Logical Arguments Not all logical arguments are valid, and the ongoing fight for equal rights proves that much progress has yet to be made. (credit: "The Steam Roller" by Library of Congress Prints and Photographs Division, public domain) Learning Objectives After completing this section, you should be able to: Apply the law of detachment to determine the conclusion of a pair of statements. Apply the law of denying the consequent to determine the conclusion for pairs of statements. Apply the chain rule to determine valid conclusions for pairs of true statements. The previous sections of this chapter provide the foundational skills for constructing and analyzing logical arguments. All logical arguments include a set of premises that support a claim or conclusion; but not all logical arguments are valid and sound. A logical argument is valid if its conclusion follows from the premises, and it is sound if it is valid and all of its premises are true. A false or deceptive argument is called a fallacy . Many types of fallacies are so common that they have been named. In 1936, Dale Carnegie published his first book, titled How to Win Friends and Influence People . It was marketed as training materials for the improvement of public speaking and negotiation skills, and the methods it presented are still used today. Carnegie famously said, “When dealing with people, remember you are not dealing with creatures of logic, but creatures of emotion.” People who put forth fallacious logical arguments often take advantage of our susceptibility to emotional appeals, to try to convince us that what they are saying is true. The study of logic helps us combat this weakness through recognition and learning to focus on the facts and structure of the argument. This section focuses on the two main forms that logical arguments can take. While inductive arguments attempt to draw a more general conclusion from a pattern of specific premises, deductive arguments attempt to draw specific conclusions from at least one or more general premises. Deductive arguments can be proven to be valid using Venn diagrams or truth tables. Inductive arguments generally cannot be proven to be true. They are judged as being strong or weak, but, like any opinion, whether you believe an argument is strong or weak often depends on your knowledge of the topic being discussed along with the evidence being provided in the premises. Hasty generalization is the name given to any fallacy that presents a weak inductive argument. Be careful! Premises may be true or false. If a premise is false, the claims made by the argument should be questioned. Law of Detachment The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, p → q , and the hypothesis, p , are true, then the conclusion q must also be true. The law of detachment is also called affirming the hypothesis (or antecedent) and modus ponens. Symbolically, it has the form ( ( p → q ) ∧ p ) → q . Law of Detachment Premise: p → q Premise: p Conclusion: ∴ q The ∴ is read as the word, “therefore.” Looking at the truth table for the conditional statement, the only time the conditional is true is when the hypothesis p is also true. The only place this happens is in the first row, where q is also true, confirming that the law of detachment is a valid argument. p q p → q T T T T F F F T T F F T Another way to verify that the law of detachment is a valid argument is to construct a truth table for the argument ( ( p → q ) ∧ p ) → q and verify that it is a tautology. p q p → q ( p → q ) ∧ p ( ( p → q ) ∧ p ) → q T T T T T T F F F T F T T F T F F T F T Venn diagrams may also be used to verify deductive arguments, which include conditional premises. Consider the statement p → q : “If you play guitar, then you are a musician.” The set of guitarists is a subset of the set of musicians, p ⊂ q . To verify that an argument is valid using a Venn diagram, draw the Venn diagram representing all the premises in the argument only, as shown in . Then verify if the conclusion is also represented by the Venn diagram of the premises. If it is, the argument is valid. If it is not, the argument is not valid. The set of guitarists is drawn as a subset of the set of musicians to represent the premise, p → q . The × represents the premise: p is true. This completes the drawing of the