calling the kibibyte a "large kilobyte" (KKB). | Byte | Wikipedia | 464 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
Modern standard definitions
The IEC adopted the IUPAC proposal and published the standard in January 1999. The IEC prefixes are part of the International System of Quantities. The IEC further specified that the kilobyte should only be used to refer to bytes.
Lawsuits over definition
Lawsuits arising from alleged consumer confusion over the binary and decimal definitions of multiples of the byte have generally ended in favor of the manufacturers, with courts holding that the legal definition of gigabyte or GB is 1 GB = (109) bytes (the decimal definition), rather than the binary definition (230, i.e., ). Specifically, the United States District Court for the Northern District of California held that "the U.S. Congress has deemed the decimal definition of gigabyte to be the 'preferred' one for the purposes of 'U.S. trade and commerce' [...] The California Legislature has likewise adopted the decimal system for all 'transactions in this state.
Earlier lawsuits had ended in settlement with no court ruling on the question, such as a lawsuit against drive manufacturer Western Digital. Western Digital settled the challenge and added explicit disclaimers to products that the usable capacity may differ from the advertised capacity. Seagate was sued on similar grounds and also settled.
Practical examples
Common uses
Many programming languages define the data type byte.
The C and C++ programming languages define byte as an "addressable unit of data storage large enough to hold any member of the basic character set of the execution environment" (clause 3.6 of the C standard). The C standard requires that the integral data type unsigned char must hold at least 256 different values, and is represented by at least eight bits (clause 5.2.4.2.1). Various implementations of C and C++ reserve 8, 9, 16, 32, or 36 bits for the storage of a byte. In addition, the C and C++ standards require that there be no gaps between two bytes. This means every bit in memory is part of a byte.
Java's primitive data type byte is defined as eight bits. It is a signed data type, holding values from −128 to 127.
.NET programming languages, such as C#, define byte as an unsigned type, and the sbyte as a signed data type, holding values from 0 to 255, and −128 to 127, respectively. | Byte | Wikipedia | 485 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
In data transmission systems, the byte is used as a contiguous sequence of bits in a serial data stream, representing the smallest distinguished unit of data. For asynchronous communication a full transmission unit usually additionally includes a start bit, 1 or 2 stop bits, and possibly a parity bit, and thus its size may vary from seven to twelve bits for five to eight bits of actual data. For synchronous communication the error checking usually uses bytes at the end of a frame. | Byte | Wikipedia | 99 | 3365 | https://en.wikipedia.org/wiki/Byte | Physical sciences | Data | null |
Boron nitride is a thermally and chemically resistant refractory compound of boron and nitrogen with the chemical formula BN. It exists in various crystalline forms that are isoelectronic to a similarly structured carbon lattice. The hexagonal form corresponding to graphite is the most stable and soft among BN polymorphs, and is therefore used as a lubricant and an additive to cosmetic products. The cubic (zincblende aka sphalerite structure) variety analogous to diamond is called c-BN; it is softer than diamond, but its thermal and chemical stability is superior. The rare wurtzite BN modification is similar to lonsdaleite but slightly softer than the cubic form.
Because of excellent thermal and chemical stability, boron nitride ceramics are used in high-temperature equipment and metal casting. Boron nitride has potential use in nanotechnology.
History
Boron nitride was discovered by chemistry teacher of the Liverpool Institute in 1842 via reduction of boric acid with charcoal in the presence of potassium cyanide.
Structure
Boron nitride exists in multiple forms that differ in the arrangement of the boron and nitrogen atoms, giving rise to varying bulk properties of the material.
Amorphous form (a-BN)
The amorphous form of boron nitride (a-BN) is non-crystalline, lacking any long-distance regularity in the arrangement of its atoms. It is analogous to amorphous carbon.
