Rational Numbers Stock gains and losses are often represented as percentages.(credit: "stock market quotes in newspaper" by Andreas Poike/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Define and identify numbers that are rational. Simplify rational numbers and express in lowest terms. Add and subtract rational numbers. Convert between improper fractions and mixed numbers. Convert rational numbers between decimal and fraction form. Multiply and divide rational numbers. Apply the order of operations to rational numbers to simplify expressions. Apply density property of rational numbers. Solve problems involving rational numbers. Use fractions to convert between units. Define and apply percent. Solve problems using percent. We are often presented with percentages or fractions to explain how much of a population has a certain feature. For example, the 6-year graduation rate of college students at public institutions is 57.6%, or 72/125. That fraction may be unsettling. But without the context, the percentage is hard to judge. So how does that compare to private institutions? There, the 6-year graduation rate is 65.4%, or 327/500. Comparing the percentages is straightforward, but the fractions are harder to interpret due to different denominators. For more context, historical data could be found. One study reported that the 6-year graduation rate in 1995 was 56.4%. Comparing that historical number to the recent 6-year graduation rate at public institutions of 57.6% shows that there hasn't been much change in that rate. Defining and Identifying Numbers That Are Rational A rational number (called rational since it is a ratio) is just a fraction where the numerator is an integer and the denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here that any integer is a rational number. An integer, n , written as a fraction of two integers is n 1 . In its most basic representation, a rational number is an integer divided by a non-zero integer, such as 3 12 . Fractions may be used to represent parts of a whole. The denominator is the total number of parts to the object, and the numerator is how many of those parts are being used or selected. So, if a pizza is cut into 8 equal pieces, each piece is 1 8 of the pizza. If you take three slices, you have 3 8 of the pizza ( ). Similarly, if in a group of 20 people, 5 are wearing hats, then 5 20 of the people are wearing hats ( ). Pizza cut in 8 slices, with 3 slices highlighted Group of 20 people, with 5 people wearing hats Another representation of rational numbers is as a mixed number, such as 2 5 8 ( ). This represents a whole number (2 in this case), plus a fraction (the 5 8 ). Two whole pizzas and one partial pizza Rational numbers may also be expressed in decimal form; for instance, as 1.34. When 1.34 is written, the decimal part, 0.34, represents the fraction 34 100 , and the number 1.34 is equal to 1 34 100 . However, not all decimal representations are rational numbers. A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal digits are 0) is a terminating decimal , as in 1.34 above. Alternately, any decimal numeral that, after a finite number of decimal digits, has digits equal to 0 for all digits following the last non-zero digit. All numbers that can be expressed as a terminating decimal are rational. This comes from what the decimal represents. The decimal part is the fraction of the decimal part divided by the appropriate power of 10. That power of 10 is the number of decimal digits present, as for 0.34, with two decimal digits, being equal to 34 100 . Another form that is a rational number is a decimal that repeats a pattern, such as 67.1313… When a rational number is expressed in decimal form and the decimal is a repeated pattern, we use special notation to designate the part that repeats. For example, if we have the repeating decimal 4.3636…, we write this as 4. 36 ¯ . The bar over the 36 indicates that the 36 repeats forever. If the decimal representation of a number does not terminate or form a repeating decimal, that number is not a rational number. One class of numbers that is not rational is the square roots of integers or rational numbers that are not perfect squares , such as 10 and 25 6 . More generally, the number b is the square root of the number a if a = b 2 . The notation for this is b = a , where the symbol is the square root sign. An integer perfect square is any integer that can be written as the square of another integer. A rational perfect square is any rational number that can be written as a fraction of two integers that are perfect squares. Sometimes you may be able to identify a perfect square from memory. Another process that may be used is to factor the number into the product of an integer with itself. Or a calculator (such as Desmos) may be used to find the square root of the number. If the calculator yields an integer, the original number was a perfect square. Using Desmos to Find the Square Root of a Number When Desmos is used, there is a tab at the bottom of the screen that opens the keyboard for Desmos. The keyboard is shown below. On the keyboard ( ) is the square root symbol ( ) . To find the square root of a number, click the square root key, and then type the number. Desmos will automatically display the value of the square root as you enter the number. Desmos keyboard with square root key Identifying Perfect Squares Which of the following are perfect squares? 45 144 We could attempt to find the perfect square by factoring. Writing all the factor pairs of 45 results in 1 × 45 , 3 × 15 , and 5 × 9 . None of the pairs is a square, so 45 is not a perfect square. Using a calculator to find the square root of 45, we obtain 6.708 (rounded to three decimal places). Since this was not an integer, the original number was not a perfect square. We could attempt to find the perfect square by factoring. Writing all the factor pairs of 144 results in 1 × 144 , 2 × 72 , 3 × 48 , 6 × 24 , 8 × 18 , and 12 × 12 . Since the last pair is an integer multiplied by itself, 144 is a perfect square. Alternately, using Desmos to find the square root of 144, we obtain 12. Since the square root of 144 is an integer, 144 is a perfect square. Introduction to Fractions Identifying Rational Numbers Determine which of the following are rational numbers: 73 4.556 3 1 5 41 17 5 . 64 ¯ Since 73 is not a perfect square, its square root is not a rational number. This can also be seen when a calculator is used. Entering 73 into a calculator results in 8.544003745317 (and then more decimal values after that). There is no repeated pattern, so this is not a rational number. Since 4.556 is a decimal that terminates, this is a rational number. 3 1 5 is a mixed number, so it is a rational number. 41 17 is an integer divided by an integer, so it is a rational number. 5.646464... is a decimal that repeats a pattern, so it is a rational number. Simplifying Rational Numbers and Expressing in Lowest Terms A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the same value with different rational numbers. For instance, 4 5 and 12 15 are the same value. If we enter them into a calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two fractions are equal. In , we see that 3 5 of the rectangle and 9 15 of the rectangle are equal areas. Two Rectangles with Equal Areas They are the same proportion of the area of the rectangle. The left rectangle has 5 pieces, three of which are shaded. The right rectangle has 15 pieces, 9 of which are shaded. Each of the pieces of the left rectangle was divided equally into three pieces. This was a multiplication. The numerator describing the left rectangle was 3 but it becomes 3 × 3 , or 9, as each piece was divided into three. Similarly, the denominator describing the left rectangle was 5, but became 5 × 3 , or 15, as each piece was divided into 3. The fractions 3 5 and 9 15 are equivalent because they represent the same portion (often loosely referred to as equal). This understanding of equivalent fractions is very useful for conceptualization, but it isn’t practical, in general, for determining when two fractions are equivalent. Generally, to determine if the two fractions a b and c d are equivalent, we check to see that a × d = b × c . If those two products are equal, then the fractions are equal also. Determining If Two Fractions Are Equivalent Determine if 12 30 and 14 35 are equivalent fractions. Applying the definition, a = 12 , b = 30 , c = 14 and d = 35 . So a × d = 12 × 35 = 420 . Also, b × c = 30 × 14 = 420 . Since these values are equal, the fractions are equivalent. That a × d = b × c indicates the fractions a b and c d are equivalent is due to some algebra. One property of natural numbers, integers, and rational numbers (also irrational numbers) is that for any three numbers a , b , and c with c ≠ 0 , if a = b , then a / c = b / c . In other words, when two numbers are equal, then dividing both numbers by the same non-zero number, the two newly obtained numbers are also equal. We can apply that to a × d and b × c , to show that a b and c d are equivalent if a × d = b × c . If a × d = b × c , and c ≠ 0 , d ≠ 0 , we can divide both sides by and obtain the following: a × d c = b × c c . We can divide out the c on the right-hand side of the equation, resulting in a × d c = b . Similarly, we can divide both sides of the equation by d and obtain the following: a × d c × d = b d . We can divide out d the on the left-hand side of the equation, resulting in a c = b d . So, the rational numbers a c and b d are equivalent when a × d = b × c . Equivalent Fractions Recall that a common divisor or common factor of a set of integers is one that divides all the numbers of the set of numbers being considered. In a fraction, when the numerator and denominator have a common divisor, that common divisor can be divided out . This is often called canceling the common factors or, more colloquially, as canceling . To show this, consider the fraction 36 63 . The numerator and denominator have the common factor 3. We can rewrite the fraction as 36 63 = 12 × 3 21 × 3 . The common divisor 3 is then divided out, or canceled, and we can write the fraction as 12 × 3 21 × 3 = 12 21 . The 3s have been crossed out to indicate they have been divided out. The process of dividing out two factors is also referred to as reducing the fraction . If the numerator and denominator have no common positive divisors other than 1, then the rational number is in lowest terms . The process of dividing out common divisors of the numerator and denominator of a fraction is called reducing the fraction . One way to reduce a fraction to lowest terms is to determine the GCD of the numerator and denominator and divide out the GCD. Another way is to divide out common divisors until the numerator and denominator have no more common factors. Reducing Fractions to Lowest Terms Express the following rational numbers in lowest terms: 36 48 100 250 51 136 One process to reduce 36 48 to lowest terms is to identify the GCD of 36 and 48 and divide out the GCD. The GCD of 36 and 48 is 12. Step 1: We can then rewrite the numerator and denominator by factoring 12 from both. 36 48 = 12 × 3 12 × 4 Step 2: We can now divide out the 12s from the numerator and denominator. 