premises. Now, examine the Venn diagram to verify if the conclusion is included in the picture. The conclusion is q . Because the × is in the set p , and p is a subset of q , × is also in q ; therefore, the law of detachment is a valid argument. Remember that an argument can be valid without being true. For the argument to be proven true, it must be both valid and sound. An argument is sound if all its premises are true. Logic Part 14: Common Argument Forms like Modus Ponens and Tollens Applying the Law of Detachment to Determine a Valid Conclusion Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of detachment to determine a valid conclusion. If Leonardo da Vinci was an artist, then he painted the Mona Lisa . Leonardo da Vinci was an artist. If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan played for the Chicago Bulls. If all fish have gills, then clown fish have gills. All fish have gills. The premises are p → q : If Leonardo da Vinci was an artist, then he painted the Mona Lisa , and p : Leonardo da Vinci was an artist. This argument has the form of the law of detachment, so, the conclusion is q : Leonardo da Vinci painted the Mona Lisa . The premises follow the form of the law of detachment, so a valid conclusion would be q . The premises are p → q : If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and p : Michael Jordan played for the Chicago Bulls. The conclusion that follows from the premises is q : Michael Jordan was not a soccer player. The premises are p → q : If all fish have gills, then clown fish have gills, and p : All fish have gills. This argument has the form of the law of detachment, so the conclusion is q : clown fish have gills. Law of Denying the Consequent Another form of a valid conditional argument is called the law of denying the consequent , or modus tollens. Recall, that the conditional statement, p → q , is logically equivalent to the contrapositive, ~ q → ~ p . So, if the conditional statement is true, then the contrapositive statement is also true. By the law of detachment, if ~ q is also true, then it follows that ~ p must also be true. Symbolically, it has the form ( ( p → q ) ∧ ~ q ) → ~ p . Law of Denying the Consequent Premise: p → q Premise: ~ q Conclusion: ∴ ~ p The conditional statement can also be described as, “If antecedent, then consequent.”This is where the law of denying the consequent gets its name. To verify if the law of denying the consequent is a valid argument, construct a truth table for the argument, ( ( p → q ) ∧ ~ q ) → ~ p , and verify that it is a tautology. p q ~ p ~ q p → q ( p → q ) ∧ ~ q ( ( p → q ) ∧ ~ q ) → ~ p T T F F T F T T F F T F F T F T T F T F T F F T T T T T To verify an argument of this form using a Venn diagram, again consider the premise: p → q : “If you play guitar, then you are a musician.” We will change the second premise to ~ q . In this case, the × represents the premise, ~ q . So, it will be placed inside the universal set of all people, but outside the set of musicians, as depicted in the Venn diagram in . Because the × is also outside the set of guitarists, the statement ~ p follows from the premises and the argument is valid. Applying the Law of Denying the Consequent to Determine a Valid Conclusion Each pair of statements represents the premises in a logical argument. Based on these premises, apply the law of denying the consequent to determine a valid conclusion. If Leonardo da Vinci was an artist, then he painted the Mona Lisa . Leonardo da Vinci did not paint the Mona Lisa . If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player. Michael Jordan was a soccer player. If all fish have gills, then clown fish have gills. Clown fish do not have gills. The premises are p → q : If Leonardo da Vinci was an artist, then he painted the Mona Lisa , and ~ q : Leonardo da Vinci did not paint the Mona Lisa . This argument has the form of the law of denying the consequent, so the conclusion is ~ p : Leonardo da Vinci was not an artist. The premises follow the form of the law of denying the consequent, so a valid conclusion would be ~ p . The premises are: p → q : If Michael Jordan played for the Chicago Bulls, then Michael Jordan was not a soccer player, and ~ q : Michael Jordan was a soccer player. The conclusion that follows from the premises is ~ p : Michael Jordan