All other forms of boron nitride are crystalline. | Boron nitride | Wikipedia | 317 | 3370 | https://en.wikipedia.org/wiki/Boron%20nitride | Physical sciences | Ceramic compounds | Chemistry |
Hexagonal form (h-BN)
The most stable crystalline form is the hexagonal one, also called h-BN, α-BN, g-BN, graphitic boron nitride and "white graphene". Hexagonal boron nitride (point group = D3h; space group = P63/mmc) has a layered structure similar to graphite. Within each layer, boron and nitrogen atoms are bound by strong covalent bonds, whereas the layers are held together by weak van der Waals forces. The interlayer "registry" of these sheets differs, however, from the pattern seen for graphite, because the atoms are eclipsed, with boron atoms lying over and above nitrogen atoms. This registry reflects the local polarity of the B–N bonds, as well as interlayer N-donor/B-acceptor characteristics. Likewise, many metastable forms consisting of differently stacked polytypes exist. Therefore, h-BN and graphite are very close neighbors, and the material can accommodate carbon as a substituent element to form BNCs. BC6N hybrids have been synthesized, where carbon substitutes for some B and N atoms. Hexagonal boron nitride monolayer is analogous to graphene, having a honeycomb lattice structure of nearly the same dimensions. Unlike graphene, which is black and an electrical conductor, h-BN monolayer is white and an insulator. It has been proposed for use as an atomic flat insulating substrate or a tunneling dielectric barrier in 2D electronics. .
Cubic form (c-BN)
Cubic boron nitride has a crystal structure analogous to that of diamond. Consistent with diamond being less stable than graphite, the cubic form is less stable than the hexagonal form, but the conversion rate between the two is negligible at room temperature, as it is for diamond. The cubic form has the sphalerite crystal structure (space group = F3m), the same as that of diamond (with ordered B and N atoms), and is also called β-BN or c-BN. | Boron nitride | Wikipedia | 439 | 3370 | https://en.wikipedia.org/wiki/Boron%20nitride | Physical sciences | Ceramic compounds | Chemistry |
Wurtzite form (w-BN)
The wurtzite form of boron nitride (w-BN; point group = C6v; space group = P63mc) has the same structure as lonsdaleite, a rare hexagonal polymorph of carbon. As in the cubic form, the boron and nitrogen atoms are grouped into tetrahedra. In the wurtzite form, the boron and nitrogen atoms are grouped into 6-membered rings. In the cubic form all rings are in the chair configuration, whereas in w-BN the rings between 'layers' are in boat configuration. Earlier optimistic reports predicted that the wurtzite form was very strong, and was estimated by a simulation as potentially having a strength 18% stronger than that of diamond. Since only small amounts of the mineral exist in nature, this has not yet been experimentally verified. Its hardness is 46 GPa, slightly harder than commercial borides but softer than the cubic form of boron nitride.
Properties
Physical
The partly ionic structure of BN layers in h-BN reduces covalency and electrical conductivity, whereas the interlayer interaction increases resulting in higher hardness of h-BN relative to graphite. The reduced electron-delocalization in hexagonal-BN is also indicated by its absence of color and a large band gap. Very different bonding – strong covalent within the basal planes (planes where boron and nitrogen atoms are covalently bonded) and weak between them – causes high anisotropy of most properties of h-BN.
For example, the hardness, electrical and thermal conductivity are much higher within the planes than perpendicular to them. On the contrary, the properties of c-BN and w-BN are more homogeneous and isotropic.
Those materials are extremely hard, with the hardness of bulk c-BN being slightly smaller and w-BN even higher than that of diamond. Polycrystalline c-BN with grain sizes on the order of 10 nm is also reported to have Vickers hardness comparable or higher than diamond. Because of much better stability to heat and transition metals, c-BN surpasses diamond in mechanical applications, such as machining steel. The thermal conductivity of BN is among the highest of all electric insulators (see table). | Boron nitride | Wikipedia | 479 | 3370 | https://en.wikipedia.org/wiki/Boron%20nitride | Physical sciences | Ceramic compounds | Chemistry |
Boron nitride can be doped p-type with beryllium and n-type with boron, sulfur, silicon or if co-doped with carbon and nitrogen. Both hexagonal and cubic BN are wide-gap semiconductors with a band-gap energy corresponding to the UV region. If voltage is applied to h-BN or c-BN, then it emits UV light in the range 215–250 nm and therefore can potentially be used as light-emitting diodes (LEDs) or lasers.