36 48 = 12 × 3 12 × 4 = 3 4 So, when 36 48 is reduced to lowest terms, the result is 3 4 . Alternately, you could identify a common factor, divide out that common factor, and repeat the process until the remaining fraction is in lowest terms. Step 1: You may notice that 4 is a common factor of 36 and 48. Step 2: Divide out the 4, as in 36 48 = 4 × 9 4 × 12 = 4 × 9 4 × 12 = 9 12 . Step 3: Examining the 9 and 12, you identify 3 as a common factor and divide out the 3, as in 9 12 = 3 × 3 3 × 4 = 3 4 . The 3 and 4 have no common positive factors other than 1, so it is in lowest terms. So, when 36 48 is reduced to lowest terms, the result is 3 4 . Step 1: To reduce 100 250 to lowest terms, identify the GCD of 100 and 250. This GCD is 50. Step 2: We can then rewrite the numerator and denominator by factoring 50 from both. 100 250 = 50 × 2 50 × 5 . Step 3: We can now divide out the 50s from the numerator and denominator. 100 250 = 50 × 2 50 × 5 = 2 5 So, when 100 250 is reduced to lowest terms, the result is 2 5 . Step 1: To reduce 51 136 to lowest terms, identify the GCD of 51 and 136. This GCD is 17. Step 2: We can then rewrite the numerator and denominator by factoring 17 from both. 51 136 = 17 × 3 17 × 8 Step 3: We can now divide out the 17s from the numerator and denominator. 51 136 = 17 × 3 17 × 8 = 3 8 So, when 51 136 is reduced to lowest terms, the result is 3 8 . Reducing Fractions to Lowest Terms Using Desmos to Find Lowest Terms Desmos is a free online calculator . Desmos supports reducing fractions to lowest terms. When a fraction is entered, Desmos immediately calculates the decimal representation of the fraction. However, to the left of the fraction, there is a button that, when clicked, shows the fraction in reduced form. Using Desmos to Reduce a Fraction Adding and Subtracting Rational Numbers Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or by finding a common denominator for the rational numbers being added or subtracted. Using Desmos to Add Rational Numbers in Fractional Form To create a fraction in Desmos, enter the numerator, then use the division key (/) on your keyboard, and then enter the denominator. The fraction is then entered. Then click the right arrow key to exit the denominator of the fraction. Next, enter the arithmetic operation (+ or –). Then enter the next fraction. The answer is displayed dynamically (calculates as you enter). To change the Desmos result from decimal form to fractional form, use the fraction button ( ) on the left of the line that contains the calculation: Fraction button on the Desmos keyboard Adding Rational Numbers Using Desmos Calculate 23 42 + 9 56 using Desmos. Enter 23 42 + 9 56 in Desmos. The result is displayed as 0.70833333333 (which is 0.708 3 ¯ ). Clicking the fraction button to the left on the calculation line yields 17 24 . Performing addition and subtraction without a calculator may be more involved. When the two rational numbers have a common denominator, then adding or subtracting the two numbers is straightforward. Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator. Symbolically, we write this as a c ± b c = a ± b c . This can be seen in the , which shows 3 20 + 4 20 = 7 20 . Partially Shaded Rectangle It is customary to then write the result in lowest terms. If c is a non-zero integer, then a c ± b c = a ± b c . Adding Rational Numbers with the Same Denominator Calculate 13 28 + 7 28 . Since the rational numbers have the same denominator, we perform the addition of the numerators, 13 + 7 , and then place the result in the numerator and the common denominator, 28, in the denominator. 13 28 + 7 28 = 13 + 7 28 = 20 28 Once we have that result, reduce to lowest terms, which gives 20 28 = 4 × 5 4 × 7 = 4 × 5 4 × 7 = 5 7 . Subtracting Rational Numbers with the Same Denominator Calculate 45 136 − 17 136 . Since the rational numbers have the same denominator, we perform the subtraction of the numerators, 45 − 17 , and then place the result in the numerator and the common denominator, 136, in the denominator. 45 136 − 17 136 − 45 − 17 136 = 28 136 Once we have that result, reduce to lowest terms, this gives 28 136 = 4 × 7 4 × 34 = 4 × 7 4 × 34 = 7 34 . When the rational numbers do not have common denominators, then we have to transform the rational numbers so that they do have common denominators. The common denominator that reduces work later in the problem is the LCM of the numerator and denominator. When adding or subtracting the rational numbers a b and c d , we perform the following steps. Step 1: Find LCM ( b , d ) . Step 2: Calculate n = LCM ( b , d ) b and m = L C M ( b , d ) d . Step 3: Multiply the numerator and denominator of a b by n , yielding a × n b × n . Step 4: Multiply the numerator and denominator of c d by m , yielding c × m d × m . Step 5: Add or subtract the rational numbers from Steps 3 and 4, since they now have the common denominators. You should be aware that the common denominator is LCM ( b , d ) . For the first denominator, we have b × n = b × L C M ( b , d ) b = L C M ( b , d ) , since we multiply and divide LCM ( b , d ) by the same number. For the same reason, d × m = d × L C M ( b , d ) b = L C M ( b , d ) . Adding Rational Numbers with Unequal Denominators Calculate 11 18 + 2 15 . The denominators of the fractions are 18 and 15, so we label b = 18 and d = 15 . Step 1: Find LCM(18,15). This is 90. Step 2 : Calculate n and m . n = 90 18 = 5 and m = 90 15 = 6 . Step 3: Multiplying the numerator and denominator of 11 18 by n = 5 yields 11 × 5 18 × 5 = 55 90 . Step 4: Multiply the numerator and denominator of 2 15 by m = 6 yields 2 × 6 15 × 6 = 12 90 . Step 5: Now we add the values from Steps 3 and 4: 55 90 + 12 90 = 67 90 . This is in lowest terms, so we have found that 11 18 + 2 15 = 67 90 . Subtracting Rational Numbers with Unequal Denominators Calculate 14 25 − 9 70 . The denominators of the fractions are 25 and 70, so we label b = 25 and d = 70 . Step 1: Find LCM(25,70). This is 350. Step 2: Calculate n and m : n = 350 25 = 14 and m = 350 70 = 5 . Step 3: Multiplying the numerator and denominator of 14 25 by n = 14 yields 14 × 14 25 × 14 = 196 350 . Step 4: Multiplying the numerator and denominator of 9 70 by m = 5 yields 9 × 5 70 × 5 = 45 350 . Step 5: Now we subtract the value from Step 4 from the value in Step 3: 196 350 − 45 350 = 151 350 . This is in lowest terms, so we have found that 14 25 − 9 70 = 151 350 . Adding and Subtracting Fractions with Different Denominators Converting Between Improper Fractions and Mixed Numbers One way to visualize a fraction is as parts of a whole, as in 5 12 of a pizza. But when the numerator is larger than the denominator, as in 23 12 , then the idea of parts of a whole seems not to make sense. Such a fraction is an improper fraction. That kind of fraction could be written as an integer plus a fraction, which is a mixed number . The fraction 23 12 rewritten as a mixed number would be 1 11 12 . Arithmetically, 1 11 12 is equivalent to 1 + 11 12 , which is read as “one and 11 twelfths.” Improper fractions can be rewritten as mixed numbers using division and remainders. To find the mixed number representation of an improper fraction, divide the numerator by the denominator. The quotient is the integer part, and the remainder becomes the numerator of the remaining fraction. Rewriting an Improper Fraction as a Mixed Number Rewrite 48 13 as a mixed number. When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite 48 13 as 3 9 13 . Converting an Improper Fraction to a Mixed Number Using Desmos Similarly, we can convert a mixed number into an improper fraction. To do so, first convert the whole number part to a fraction by writing the whole number as itself divided by 1, and then add the two fractions. Alternately, we can multiply the whole number part and the denominator of the fractional part. Next, add that product to the numerator. Finally, express the number as that product divided by the denominator. Rewriting a Mixed Number as an Improper Fraction Rewrite 5 4 9 as an improper fraction. Step 1: Multiply the integer part, 5, by the denominator, 9, which gives 5 × 9 = 45 . Step 2: Add that product to the numerator, which gives 45 + 4 = 49 . Step 3: Write the number as the sum, 49, divided by the denominator, 9, which gives 49 9 . Using Desmos to Rewrite a Mixed Number as an Improper Fraction Desmos can be used to convert from a mixed number to an improper fraction. To do so, we use the idea that a mixed number, such as 5 6 11 , is another way to represent 5 + 6 11 . If 5 + 6 11 is entered in Desmos, the result is the decimal form of the number. However, clicking the fraction button to the left will convert the decimal to an improper fraction, 61 11 . As an added bonus, Desmos will automatically reduce the fraction to lowest terms. Converting Rational Numbers Between Decimal and Fraction Forms Understanding what decimals represent is needed before addressing conversions between the fractional form of a number and its decimal form , or writing