did not play for the Chicago Bulls. The premises are p → q : If all fish have gills, then clown fish have gills, and ~ q : Clown fish do not have gills. This argument has the form of the law denying the consequent, so the conclusion is ~ p : Some fish do not have gills. Chain Rule for Conditional Arguments The chain rule for conditional arguments is another form of a valid conditional argument. It is also called hypothetical syllogism or the transitivity of implication. Recall that the conditional statement p → q can also be read as p implies q . This is where the name transitivity of implication comes from. The transitive property for numbers states that, if 3 < 4 and 4 < 5 , then it follows that 3 < 5. The chain rule extends this property to conditional statements. If the premises of the argument consist of two conditional statements, with the form “ p → q ” and “ q → r , ” then it follows that p → r . Symbolically, it has the form ( ( p → q ) ∧ ( q → r ) ) → ( p → r ) . Chain Rule for Conditional Arguments Premise: p → q Premise: q → r Conclusion: ∴ p → r To verify the chain rule for conditional arguments, construct a truth table for the argument, ( ( p → q ) ∧ ( q → r ) ) → ( p → r ) , and verify that it is a tautology. p q r p → q q → r ( p → q ) ∧ ( p → r ) p → r ( ( p → q ) ∧ ( q → r ) ) → ( p → r ) T T T T T T T T T T F T F F F T T F T F T F T T T F F F T F F T F T T T T T T T F T F T F F T T F F T T T T T T F F F T T T T T To verify an argument of this form using a Venn diagram, again consider the premise p → q : “If you play guitar, then you are a musician,” but change the second premise to q → r : “If you are a musician, then you are an artist.” In this case, the set p of guitarists is a subset of the set r of artists, and it follows that if you are a guitarist, then you are an artist. Therefore, the conclusion p → r follows from the premises and the chain rule for logical arguments is valid. See . Applying the Chain Rule for Conditional Arguments to Determine a Valid and Sound Conclusion Each pair of statements represents true premises in a logical argument. Based on these premises, apply the chain rule for conditional arguments to determine a valid and sound conclusion. If my roommate goes to work, then my roommate will get paid. If my roommate gets paid, then my roommate will pay their bills. If robins can fly, then some birds can fly. If some birds can fly, then we will watch birds fly. If Irma is a teacher, then Irma has a college degree. If Irma has a college degree, then Irma graduated from college. The premises are p → q : “If my roommate goes to work, then they will get paid,” and q → r : “If my roommate gets paid, then my roommate will pay their bills.” This argument has the form of the chain rule for conditional arguments, so the valid conclusion will have the form “ p → r . ” Because all the premises are true, the valid and sound conclusion of this argument is: “If my roommate goes to work, then my roommate will pay their bills.” The premises are p → q : “If robins can fly, then some birds can fly,” and q → r : “If some birds can fly, then we will watch them fly.” This argument has the form of the chain rule for conditional arguments, so, the valid conclusion will have the form “ p → r . ” Because all the premises are true, the valid and sound conclusion of this argument is: “If robins can fly, then we will watch birds fly.” The premises are p → q : (see line 1 of solution 1 and 2 above) “If Irma is a teacher, then Irma has a college degree,” and q → r : “If Irma has a college degree, then Irma graduated from college.” This argument has the form of the chain rule for conditional arguments, so the valid conclusion will have the form “ p → r . ” Because all the premises are true, the valid and sound conclusion of this argument is: “If Irma is a teacher, then Irma graduated from college.” Check Your Understanding Projects Logic Gates Logic gates are the basis for all digital circuits. Research and document the following terms: logic gate, OR gate, AND gate, and NOT gate. Construct a diagram of a NAND gate, NOR gate, and a XOR gate by using at least two of the following gates: AND, OR, and NOT. Digital electronics use a 1 for true or on, and a 0 for false or off. Create a truth table documenting all possible cases using 0s and 1s for