Little is known on melting behavior of boron nitride. It degrades at 2973 °C, but melts at elevated pressure.
Thermal stability
Hexagonal and cubic BN (and probably w-BN) show remarkable chemical and thermal stabilities. For example, h-BN is stable to decomposition at temperatures up to 1000 °C in air, 1400 °C in vacuum, and 2800 °C in an inert atmosphere. The reactivity of h-BN and c-BN is relatively similar, and the data for c-BN are summarized in the table below.
Thermal stability of c-BN can be summarized as follows:
In air or oxygen: protective layer prevents further oxidation to ~1300 °C; no conversion to hexagonal form at 1400 °C.
In nitrogen: some conversion to h-BN at 1525 °C after 12 h.
In vacuum (): conversion to h-BN at 1550–1600 °C.
Chemical stability
Boron nitride is not attacked by the usual acids, but it is soluble in alkaline molten salts and nitrides, such as LiOH, KOH, NaOH-, , , , , or , which are therefore used to etch BN.
Thermal conductivity
The theoretical thermal conductivity of hexagonal boron nitride nanoribbons (BNNRs) can approach 1700–2000 W/(m⋅K), which has the same order of magnitude as the experimental measured value for graphene, and can be comparable to the theoretical calculations for graphene nanoribbons. Moreover, the thermal transport in the BNNRs is anisotropic. The thermal conductivity of zigzag-edged BNNRs is about 20% larger than that of armchair-edged nanoribbons at room temperature. | Boron nitride | Wikipedia | 473 | 3370 | https://en.wikipedia.org/wiki/Boron%20nitride | Physical sciences | Ceramic compounds | Chemistry |
Mechanical properties
BN nanosheets consist of hexagonal boron nitride (h-BN). They are stable up to 800°C in air. The structure of monolayer BN is similar to that of graphene, which has exceptional strength, a high-temperature lubricant, and a substrate in electronic devices.
The anisotropy of Young's modulus and Poisson's ratio depends on the system size. h-BN also exhibits strongly anisotropic strength and toughness, and maintains these over a range of vacancy defects, showing that the anisotropy is independent to the defect type.
Natural occurrence
In 2009, cubic form (c-BN) was reported in Tibet, and the name qingsongite proposed. The substance was found in dispersed micron-sized inclusions in chromium-rich rocks. In 2013, the International Mineralogical Association affirmed the mineral and the name.
Synthesis
Preparation and reactivity of hexagonal BN
Hexagonal boron nitride is obtained by the treating boron trioxide () or boric acid () with ammonia () or urea () in an inert atmosphere:
(T = 900 °C)
(T = 900 °C)
(T > 1000 °C)
(T > 1500 °C)
The resulting disordered (amorphous) material contains 92–95% BN and 5–8% . The remaining can be evaporated in a second step at temperatures in order to achieve BN concentration >98%. Such annealing also crystallizes BN, the size of the crystallites increasing with the annealing temperature.
h-BN parts can be fabricated inexpensively by hot-pressing with subsequent machining. The parts are made from boron nitride powders adding boron oxide for better compressibility. Thin films of boron nitride can be obtained by chemical vapor deposition from borazine. ZYP Coatings also has developed boron nitride coatings that may be painted on a surface. Combustion of boron powder in nitrogen plasma at 5500 °C yields ultrafine boron nitride used for lubricants and toners.
Boron nitride reacts with iodine fluoride to give in low yield.
Boron nitride reacts with nitrides of lithium, alkaline earth metals and lanthanides to form nitridoborates. For example:
Intercalation of hexagonal BN | Boron nitride | Wikipedia | 505 | 3370 | https://en.wikipedia.org/wiki/Boron%20nitride | Physical sciences | Ceramic compounds | Chemistry |
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