a number in decimal notation . The decimal number 4.557 is equal to 4 557 1,000 . The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were three decimal digits in 4.557, we divided 557 by 10 3 = 1000 . Decimal representations may be very long. It is convenient to round off the decimal form of the number to a certain number of decimal digits. To round off the decimal form of a number to n (decimal) digits, examine the ( n + 1 )st decimal digit. If that digit is 0, 1, 2, 3, or 4, the number is rounded off by writing the number to the n th decimal digit and no further. If the ( n + 1 )st decimal digit is 5, 6, 7, 8, or 9, the number is rounded off by writing the number to the n th digit, then replacing the n th digit by one more than the n th digit. Rounding Off a Number in Decimal Form to Three Digits Round 5.67849 to three decimal digits. The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. So, 5.67849 rounded off to three decimal places is 5.678. Rounding Off a Number in Decimal Form to Four Digits Round 45.11475 to four decimal digits. The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. So, 45.11475 rounded off to four decimal places is 45.1148. To convert a rational number in fraction form to decimal form, use your calculator to perform the division. Converting a Rational Number in Fraction Form into Decimal Form Convert 47 25 into decimal form. Using a calculator to divide 47 by 25, the result is 1.88. Converting a terminating decimal to the fractional form may be done in the following way: Step 1: Count the number of digits in the decimal part of the number, labeled n . Step 2: Raise 10 to the n th power. Step 3: Rewrite the number without the decimal. Step 4: The fractional form is the number from Step 3 divided by the result from Step 2. This process works due to what decimals represent and how we work with mixed numbers. For example, we could convert the number 7.4536 to fractional from. The decimal part of the number, the .4536 part of 7.4536, has four digits. By the definition of decimal notation, the decimal portion represents 4,536 10 4 = 4,536 10,000 . The decimal number 7.4536 is equal to the improper fraction 7 4,536 10,000 . Adding those to fractions yields 74,536 10,000 . Converting from Decimal Form to Fraction Form with Terminating Decimals Convert 3.2117 to fraction form. Step 1: There are four digits after the decimal point, so n = 4 . Step 2: Raise 10 to the fourth power, 10 4 = 10,000 . Step 3: When we remove the decimal point, we have 32,117. Step 4: The fraction has as its numerator the result from Step 3 and as its denominator the result of Step 2, which is the fraction 32,117 10,000 . The process is different when converting from the decimal form of a rational number into fraction form when the decimal form is a repeating decimal. This process is not covered in this text. Multiplying and Dividing Rational Numbers Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product. Symbolically, a b × c d = a × c b × d . As always, rational numbers should be reduced to lowest terms. If b and d are non-zero integers, then a b × c d = a × c b × d . Multiplying Rational Numbers Calculate 12 25 × 10 21 . Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator. 12 25 × 10 21 = 12 × 10 25 × 21 = 120 525 This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15. 120 525 = 15 × 8 15 × 35 = 8 35 Multiplying Fractions As with multiplication, division of rational numbers can be done using a calculator. Dividing Decimals with a Calculator Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary. Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474. Before discussing division of fractions without a calculator, we should look at the reciprocal of a number. The reciprocal of a number is 1 divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. For the fraction a b , the reciprocal is b a . An important feature for a number and its reciprocal is that their product is 1. When dividing two fractions by hand, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication. Symbolically, b a ÷ c d = a b × d c = a × d b × c . As before, reduce to lowest terms. If b , c and d are non-zero integers, then b a ÷ c d = a b × d c = a × d b × c . Dividing Rational Numbers Calculate 4 21 ÷ 6 35 . Calculate 1 8 ÷ 5 28 . Step 1: Find the reciprocal of the number being divided by 6 35 . The reciprocal of that is 35 6 . Step 2: Multiply the first fraction by that reciprocal. 4 21 ÷ 6 35 = 4 21 × 35 6 = 140 126 The answer, 140 126 is not in lowest terms. The GCD of 140 and 126 is 14. Factoring and canceling gives 140 126 = 14 × 10 14 × 9 = 10 9 . Step 1: Find the reciprocal of the number being divided by, which is 5 28 . The reciprocal of that is 28 5 . Step 2: Multiply the first fraction by that reciprocal: 1 8 ÷ 5 28 = 1 8 × 28 5 = 28 40 The answer, 28 40 , is not in lowest reduced form. The GCD of 28 and 40 is 4. Factoring and canceling gives 28 40 = 4 × 7 4 × 10 = 7 10 . Dividing Fractions Applying the Order of Operations to Simplify Expressions The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations . The order of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym PEMDAS: P Parentheses E Exponents M/D Multiplication and division A/S Addition and subtraction The first step in calculating using the order of operations is to perform operations inside the parentheses. Moving down the list, next perform all exponent operations moving from left to right. Next (left to right once more), perform all multiplications and divisions. Finally, perform the additions and subtractions. Applying the Order of Operations with Rational Numbers Correctly apply the rules for the order of operations to accurately compute ( 5 7 − 2 7 ) × 2 3 . Step 1: To calculate this, perform all calculations within the parentheses before other operations. ( 5 7 − 2 7 ) × 2 3 = ( 3 7 ) × 2 3 Step 2: Since all parentheses have been cleared, we move left to right, and compute all the exponents next. ( 3 7 ) × 2 3 = ( 3 7 ) × 8 Step 3: Now, perform all multiplications and divisions, moving left to right. ( 3 7 ) × 8 = 24 7 Applying the Order of Operations with Rational Numbers Correctly apply the rules for the order of operations to accurately compute 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 2 3 + 5 ) ) 2 . To calculate this, perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. Step 1: The innermost parentheses contain 2 3 + 5 . Calculate that first, dividing after finding the common denominator. 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 2 3 + 5 ) ) 2 = 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 2 3 + 5 1 ) ) 2 = 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 2 3 + 15 3 ) ) 2 = 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 17 3 ) ) 2 Step 2: Calculate the exponent in the parentheses, ( 5 9 ) 2 . 4 + 2 3 ÷ ( ( 5 9 ) 2 − ( 17 3 ) ) 2 = 4 + 2 3 ÷ ( ( 25 81 ) − ( 17 3 ) ) 2 Step 3: Subtract inside the parentheses is done, using a common denominator. 4 + 2 3 ÷ ( ( 25 81 ) − ( 17 3 ) ) 2 4 + 2 3 ÷ ( ( 25 81 ) − ( 17 × 27 3 × 27 ) ) 2 4 + 2 3 ÷ ( ( 25 81 ) − ( 459 81 ) ) 2 4 + 2 3 ÷ ( ( − 434 81 ) ) 2 Step 4: At this point, evaluate the exponent and divide. 4 + 2 3 ÷ ( ( − 434 81 ) ) 2 4 + 2 3 ÷ ( 188,356 6,561 ) = 4 + 2 3 × ( 6,561 188,356 ) = 4 + 2,187 94,178 Step 5: Add. 4 + 2,187 94,178 = 378,899 94,178 Had this been done on a calculator, the decimal form of the answer would be 4.0232 (rounded to four decimal places). Order of Operations Using Fractions Applying the Density Property of Rational Numbers Between any two rational numbers, there is another rational number. This is called the density property of the rational numbers. Finding a rational number between any two rational numbers is very straightforward. Step 1: Add the two rational numbers. Step 2: Divide that result by 2. The result is always a rational number. This follows what we know about rational numbers. If two fractions are added, then the result is a fraction. Also, when a fraction is divided by a fraction (and 2 is a fraction), then we get another fraction. This two-step process will give a rational number, provided the first two numbers were rational. Applying the Density Property of Rational Numbers Demonstrate the density property of rational numbers by finding a rational number between 4 11 and 7 12 . To find a rational number between 4 11 and 7 12 : Step 1: Add the fractions. 4 11 + 7 12 = 4 × 12 11 × 12 + 7 × 11 12 × 11 = 48 132 + 77 132 = 125 132 Step 2: Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is 1 2 , as seen below. 125 132 ÷ 2 = 125 132 × 1 2 = 125 264 So, one rational number between 4 11 and 7 12 is 125 264 . We could check that the number we found is between the other two by finding the decimal representation of the numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and 0.5833333…. Here it is clear that 125 264 is between 4 11 and 7 12 . Solving Problems Involving Rational Numbers Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines. Mixing Soil for Vegetables James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that 2 5 of the soil can be topsoil, but 2 5 needs to be peat moss and 1 5 has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use? In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction required of the component. Step 1: The required fraction of topsoil is 2 5 , so James needs 60 × 2 5 cubic feet of topsoil. Performing the multiplication, James needs 60 × 2 5 = 120 5 = 24 (found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of topsoil. Step 2: The required fraction of peat moss is also 2 5 , so he also needs 60 × 2 5 cubic feet, or 60 × 2 5 = 120 5 = 24 cubic feet of peat moss. Step 3: The required fraction of compost is 1 5 . For the compost, he needs 60 × 1 5 = 60 5 = 12 cubic feet. Determining the Number of Specialty Pizzas At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas? One-third of the whole are specialty pizzas, so we need one-third of 273, which gives 1 3 × 273 = 273 3 = 91 , found by dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas. Finding a Fraction of a Total Using Fractions to Convert Between Units