the NAND gate, NOR gate and XOR gate. Use a truth table to explain how XOR is related to the biconditional statement. Logical Fallacies Fallacies are false or deceptive logical arguments. Research and document the structure of five of the following named fallacies: hasty generalization, limited choice, false cause, appeal to popularity, appeal to emotion, appeal to authority, personal attack, gamblers' ruin, slippery slope, and circular reasoning. Create a presentation highlighting one of the five fallacies researched in the previous question. The presentation must include an introductory slide with the title of the fallacy and the form or structure of the argument. The second slide must include an example of this fallacy as used in a commercial, a political cartoon or a current event or new article. The third slide must include an explanation of why the example on slide to is a representative example of the fallacy. The last slide must include citations for any materials used. No textbooks should be used as reference. Careers in Logic Lawyers, mathematicians, and computer programmers are a few of the careers that require knowledge of logic. What career are you interested in? Research how knowledge of logic applies to your chosen field of study. Then, write a cover letter for a position in your field you'd like to apply to. In the cover letter, include how your knowledge of logic qualifies you for the position you are applying for. If you do not think logic is important for your given career choice, find a position where logic is an essential element of the position and complete the project by pretending you are writing a cover letter for that job. Key Terms sound fallacy deductive arguments law of detachment law of denying the consequent chain rule for conditional arguments Key Concepts A logical argument uses a series of facts or premises to justify a conclusion or claim. It is valid if its conclusion follows from the premises, and it is sound if it is valid, and all of its premises are true. The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, p → q is true and the hypothesis, p is true, then the conclusion q must also be true. Law of Detachment Premise: p → q Premise: p Conclusion: ∴ q Know how to apply the law of detachment to determine the conclusion of a pair of statements. The law of denying the consequent is a valid form of a conditional argument that asserts that if both the conditional, p → q is true and the negation of the conclusion, ~ q is true, then the negation of the hypothesis ~ p must also be true. Law of Denying the Consequent Premise: p → q Premise: ~ q Conclusion: ∴ ~ p Know how to apply the law of denying the consequent to determine the conclusion for pairs of statements. The chain rule for conditional arguments is a valid form of a conditional argument that asserts that if the premises of the argument have the form, p → q and q → r , then it follows that p → r . Chain Rule for Conditional Arguments Premise: p → q Premise: q → r Conclusion: ∴ p → r Know how to apply the chain rule to determine valid conclusions for pairs of true statements. Video Logic Part 14: Common Argument Forms like Modus Ponens and Tollens Chapter Review Statements and Quantifiers Compound Statements Constructing Truth Tables Truth Tables for the Conditional and Biconditional Equivalent Statements De Morgan’s Laws Logical Arguments Chapter Test

Math


Introduction Encryption of computers and messages use very large prime numbers. (credit: modification of work "Jefferson cylinder cipher (replica)" by Daderot/Wikimedia Commons, Public Domain) Encryption is used to secure online banking, for secure online shopping, and for browsing privately using VPNs (Virtual Private Networks). We need encryption (using prime numbers) for a secure exchange of information. For a prime number to be useful for encryption, though, it has to be large. Encryption uses a composite number that is the product of two very large primes. In order to break the encryption, one must determine the two primes that were used to form the composite number. If the two prime numbers used are sufficiently large, even the fastest computer cannot determine those two prime numbers in a reasonable amount of time. It would take a computer 300 trillion years to crack the current encryption standard.

Math


Prime and Composite Numbers Computers are protected using encryption based on prime numbers. (credit: “Data Security” by Blogtrepreneur/Flickr, CC BY 2.0) After completing this section, you should be able to: Apply divisibility rules. Define and identify numbers that are prime or composite. Find the prime factorization of composite numbers. Find the greatest common divisor. Use the greatest common divisor to solve application problems. Find the least common multiple. Use the least common multiple to solve application problems. Encryption, which is needed for the secure exchange of information (i.e., online banking or shopping) is based on prime numbers. Encryption uses a composite number that is the product of two very large prime numbers. To break the encryption, the two primes that were used to form the composite number need to be determined. If the two prime numbers used are sufficiently large, even the fastest computer cannot determine those two prime numbers in a reasonable amount of time. It would take a computer 300 trillion years to crack the current encryption standard. Applying Divisibility Rules Before we begin our investigation of divisibility, we need to know some facts about important sets of numbers: The counting numbers are referred to as the natural numbers . This set of numbers, { 1 , 2 , 3 , 4 , … } , is denoted with the symbol ℕ . Another important set of numbers is the integers . The integers are the natural numbers, along with 0, and the negatives of the natural numbers. This set is often written as { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } . We denote the integers with the symbol ℤ. Notice that ℕ is a proper subset of ℤ, or, ℕ ⊂ ℤ. All the ideas of this section apply to the natural numbers, while only some apply to all the integers. Divisibility is when the integer n is divisible by m , if n can be written as m times another integer. Equivalently, there is no remainder when n is divided by m . There are many occasions when separating items into equal groups comes into play to ensure an equal distribution of whole items. For example, Francis, a preschool art teacher, has 15 students in one class. Francis has 225 sheets of construction paper and wants to provide each student with an equal number of pieces. To know if he will use all the construction paper, Francis is really asking if 225 can be evenly divided into 15 groups. Determining if a Number Divides Another Number Determine if 36 is divisible by 4. We could divide 36 by 4 and see if there is a remainder, or we could see if we can write 36 as 4 times another integer. If we divide 36 by 4, we see 36 ÷ 4 = 9 with no remainder. We see that 36 is divisible by 4. We can write 36 as 4 times another integer, 36 = 4 × 9 . By the definition of divisibility, 36 is divisible by 4. You can quickly check if a number is divisible by 2, 3, 4, 5, 6, 9, 10, and 12. Each has an easy-to-identify feature, or rule, that indicates the divisibility by those numbers, as shown in the following table. Divisor Rule 2 Last digit is even 3 Add the digits of the number together. If that sum is divisible by 3, then so is the original number 4 Look at only the last two digits. If this number is divisible by 4, so is the original number 5 Look at only the last digit. If it is a 5 or a 0, then the original number is divisible by 5 6 If the number passes the rule for divisibility by 2 and for 3, then the number is divisible by 6 9 Add the digits of the number together. If that number is divisible by 9, then so is the original number 10 Look at only the last digit. If it is a 0, then the original number is divisible by 10 12 If the number passes the rule for 3 and 4, the number is divisible by 12 Using Divisibility Rules Using divisibility rules, determine if 245 is divisible by 5. Since the last digit is a 5, the number 245 is divisible by 5 because the rule states if the last digit of the number is a 5 or a 0, then the original number is divisible by 5. Using Divisibility Rules Using divisibility rules, determine if 25,983 is divisible by 9. The divisibility rule for 9 is when the digits of the number are added, the sum is divisible by 9. So, we calculate the sum of the digits. 