A common application of fractions is called unit conversion , or converting units , which is the process of changing from the units used in making a measurement to different units of measurement. For instance, 1 inch is (approximately) equal to 2.54 cm. To convert between units, the two equivalent values are made into a fraction. To convert from the first type of unit to the second type, the fraction has the second unit as the numerator, and the first unit as the denominator. From the inches and centimeters example, to change from inches to centimeters, we use the fraction 2.54 cm 1 in . If, on the other hand, we wanted to convert from centimeters to inches, we’d use the fraction 1 in 2.54 cm . This fraction is multiplied by the number of units of the type you are converting from , which means the units of the denominator are the same as the units being multiplied. Converting Liters to Gallons It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons. We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by 1 gal 0.264172 gal / 1 liter . Notice the gallon part is in the numerator since we’re converting to gallons, and the liter part is in the denominator since we are converting from liters. Performing this and rounding to three decimal places, we find that 14 liters is 14 liter × 0.264172 gal 1 liter = 3.69841 gal . Converting Centimeters to Inches It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches. We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make sure the centimeter part of the equivalence is in the denominator, 1 in 2.54 cm . To convert the 100 cm to inches, multiply 100 by 1 in 2.54 cm . Notice the inch part is in the numerator since we’re converting to inches, and the centimeter part is in the denominator since we are converting from centimeters. Performing this and rounding to three decimal places, we obtain 100 cm × 1 in 2.54 cm = 39.370 in . This means 100 cm equals 39.370 in. Converting Units Defining and Applying Percent A percent is a specific rational number and is literally per 100. n percent, denoted n %, is the fraction n 100 . Rewriting a Percentage as a Fraction Rewrite the following as fractions: 31% 93% Using the definition and n = 31 , 31% in fraction form is 31 100 . Using the definition and n = 93 , 93% in fraction form is 93 100 . Rewriting a Percentage as a Decimal Rewrite the following percentages in decimal form: 54% 83% Using the definition and n = 54 , 54% in fraction form is 54 100 . Dividing a number by 100 moves the decimal two places to the left; 54% in decimal form is then 0.54. Using the definition and n = 83 , 83% in fraction form is 83 100 . Dividing a number by 100 moves the decimal two places to the left; 83% in decimal form is then 0.83. You should notice that you can simply move the decimal two places to the left without using the fractional definition of percent. Percent is used to indicate a fraction of a total. If we want to find 30% of 90, we would perform a multiplication, with 30% written in either decimal form or fractional form. The 90 is the total , 30 is the percentage , and 27 (which is 0.30 × 90 ) is the percentage of the total . n % of x items is n 100 × x . The x is referred to as the total , the n is referred to as the percent or percentage , and the value obtained from n 100 × x is the part of the total and is also referred to as the percentage of the total . Finding a Percentage of a Total Determine 40% of 300. Determine 64% of 190. The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain 0.40 × 300 = 120 . The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain 0.64 × 190 = 121.6 . In the previous situation, we knew the total and we found the percentage of the total. It may be that we know the percentage of the total, and we know the percent, but we don't know the total. To find the total if we know the percentage of the total, use the following formula. If we know that n % of the total is x , then the total is 100 × x n . Finding the Total When the Percentage and Percentage of the Total Are Known What is the total if 28% of the total is 140? What is the total if 6% of the total is 91? 28 is the percentage, so n = 28 . 28% of the total is 140, so x = 140 . Using those we find that the total was 100 × 140 28 = 500 . 6 is the percentage, so n = 6 . 6% of the total is 91, so x = 91 . Using those we find that the total was 100 × 91 6 = 1,516.6 . The percentage can be found if the total and the percentage of the total is known. If you know the total, and the percentage of the total, first divide the part by the total. Move the decimal two places to the right and append the symbol %. The percentage may be found using the following formula. The percentage, n , of b that is a is a b × 100 % . Finding the Percentage When the Total and Percentage of the Total Are Known Find the percentage in the following: Total is 300, percentage of the total is 60. Total is 440, percentage of the total is 176. The total is 300; the percentage of the total is 60. Calculating yields 0.2. Moving the decimal two places to the right gives 20. Appending the percentage to this number results in 20%. So, 60 is 20% of 300. The total is 440; the percentage of the total is 176. Calculating yields 0.4. Moving the decimal two places to the right gives 40. Appending the percentage to this number results in 40%. So, 176 is 40% of 440. Solve Problems Using Percent In the media, in research, and in casual conversation percentages are used frequently to express proportions. Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using percentages comes down to identifying which of the three components of a percentage you are given, the total, the percentage, or the percentage of the total. If you have two of those components, you can find the third using the methods outlined previously. Percentage of Students Who Are Sleep Deprived A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night? The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were sleep deprived was 0.70 × 400 = 280 ; 280 students on the study were sleep deprived. Amazon Prime Subscribers There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers? We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, which is 328.2 million. Performing this division and rounding to three decimal places yields 126 328.2 = 0.384 . The decimal is moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents are Amazon Prime subscribers. Finding the Percentage When the Total and Percentage of the Total Are Known Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student-athletes are at the university? We need to find the total number of student-athletes at Evander’s university. Step 1: Identify what we know. We know the percentage of students who receive some sort of scholarship, 73%. We also know the number of athletes that form the part of the whole, or 219 student-athletes. Step 2: To find the total number of student-athletes, use 100 × x n , with x = 219 and n = 73 . Calculating with those values yields 100 × 219 73 = 300 . So, there are 300 total student-athletes at Evander’s university Check Your Understanding Key Terms density property of rational numbers improper fraction lowest terms mixed number rational number repeating decimal terminating decimal Key Concepts Rational numbers are fractions of integers, and can always be written as an integer divided by an integer. The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced. An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number. Using division and remainder, an improper fraction may be written as a mixed number. A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number. The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers. Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator. Multiplication and division of rational numbers can be performed without regard to common denominators. Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers. Formulas a c ± b c = a ± b c a b × c d = a × c b × d a b × c d = a × c b × d a b ÷ c d = a d × d c = a × d b × c Videos Introduction to Fractions Equivalent Fractions Reducing Fractions to Lowest Terms Using Desmos to Reduce a Fraction Adding and Subtracting Fractions with Different Denominators Converting an Improper Fraction to a Mixed Number Using Desmos Multiplying Fractions Dividing Fractions Order of Operations Using Fractions Finding a Fraction of a Total Converting Units

Math


Irrational Numbers The Pythagoreans were a philosophical sect of ancient Greece, often associated with mathematics. (credit: Fedor Andreevich Bronnikov (1827-1902) “Hymn of the Pythagoreans to the Rising Sun,” 1877, oil on canvas/Wikimedia, public domain) Learning Objectives After completing this section, you should be able to: Define and identify numbers that are irrational. Simplify irrational numbers and express in lowest terms. Add and subtract irrational numbers. Multiply and divide irrational numbers. Rationalize fractions with irrational denominators. The Pythagoreans were a philosophical sect in ancient Greece. Their philosophy included reincarnation and purifying the mind through the study and contemplation of mathematics and science. One of their principles was the cosmos is ruled by order, specifically mathematics and music. They even held mystic beliefs about specific numbers and figures. For example, the number 1 was associated with the mind and essence. Four represented justice, as it is the first product of two even numbers. Most famously, though, is the association with the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the shorter sides of the triangle (the legs) equals the square of the longer side (the hypotenuse). Even the ancient Egyptians used this relationship, as triangles with side measures 3, 4, and 5 were often used in surveying following the flooding of the Nile. There is a story of a Pythagorean, Hippasus, discovering that not all numbers could be expressed as fractions. In other words, not all numbers were rational numbers. The story ends with Hippasus, who shared this, or in some versions discovered it, put to death by drowning for sharing this fact, that not all quantities could be expressed as the ratio of two natural numbers. As colorful as that story may be, it is most likely false, as there are no contemporary sources to corroborate it. But it does seem to mark the discovery that not all quantities or measures were fractions of numbers. And so, irrational numbers were discovered. The Philosophy of the Pythagoreans Defining and Identifying Numbers That Are Irrational We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. Irrational numbers are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern. One collection of irrational numbers is square roots of numbers that aren’t perfect squares . x is the square root of the number a , denoted a , if x 2 = a . The number a is the perfect square of the integer n if a = n 2 . The rational number a b is a perfect square if both a and b are perfect squares. One method of determining if an integer is a perfect square is to examine its prime factorization. If, in that factorization, all the prime factors are raised to even powers, the integer is a perfect square. Another method is to attempt to factor the integer into an integer squared. It is possible that you recognize the number as a perfect square (such as 4 or 9). Or, if you have a calculator at hand, use the calculator to determine if the square root of the integer is an integer. Identifying Perfect Squares Determine which of the following are perfect squares. 