2 + 5 + 9 + 8 + 3 = 27 . Since 27 is divisible by 9, so is the original number 25,983. Using Divisibility Rules Can 298 coins be stacked into 6 stacks with an equal number of coins in each stack? In order for the coins to be in equal-sized stacks, 298 would need to be divisible by 6. The divisibility rule for 6 is that the number passes the divisibility rules for both 2 and 3. Since the last digit is even, 298 is divisible by 2. To determine if 298 is divisible by 3, we first add the digits of the number: 2 + 9 + 8 = 19 . Since 19 is not divisible by 3, neither is 298. Because 298 is not divisible by 3, it is also not divisible by 6, which means they cannot be put into 6 equal stacks of coins. Using Divisibility Rules Using divisibility rules, determine if 4,259 is divisible by 10. The divisibility rule for 10 is that the last digit of the number is 0. Since the last digit of 4,259 is not 0, then 4,259 is not divisible by 10. Using Divisibility Rules Using divisibility rules, determine if 936,276 is divisible by 4. The divisibility rule for 4 is to check the last two digits of the number. If the number formed by the last two digits of the original number is divisible by 4, then so is the original number. The last two digits make the number 76 and 76 is divisible by 4, since 76 = 4 × 19 . Since 76 is divisible by 4, so is 936,276. Divisibility Rules Prime and Composite Numbers Sometimes, a natural number has only two unique divisors, 1 and itself. For instance, 7 and 19 are prime . In other words, there is no way to divide a prime number into groups with an equal number of things, unless there is only one group, or those groups have one item per group. Other natural numbers have more than two unique divisors, such as 4, or 26. These numbers are called composite . The number 1 is special; it is neither prime nor composite. To determine if a number is prime or composite, you have to determine if the number has any divisors other than 1 and itself. The divisibility rules are useful here, and can quickly show you if a number has a divisor on that list. However, if none of those divide the number, you still have to check all other possible prime divisors. What are the prime numbers that are possibly divisors of the number you are checking? You need only check the prime numbers up to the square root of the number in question. For instance, if you want to know if 2,117 is prime, you need to determine if any primes up to the square root of 2,117 (which is 46.0 when rounded to one decimal place) divide 2,117. If any of those primes are divisors of the number in question, then the number is composite. If none of those primes work, then the number is itself prime. We can check divisibility with whatever tool we wish. Divisibility rules are quick for some prime divisors (2 and 5 come to mind) but aren't quick for other values (like 11). In place of divisibility rules, we could just use a calculator. If the prime number divides the number evenly (that is, there is no decimal or fractional part), then the number is divisible by that prime. is a quick list of the prime numbers up to 50. There are 15 prime numbers less than 50. 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Prime Numbers Less than 50 Determining If a Number Is Prime or Composite Determine if 2,117 is prime or composite. The square root of 2,117 is 46.0 (rounded to one decimal place). So, we need to check if 2,117 is divisible by any prime up to 46. Step 1: First we’ll use the rules of divisibility we learned earlier: We can tell 2,117 is not divisible by 2, as the last digit isn't even. 2,117 is not divisible by 5 (the last digit isn't 0 or 5). Add the digits of 2,117 to get 11, which is not divisible by 3. So, 2,117 is also not divisible by 3. Step 2: Now we repeat the process for all the primes up to 46. Using a calculator, we find that 2,117 divided by the prime numbers 7, 11, 13, 17, 19, and 23 results in a remainder, a decimal part. So, we know that 2,117 is not divisible by these prime numbers. (You should check these results yourself.) Moving on, we check the next prime: 29. Using the calculator to divide 2,117 by 29 results in 73. Since there is no decimal part, 2,117 is divisible by 29. This means that 2,117 is not a prime number, but rather, a composite number. Writing 2,117 as the product of 29 and another natural number, 2,117 = 29 × 73 . Determining if a Number Is Prime or Composite Determine if 423 is prime or composite. The square root of 423 is 20.57 (rounded to two decimal places). So, we need to check if 423 is divisible by any prime up to 20. Step 1: Check 2. We can tell 423 is not divisible by 2, as the last digit isn't even. Step 2: Check 5. It is not divisible by 5 (the last digit isn't 0 or 5). Step 3: Check 3. To check if 423 is divisible by 3, we use the divisibility rule for 3. When we take the sum of the digits of 423, the result is 9. Since 9 is divisible by 3, so is 423. Since 423 is divisible by 3, then 423 is a composite number. Writing 423 as the product of 3 and another natural number, 423 = 3 × 141 . Determining if a Number Is Prime or Composite Determine if 1,034 is prime or composite A quick inspection of 1,034 shows it is divisible by 2 since the last digit is even, and so 1,034 is a composite number. Determining if a Number Is Prime or Composite Determine if 2,917 is prime or composite. The square root of 2,917 is 50.01 (rounded to two decimal