45 81 9 28 144 400 The prime factorization of 45 is 45 = 3 2 × 5 . Since the 5 is not raised to an even power, 45 is not a perfect square. The prime factorization of 81 is 3 4 . All the prime factors are raised to even powers, so 81 is a perfect square. We must determine if both the numerator and denominator of 9 28 are perfect squares for the rational number to be a perfect square. The numerator is 9, and as mentioned above, 9 is a perfect square (it is 3 squared). Now we check the prime factorization of the denominator, 28, which is 28 = 2 2 × 7 . Since 7 is not raised to an even power, 28 is not a perfect square. Since the denominator is not a perfect square, 9 28 is not a perfect square. We must determine if both the numerator and denominator of 144 400 are perfect squares for the rational number to be a perfect square. The numerator is 144. The prime factorization of 144 is 144 = 2 4 × 3 2 . Since all the prime factors of 144 are raised to even powers, 144 is a perfect square. Now we check the prime factorization of the denominator, 400, which is 400 = 2 4 × 5 2 . Since all the prime factors of 400 are raised to even powers, 400 is a perfect square. Since the numerator and denominator of 144 400 are perfect squares, 144 400 is a perfect square. Using Desmos to Determine if a Number Is a Perfect Square Desmos may be used to determine if a number is a perfect square by using its square root function. When Desmos is opened, there is a tab in the lower left-hand corner of the Desmos screen. This tab opens the Desmos keypad, shown in . Desmos keyboard with square root key circles There you find the key for the square root, which is circled in . To find the square root of a number, click the square root key, which begins a calculation, and then enter the value for which you want a square root. If the result is an integer, then the number is a perfect square. Using Desmos to Find the Square Root of a Number Another collection of irrational numbers is based on the special number, pi , denoted by the Greek letter π , which is the ratio of the circumference of the diameter of the circle ( ). Circle with radius, diameter, and circumference labeled Any multiple or power of π is an irrational number. Any number expressed as a rational number times an irrational number is an irrational number also. When an irrational number takes that form, we call the rational number the rational part , and the irrational number the irrational part . It should be noted that a rational number plus, minus, multiplied by, or divided by any irrational number is an irrational number. Identifying Irrational Numbers Identify which of the following numbers are irrational. 35 0. 15 ¯ 121 4 π 35 can be factored as 5 × 7 , showing that 35 is not the square of an integer or a rational number. This mean its square root is an irrational number. Since 0. 15 ¯ is a decimal with a repeating pattern, it is rational, so it is not an irrational number. 121 = 11 2 . Since 121 is the square of an integer, its square root is a rational number. Since 4 π is a multiple of pi, it is irrational. In this case, the rational part of the number is 4, while the irrational part is π . Euler-Mascheroni Constant Determining if a number is rational or irrational is not trivial. There are numbers that defied such classification for quite a long time. One such is the Euler-Mascheroni constant. The Euler-Mascheroni constant is used in mathematics, and is primarily associated with the natural logarithm, which is a mathematical function. The constant has been around since around 1790. However, it was unknown if this constant was rational or irrational until 2013, at which point it was proven to be irrational. Simplifying Square Roots and Expressing Them in Lowest Terms To simplify a square root means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root. The number inside the square root symbol is referred to as the radicand . So in the expression a the number a is referred to as the radicand. Before discussing how to simplify a square root, we need to introduce a rule about square roots. The square root of a product of numbers equals the product of the square roots of those number. Written symbolically, a × b = a × b . For any two numbers a and b , a × b = a × b . Using this formula, we can factor an integer inside a square root into a perfect square times another integer. Then the square root can be applied to the perfect square, leaving an integer times the square root of another integer. If the number remaining under the square root has no perfect square factors, then we’ve simplified the irrational number into lowest terms. To simplify the irrational number into lowest terms when n is an integer: Step 1: Determine the largest perfect square factor of n , which we denote a 2 . Step 2: Factor n into a 2 × b . Step 3: Apply a 2 × b = a 2 × b . Step 4: Write n in its simplified form, a b . When a square root has been simplified in this manner, a is referred to as the rational part of the number, and b is referred to as the irrational part. Simplifying a Square Root Simplify the irrational number 180 and express in lowest terms. Identify the rational and irrational parts. Begin by finding the largest perfect square that is a factor of 180. We can do this by writing out the factor pairs of 180: 1 × 180 2 × 90 3 × 60 4 × 45 5 × 36 6 × 30 9 × 20 10 × 18 12 × 15 Looking at the list of factors, the perfect squares are 4, 9, and 36. The largest is 36, so we factor the into 36 × 5 = 6 2 × 5 . In the formula, a = 6 and b = 5 . Apply a 2 × b = a 2 × b . 6 2 × 5 = 6 2 × 5 The simplified form of 180 is 6 5 . In this example, the 6 is the rational part, and the 5 is the irrational part. Simplifying Square Roots Simplifying a Square Root Simplify the irrational number 330 and express in lowest terms. Identify the rational and irrational parts. Begin by finding the largest perfect square that is a factor of 330. We can do this by writing out the factor pairs of 330: 1 × 330 2 × 165 3 × 110 5 × 66 6 × 55 10 × 33 11 × 30 15 × 22 Looking at the list of factors, there are no perfect squares other than 1, which means 330 is already expressed in lowest terms. In this case, 1 is the rational part, and 330 is the irrational part. Though we could write this as 1 330 , but the product of 1 and any other number is just the number. Simplifying a Square Root Simplify the irrational number 2,548 and express in lowest terms. Identify the rational and irrational parts. Begin by finding the largest perfect square that is a factor of 2,548. We can do this by writing out the factor pairs of 2,548: 1 × 2548 2 × 1274 4 × 637 7 × 364 13 × 196 14 × 182 26 × 98 28 × 91 49 × 52 Looking at the list of factors, the perfect squares are 4, 49, and 196. The largest is 196, so we factor the 2,548 into 196 × 13 = 14 2 × 13 . In the formula, a = 14 and b = 5 . Apply a 2 × b = a 2 × b . 14 2 × 13 = 14 2 × 13 The simplified form of 2,548 is 14 13 . In this example, 14 is the rational part, and 13 is the irrational part. Simplifying Square Roots Adding and Subtracting Irrational Numbers Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a calculator, enter the operation and a decimal representation will be given. However, there are times when two irrational numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the irrational numbers are the same. To add or subtract two irrational numbers that have the same irrational part, add or subtract the rational parts of the numbers, and then multiply that by the common irrational part. Let our first irrational number be a × x , where a is the rational and x the irrational parts. Let the other irrational number be b × x , where b is the rational and x the irrational parts. Then a × x ± b × x = ( a ± b ) × x . Subtracting Irrational Numbers with Similar Irrational Parts If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. 3 7 − 8 7 Since these two irrational numbers have the same irrational part, 7 , we can subtract without using a calculator. The rational part of the first number is 3. The rational part of the second number is 8. Using the formula yields 3 7 − 8 7 = ( 3 − 8 ) × 7 = − 5 7 . Adding Irrational Numbers with Similar Irrational Parts If possible, add the following irrational numbers without using a calculator. If this is not possible, state why. 35 π + 17 π Since these two irrational numbers have the same irrational part, π , the addition can be performed without using a calculator. The rational part of the first number is 35. The rational part of the second number is 17. Using the formula yields 35 π + 17 π = ( 35 + 17 ) × π = 52 π . Subtracting Irrational Numbers with Different Irrational Parts If possible, subtract the following irrational numbers without using a calculator. If this is not possible, state why. 19 3 − 5.6 7 The two numbers being subtracted do not have the same irrational part, so the operation cannot be performed. Multiplying and Dividing Irrational Numbers Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain the form of the irrational number as a rational part times an irrational part. The process is similar to adding and subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to lowest terms. When multiplying two square roots, use the following formula. It is the same formula presented during the discussion of simplifying square roots. For any two positive numbers a and b , a × b = a × b . When dividing two square roots, use the following formula. For any two positive numbers a and b , with b not equal to 0, a ÷ b = a b = a b . To multiply or divide irrational numbers with similar irrational parts, do the following: Step 1: Multiply or divide the rational parts. Step 2: If necessary, reduce the result of Step 1 to lowest terms. This becomes the rational part of the answer. Step 3: Multiply or divide the irrational parts. Step 4: If necessary, reduce the result from Step 3 to lowest terms. This becomes the irrational part of the answer. Step 5: The result is the product of the rational and irrational parts. Dividing Irrational Numbers with Similar Irrational Parts Perform the following operations without a calculator. Simplify if possible. 3 15 ÷ ( 8 3 ) 14.7 135 ÷ ( 3 5 ) . In this division problem, 3 15 ÷ ( 8 3 ) , notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above. Step 1: Divide the rational parts. 3 ÷ 8 = 3 8 Step 2: If necessary, reduce the result of Step 1 to lowest terms. The 3 and 8 have no common factors, so 3 8 is already in lowest terms. Step 3: Divide the irrational parts. 15 ÷ 3 = 15 3 = 15 3 Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields 5 . Step 5: The result is the product of the rational and irrational parts, which is 3 8 5 . In this division problem, 14.7 135 ÷ ( 3 5 ) , notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above. Step 1: Divide the rational parts. 14.7 ÷ 3 = 4.9 Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal so will not be reduced. Step 3: Divide the irrational parts. 135 ÷ 5 = 135 5 = 135 5 Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand can be reduced, which yields 135 5 = 27 = 9 × 3 = 3 3 . Step 5: The result is the product of the rational and irrational parts, which is 4.9 × 3 3 = 14.7 3 . Multiplying Irrational Numbers with Similar Irrational Parts Perform the following operations without a calculator. Simplify if possible. ( 19 3 ) × ( 5.6 12 ) 13 π × 8 π In this multiplication problem, ( 19 3 ) × ( 5.6 12 ) , notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above. Step 1: Multiply the rational parts. 19 × 5.6 = 106.4 Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced. Step 3: Multiply the irrational parts. 3 × 12 = 3 × 12 = 36 Step 4: If necessary, reduce the result from Step 3 to lowest terms. The radicand is 36, which is the square of 6. The irrational part reduces to 36 = 6 . Step 5: The result is the product of the rational and irrational parts, which is 106.4 × 6 = 638.4 . Notice that sometimes multiplying or dividing irrational numbers can result in a rational number. In this multiplication problem, 13 π × 8 π , notice that the irrational parts of these numbers are the same, π . Follow the process above. Step 1: Multiply the rational parts. 13 × 8 = 104 Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer. Step 3: Multiply the irrational parts. π × π = π 2 Step 4: If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced. Step 5: The result is the product of the rational and irrational parts, which is 104 π 2 . Rationalizing Fractions with Irrational Denominators Fractions often represent that some amount is being equally divided into some number of parts. But to conceptualize a fraction in that manner, the denominator needs to be an integer. An irrational number in the denominator interferes with that interpretation of a fraction. Fractions that have denominators that are just the square root of an integer can be altered into fractions with integer denominators using a process called rationalizing the denominator . The process relies on the following property of square roots: a × a = a and the following property of fractions: a b = a c b c for any non-zero number c . Using these two properties, when a fraction has a square root in the denominator, we can eliminate that square root. Multiply the numerator and denominator by that square root from the denominator, a b = a b b × b . Then apply a × a = a to the denominator, yielding a b b × b = a b b . Notice that there is no longer a square root in the denominator, which allows for interpreting the fraction as dividing a whole into equal parts. Rationalizing the Denominator Rationalizing the Denominator Rationalize the denominator of the following: 5 7 3 6 2 10 The square root in the denominator is 7 . In order to rationalize the denominator of 5 7 , we need to multiply the numerator and denominator by 7 and simplify. 5 7 = 5 7 7 × 7 = 5 7 7 The square root is in simplified form, so the final answer is 5 7 7 . The square root in the denominator is 10 . Step 1: In order to rationalize the denominator of 3 6 2 10 , we need to multiply the numerator and denominator by 10 and simplify. 3 6 2 10 = 3 6 × 10 2 10 × 10 = 3 60 2 × 10 = 3 60 20 Step 2: The 60 under the square root can be factored into the following factor pairs: 1 × 60 2 × 30 3 × 20 4 × 15 5 × 12 6 × 10 Step 3: The largest square factor of 60 is 4, so we simplify the 60 in the numerator into 2 15 . We also cancel any common factors. 3 60 20 = 3 × 2 15 20 = 6 15 20 = 3 15 10 This is completely simplified. There are occasions when the denominator is irrational but is the sum of two numbers where one or both involve square roots. For instance, 5 4 + 3 . The process used earlier required that the denominator was the square root of a number and would not work here. However, this type of denominator can be rationalized. In order to rationalize such a denominator, we will multiply the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate of a + b is a – b . We say that a + b and a ‒ b are conjugate numbers . So, the conjugate of − 3 + 10 is just − 3 − 10 . But why is this of interest? The reason is because it leads to the difference of squares formula, which is used to factor the difference of two squares. Or, for our purposes, in reverse it allows us to eliminate a square root. For any two numbers, a and b , a 2 − b 2 = ( a − b ) ( a + b ) . Looking at that formula, you should see that the two factors on the right-hand side of the equals sign are conjugates of one another. So, for our purposes, we’re interested in ( a − b ) ( a + b ) = a 2 − b 2 . This tells us that when we multiply a + b by its conjugate, we get a squared minus b squared, or a 2 − b 2 . But how is this useful? Let’s return to the fraction above, 5 4 + 3 . The denominator is 4 + 3 . Its conjugate is 4 − 3 . According to the formula, and letting a = 4 and b = 3 , we see that ( 4 + 3 ) ( 4 − 3 ) = 4 2 − ( 3 ) 2 . But ( 3 ) 2 is just 3. That means the product is 16 − 3 or 13. This no longer has a square root. We use this to rationalize the denominator. We will also need the distributive property of numbers. For any three numbers a , b , and c , a × ( b ± c ) = a × b ± a × c . This is called the distributive property. Rationalizing the Denominator Using Conjugates Rationalize the denominator of 4 6 + 10 . Step 1: We recognize that the denominator is the sum of two numbers where one or both involve square roots. This means the conjugate can be used to remove the square root from the denominator. Step 2: To do so, we multiply the numerator and the denominator each by the conjugate of the denominator. Since the denominator is 6 + 10 , the conjugate we will use is 6 − 10 . Step 3: The conjugate is multiplied by the numerator and the denominator. 4 6 + 10 × 6 − 10 6 − 10 Step 4: Remembering how a number times its conjugate works, this becomes 4 6 + 10 × 6 − 10 6 − 10 = 4 × ( 6 − 10 ) 6 2 − ( 10 ) 2 . Step 5: In the numerator, we apply the distributive property. Using it yields 4 × ( 6 − 10 ) 6 2 − ( 10 ) 2 = 24 − 4 10 36 − 10 = 24 − 4 10 26 . Step 6: Notice that the denominator no longer contains a square root. It has been rationalized. If desired, this can then be written as a rational number minus an irrational number, by recalling that a − b c = a c − b c . Applying that to the answer, we have 4 6 + 10 = 24 − 4 10 26 = 24 26 − 4 10 26 . Step 7: With a bit of cancellation, this reduces to 4 6 + 10 = 24 26 − 4 10 26 = 12 13 − 2 10 13 . Rationalizing the Denominator Check Your Understanding Key Terms conjugate numbers difference of squares irrational numbers lowest terms rationalize the denominator Key Concepts Irrational numbers are numbers that cannot be written as an integer divided by another integer. One example is pi, denoted π . Another collection of irrational numbers are natural numbers that are not perfect squares. Some irrational numbers can be written as a rational part multiplied by an irrational part. If two irrational numbers have the same irrational parts, they can be added or subtracted. When irrational numbers are similar, on can multiply and divide the numbers without a calculator. Since a × b = a × b , and a ÷ b = a b = a b , products and quotients of square roots can be determined. Because a 2 = a and a × b = a × b , it is possible to simplify square root expressions so the radicand contains no perfect square factors. When a fraction has an irrational number as its denominator, it is possible to convert the denominator into a rational number using its conjugate. Doing so involves multiplying the numerator and denominator by the conjugate of the denominator, and then applying the difference of squares formula. With a single square root term Using conjugate numbers for two term denominators Formulas a × b = a × b a × x ± b × x = ( a ± b ) × x a ÷ b = a b = a b a 2 − b 2 = ( a − b ) ( a + b ) Videos The Philosophy of the Pythagoreans Using Desmos to Find the Square Root of a Number Simplifying Square Roots Rationalizing the Denominator

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Real Numbers Quick mental math involves using the known properties of real numbers. Learning Objectives After completing this section, you should be able to: Define and identify numbers that are real numbers. Identify subsets of the real numbers. Recognize properties of real numbers. Have you ever been impressed by the speed at which someone can do math in their head? Most of us at one time or another have witnessed a person speed through mental math, an impressive feat that often bests calculators. One such person is Neelkantha Bhanu Prakash. As of September 20, 2020, he is considered the world’s fastest human calculator. He currently holds four world records. How does someone do that, though? Have they memorized lots of arithmetic facts? Are they simply brilliant? The answer isn't simple so much as it is about knowledge. Real numbers behave in some very regular ways, following rules that can be learned. In this section, those rules are explored. Watch the video of Arthur Benjamin’s TED Talk to learn about another mathematician with remarkable mental abilities. Arthur Benjamin TED Talk, Faster Than a Calculator Defining and Identifying Real Numbers Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is 3 8 units. Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other side length of 3, the long side of the triangle will have length 10 units, as shown in . Right triangle Of course, if we name something the real numbers, there must be numbers that aren't real. Otherwise, they’d just be called the numbers. One such not real number, one that cannot be a length, is − 1 . It is part of a collection of numbers called the complex numbers, it is denoted with the letter i . As an extension, the square root of any negative number is not a real number, but instead a complex number. To determine if a number is real, check to see if there are any negatives under a square root or any i ' s . If there are any present, the number is not real. Identifying Real Numbers Determine if each of the following are real numbers: 4 3 7 13.3381 17 − 8 4 3 7 is a real number, as there are no negatives under the square roots, nor is there any factor of i . 13.3381 is a rational number, and so it is a real number. 17 − 8 is not a real number, as there is a negative number under the square root. Identifying Subsets of Real Numbers The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real numbers have named subsets, as shown in the table below. Set Name Set Symbol Set Description Natural Numbers ℕ The counting numbers Whole Numbers The counting numbers and 0 Integers ℤ The natural numbers, their negatives, and 0 Rational Numbers ℚ Fractions of integers Irrational Numbers ℙ Numbers that cannot be written as a fraction of integers Real Numbers ℝ The union of the rational and irrational numbers, all possible physical lengths, and their negatives When we categorize numbers using these sets, we use the smallest set that they belong to. For instance, −7 is an integer, and a rational number, and a real number. The smallest set to which −7 belongs is integer, so we’d say it belongs to the integers. We can also represent the relationships between the different sets of real numbers using set notation. All natural numbers are integers, but there are integers that are not natural numbers, so ℕ ⊂ ℤ . Similarly, every integer is a rational number, but there are rational numbers that are not integers, so ℤ ⊂ ℚ . The same is true of the rational numbers and the real numbers, so ℚ ⊂ ℝ . There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set A , we should note that the following are true: ℚ ∪ A = ℝ and ℚ ∩ A = ∅ . Recall that this means the irrationals are the complement of the rational numbers in the universal set of real numbers. Categorizing Numbers Identify all subsets of the real numbers to which the following real numbers belong: 14 − 14.223 17 14 is a natural number, integer, and rational number. − 14.223 is a rational number. 17 is an irrational number. Categorizing Numbers within a Venn Diagram Place the following numbers correctly in the Venn diagram ( ). − 4 2 − 10 37 150 41 1 20 4 π Since − 4 2 is irrational, it belongs in the real numbers, but outside the rational numbers ( ). Since −10 is an integer, it belongs in the integers but outside the natural numbers ( ). Since 37 150 is a rational number, it belongs in the rational numbers but not in the integers ( ). Since 41 is a natural number, it belongs in the natural numbers circle ( ). Since 1 20 is a rational number, it belongs in the rational numbers but not in the integers ( ). Since 4 π is irrational, it belongs in the real numbers, but outside the rational numbers ( ). Identifying Sets of Real Numbers Recognizing Properties of Real Numbers The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers . Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying 4 × 13 × 25 , you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier. The table below is a partial list of properties of real numbers. Property Example In Words Distributive property a × ( b + c ) = a × b + a × c 5 × ( 3 + 4 ) = 5 × 3 + 5 × 4 Multiplication distributes across addition Commutative property of addition a + b = b + a 3 + 7 = 7 + 3 Numbers can be added in any order Commutative property of multiplication a × b = b × a 10 × 4 = 4 × 10 Numbers can be multiplied in any order Associative property of addition a + ( b + c ) = ( a + b ) + c 4 + ( 3 + 8 ) = ( 4 + 3 ) + 8 Doesn't matter which pair of numbers is added first Associative property of multiplication a × ( b × c ) = ( a × b ) × c 2 × ( 5 × 7 ) = ( 2 × 5 ) × 7 Doesn't matter which pair of numbers is multiplied first Additive identity property a + 0 = a 17 + 0 = 17 Any number plus 0 is the number Multiplicative identity property a × 1 = a 21 × 1 = 21 Any number times one is the number Additive inverse property a + ( − a ) = 0 14 + ( − 14 ) = 0 Every number plus its negative is 0 Multiplicative inverse property a × ( 1 a ) = 1 , provided a ≠ 0 3 × ( 1 3 ) = 1 Every non-zero number times its reciprocal is 1 The names of the properties are suggestive. The commutative properties , for example, suggest commuting, or moving. Associative properties suggest which items are associated with others, or if order matters in the computation. The distributive property addresses how a number is distributed across parentheses. Identifying Properties of Real Numbers In each of the following, identify which property of the real numbers is being applied. 4 + ( 8 + 13 ) = ( 4 + 8 ) + 13 34 × ( 1 34 ) = 1 14 + 27 = 27 + 14 Here, the pair of numbers that is added first is switched. This is the associative property of addition. Here, a number is multiplied by its reciprocal, resulting in 1. This is the multiplicative inverse property. Here, the order in which numbers are added is switched. This is the commutative property of addition. Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of 10, or multiply to a multiple of 10 or 100. Using Properties of Real Numbers in Calculations Use properties of the real numbers and mental math to calculate the following: 2 × 13 × 50 13 + 84 + 27 9 × 16 × 11 Notice that 2 × 50 = 100 , so that becomes the multiplication to do first. Use the commutative property of multiplication to change the order of the numbers being multiplied. 2 × 13 × 50 = 2 × 50 × 13 = 100 × 13 = 1,300 Notice that 13 + 27 = 40 , so that becomes the addition to do first. Use the commutative property of addition to change the order in which the numbers are added. 13 + 84 + 27 = 13 + 27 + 84 = 40 + 84 = 124 Notice that 9 × 11 = 99 . Using that, the problem can be changed to 99 × 16 . That, however, doesn't look easy at all. But 99 = ( 100 − 1 ) . Using the distributive property, we rewrite and expand this as 99 × 16 = ( 100 − 1 ) × 16 = 100 × 16 − 1 × 16 = 1,600 − 16 . The last step is subtraction, so the final answer is 1,584. So, multiplying by 99 is the same as multiplying by 100, and then subtracting the other number once. Videos Properties of the Real Numbers 1 Properties of the Real Numbers 2 Properties of the Real Numbers 3 Check Your Understanding Key Terms complex number imaginary number real number Key Concepts Real numbers is the collection of all rational and irrational numbers. Conceptually, it is the collection of all values that can be represented on a number line, or, as a length along with sign. The subsets of the real numbers include the natural numbers, integers, rational numbers and irrational numbers. The natural numbers are a subset of the integers, which is a subset of the rational numbers. The rational and irrational numbers are disjoint sets. The real numbers, due to order of operation rules and that performing arithmetic operations on real number always results in a real number, have arithmetic properties that apply in all cases. There include the distributive property, the commutative property, and the associative property. Also, every real number has an additive inverse and, except for zero (0), have a multiplicative inverse. Formulas a × ( b + c ) = a × b + a × c a + b = b + a a × b = b × a a + ( b + c ) = ( a + b ) + c a × ( b × c ) = ( a × b ) × c a + 0 = a a × 1 = a a + ( − a ) = 0 a × ( 1 a ) = 1 Videos Arthur Benjamin TED talk, Faster than a Calculator Identifying Sets of Real Numbers Properties of the Real Numbers #1 Properties of the Real Numbers #2 Properties of the Real Numbers #3

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Clock Arithmetic If a credit card number is entered incorrectly, error checking algorithms will often catch the mistake. (credit: modification of work “Senior couple at home checking finance on credit card from above” by Nenad Stojkovic/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Add, subtract, and multiply using clock arithmetic. Apply clock arithmetic to calculate real-world applications. Online shopping requires you to enter your credit card number, which is then sent electronically to the vendor. Using an ATM involves sliding your bank card into a reader, which then reads, sends, and verifies your card. Swiping or tapping for a purchase in a brick–and-mortar store is how your card sends its information to the machine, which is then communicated to the store’s computer and your credit card company. This information is read, recorded, and transferred many times. Each instance provides one more opportunity for error to creep into the process, a misrecorded digit, transposed digits, or missing digits. Fortunately, these card numbers have a built-in error checking system that relies on modular arithmetic, which is often referred to as clock arithmetic. In this section, we explore clock, or modular, arithmetic. Determining the Day of the Week for Any Date in History Adding, Subtracting, and Multiplying Using Clock Arithmetic When we do arithmetic, numbers can become larger and larger. But when we work with time, specifically with clocks, the numbers cycle back on themselves. It will never be 49 o’clock. Once 12 o’clock is reached, we go back to 1 and repeat the numbers. If it's 11 AM and someone says, “See you in four hours,” you know that 11 AM plus 4 hours is 3 PM, not 15 AM (ignoring military time for now). Math worked on the clock, where numbers restart after passing 12, is called clock arithmetic . Clock arithmetic hinges on the number 12. Each cycle of 12 hours returns to the original time ( ). Imagine going around the clock one full time. Twelve hours pass, but the time is the same. So, if it is 3:00, 14 hours later and two hours later both read the same on the clock, 5:00. Adding 14 hours and adding 2 hours are identical. As is adding 26 hours. And adding 38 hours. Clock showing 3:00 with arrow going around the clock one full time, or 12 hours What do 2, 14, 26, and 38 have in common in relation to 12? When they are divided by 12, they each have a remainder of 2. That's the key. When you add a number of hours to a specific time on the clock, first divide the number of hours being added by 12 and determine the remainder. Add that remainder to the time on the clock to know what time it will be. A good visualization is to wrap a number line around the clock, with the 0 at the starting time. Then each time 12 on the number line passes, the number line passes the starting spot on the clock. This is referred to as modulo 12 arithmetic. Even though the process says to divide the number being added by 12, first perform the addition; the result will be the same if you add the numbers first, and then divide by 12 and determine the remainder. In general terms, let n be a positive integer. Then n modulo 12, written ( n mod 12), is the remainder when n is divided by 12. If that remainder is x , we would write n = x (mod 12). Caution: 12 mod 12 is 0. So, if a mod 12 problem ends at 0, that would be 12 on the clock. Determining the Value of a Number modulo 12 Find the value of the following numbers modulo 12: 34 539 156 To determine the value of a number modulo 12, divide the number by 12 and record the remainder. To find the value 34 modulo 12: Step 1: Determine the remainder when 34 is divided by 12 using long division. The largest multiple of 12 that is less than or equal to 34 is 24, which is the product of 12 and 2. 12 34 2 24 _ Step 2: Performing the subtraction yields 10. 12 34 2 24 _ 10 Since that subtraction resulted in a number less than 12, that is the remainder, 10. The value of 34 modulo 12 is 10, or 34 = 10 (mod 12). To find the value 539 modulo 12: Step 1: Determine the remainder when 539 is divided by 12 using long division. We first look to the first two digits of 539, 53. The largest multiple of 12 that is less than or equal to 53 is 48, which is the product of 12 and 4. 12 539 4 48 _ Step 2: Performing the subtraction results in 5. 12 539 4 48 _ 5 Step 3: Now, the 9 is brought down. 12 539 4 48 _ 59 Step 4: The largest multiple of 12 that is less than or equal to 59 is once more 48 itself, which is 12 × 4 . 12 539 44 48 _ 59 48 _ Step 5: Finishing the process, the 48 is subtracted from the 59, yielding 11. 12 539 44 48 _ 59 48 _ 11 We've used all the digits of 539, and the last subtraction resulted in a number less than 12, so that number, 11, is the remainder. The value of 539 modulo 12 is 11, or, 539 = 11 (mod 12). To find the value 156 modulo 12: Step 1: Determine the remainder when 156 is divided by 12 using long division. We first look to the first two digits of 156, 15. The largest multiple of 12 that is less than or equal to 15 is 12 itself, which is the product of 12 and 1. 12 156 1 12 _ Step 2: Performing the subtraction results in 3. 12 156 1 12 _ 3 Step 3: Now, the 6 is brought down. 12 156 1 12 _ 36 Step 4: The largest multiple of 12 that is less than or equal to 36 is 36 itself, which is 12 × 3 . 12 156 13 12 _ 36 36 _ Step 5: Finishing the process, the 36 is subtracted from the 36, yielding 0. 12 156 13 12 _ 36 36 _ 0 We've used all the digits of 156, and the last subtraction resulted in a number less than 12, so that number, 0, is the remainder. The value of 156 modulo 12 is 0, or, 156 = 0 (mod 12). We should note here that, had we been speaking of time, the 0 would be interpreted as 12:00. Using Desmos to Determine the Value of a Number module 12 Desmos may be used to determine the value of a number modulo 12. It is flexible enough to find the value of a number modulo of any other integer you want. To determine the value of n modulo 12, type mod( n ,12) into Desmos. The result will be displayed immediately. This can be used to find 539 modulo 12, as shown in the . Display of 539 modulo 12 Clock arithmetic is modulo 12 arithmetic but applied to time. As time is divided into 12 hours that repeat a cycle, we use modulo 12 for clock arithmetic. Clock Arithmetic Adding with Clock Arithmetic If it's 3:00, what time will it be in 89 hours? To find that future time, we may determine the value of 89 (mod 12), either by long division or by using a calculator, such as Desmos. Then add the result to 3:00. Entering mod(89,12) in Desmos results in 5. Adding 5 hours, which was 89 (mod12), to 3:00 results in 8:00. Subtracting time on the clock works in much the same way as addition. Find the value of the number of hours being subtracted modulo 12, then subtract that from the original time. Subtracting with Clock Arithmetic If it is 4:00 now, what time was it 67 hours ago? To find that past time, we may determine the value of 67 (mod 12), either by long division or by using a calculator, such as Desmos. Then subtract the result to 4:00. Entering mod(67,12) in Desmos results in 7. Subtracting 7 hours from 4:00 results in ‒3:00. We know, though, that time is not represented with negative times. This value, ‒3:00, indicates three hours before 12:00, which is 9:00. So, 67 hours before 4:00 was 9:00. We see this in the . Clock showing 7 hours subtracted from 4:00 Recall that clock arithmetic was referred to as modulo 12 arithmetic. Multiplying in modulo 12 also relies on the remainder when dividing by 12. To multiply modulo 12 is just to multiply the two numbers, and then determine the remainder when divided by 12. Multiplying modulo 12 What is the product of 11 and 45 modulo 12? We begin by multiplying 11 and 45, which is 495. Next, we find 495 modulo 12, either by dividing the result by 12 to determine the remainder, or by using a calculator. Entering mod(495,12) in Desmos yields 3. Had long division been used, the remainder would be 3. So 11 × 45 = 3 modulo 12. Calculating Real-World Applications with Clock Arithmetic Applying Clock Arithmetic Suppose it is 3:00, and you decide to check your email every 5 hours. What time will it be when you check your email the ninth time? If you check your email every 5 hours nine times, that ninth check will occur 45 hours after 3:00, which is an addition of 45 hours to 3:00. So, we find 45 modulo 12, which is 9. Nine hours after 3:00 is 12:00. It will be 12:00 when you check your email the ninth time. Clock arithmetic processes can be applied to days of the week. Every 7 days the day of the week repeats, much like every 12 hours the time on the clock repeats. The only difference will be that we work with remainders after dividing by 7. In technical terms, this is referred to as modulo 7 . More generally, let n be a positive integer. Then n modulo 7, written n mod 7, is the remainder when n is divided by 7. If that value is x , we may write n = x (mod 7). Applying Clock Arithmetic to Days of the Week Your family has a cat, and no one wants to empty the litter box. However, it has to be done daily. The six of you agree to take turns, so everyone has to empty the litter box every 6 days. You empty the box on a Thursday. What day will you empty the box for the 10th time? The first time you emptied the litter box was on a Thursday. So,the 10th time you empty the litter box will be 9 times later (you've already had your first turn, so 9 turns left!). This will happen 54 (9 times 6) days later. Finding the value of 54 modulo 7, using division to determine the remainder or using a calculator to find the value of 54 modulo 7 gives the answer 5. Five days after a Thursday is Tuesday. Check Your Understanding Key Terms clock arithmetic modulo 7 modulo 12 Key Concepts Clock arithmetic uses the idea that after 12 o’clock comes 1 o’clock. For clock arithmetic, this means that every time 12 is passed in an arithmetic process, the next number is 1, not 13. To determine the clock result of an arithmetic operation, divide the final result by 12 and keep the remainder. If the remainder is 0, then the time is 12 o’clock. Clock arithmetic is technically called modulo 12 arithmetic. To perform modulo 12 arithmetic, calculate the expression, then divide the result by 12. The modulo 12 result is the remainder. Days, in our system, pass in groups of seven. To calculate in day arithmetic, modulo 7 is used. To perform modulo 7 arithmetic, calculate the expression, then divide the result by 7. The modulo 7 result is the remainder. Videos Determining the Day of the Week for Any Date in History Clock Arithmetic

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