places). So, we need to check if 2,917 is divisible by any prime up to 50. Step 1: Check 2. We can tell 2,917 is not divisible by 2, as the last digit isn't even. Step 2: Check 5. It is it divisible by 5 (the last digit isn't 0 or 5). Step 3: Check 3. Using the divisibility rule for 3, we take the sum of the digits of 2,917, which is 19. Since 19 is not divisible by 3, neither is 2,917. Step 4: Check the rest of the primes up to 50 using a calculator. When 2,917 is divided by every prime number up to 50, the result has a decimal part. Since no prime up to 50 divides 2,917, it is a prime number. ILLEGAL PRIMES Large primes are a hot commodity. Using two very large primes (some have more than 22 million digits!) is necessary for secure encryption. Anyone who has a new prime that is large enough can use that prime to create a new encryption. Of course, whoever discovers a large prime could sell it to a security company. These primes are so useful for encryption, it is necessary to protect that intellectual property. In fact, at least one prime number was declared illegal. Illegal Prime Number Sophie Germain Sophie Germain (credit: “Sophie Germain at 14 years,” Illustration from histoire du socialisme, approx. 1880, Wikimedia Commons, public domain) Born into a wealthy French family in 1776, Sophie Germain discovered and fell in love with mathematics by browsing her father’s books. Clandestine study, hard and tenacious work, and a mathematical mindset did not lead to college, however, as she was not allowed to attend. She did manage, through friends, to obtain problem sets and submit brilliant solutions under the name Monsieur LaBlanc. One of her great interests was number theory, which is the study of properties of integers. One of her theorems, titled “Sophie Germain’s Theorem,” partially solved one of the great mathematical mysteries, Fermat’s Last Theorem. From this she discovered what are now known as Sophie Germain Primes. A Sophie Germain Prime is a prime number that can be written in the form 2 p + 1 , where p is a prime number. For instance, 23 is prime: 2 ( 23 ) + 1 = 47 , which is prime, so 47 is a Sophie Germain Prime. It should be noted that 2 p + 1 , where p is a prime number, may or may not be prime (check for p = 7 !). Finding the Prime Factorization of Composite Numbers Before we can start with prime factorization, let’s remind ourselves what it means to factor a number. We factor a number by identifying two (or more) numbers that, when multiplied, result in the original number. For instance, 3 and 24, when multiplied, give 72. So, 72 can be factored into 3 × 24 . Notice that we could have factored the 72 differently, say as 72 = 6 × 12 , or 72 = 2 × 36 , or even as 72 = 3 × 4 × 6 . Finding the prime factorization of a composite number means writing the number as the product of all of its prime factors. For example, 80 = 2 × 2 × 2 × 2 × 5 . Notice that all the numbers being multiplied on the right-hand side are prime numbers. Sometimes prime numbers repeat themselves in the factorization. When prime factors do repeat, we may write the prime factorization using exponents , as in 80 = 2 4 × 5 . In that equation, the 2 is raised to the 4th power. The 4 is the exponent, and the 2 is the base . More generally, in the exponent notation a b , the number represented by a is the base, and the number represented by b is the exponent. One has to wonder if finding the prime factorization could result in different factorizations. The Fundamental Theorem of Arithmetic tells us that there is only one prime factorization for a given natural number. Fundamental Theorem of Arithmetic Every natural number, other than 1, can be expressed in exactly one way, apart from the arrangement, as a product of primes. The process of finding the prime factorization of a number is iterative, which means we do a step, then repeat it until we cannot do the step any longer. The step we use is to identify one prime factor of the number, then write the number as the prime factor times another factor. We repeat this step on the other, newly found, factor. This step is repeated until no more primes can be factored from the remaining factor. This is easier to see and explain with an example. Finding the Prime Factorization Find the prime factorization of 140. Step 1: Identify a prime number that divides 140. Since 140 is even (the last digit is even), 2 divides 140. We then factor the 2 out of the 140, giving us 140 = 2 × 70 . Step 2: With the other factor, 70, find a prime factor of 70. Since 70 is also even, 2 divides 70. We factor the 2 out of the 70 and the factorization is now 140 = 2 × 2 × 35 = 2 2 × 35 . Notice that we expressed the two factors of 2 as 2 2 . Step 3: Look to the remaining factor, 35. The last digit of 35 is 5, so 5 is a factor of 35. We factor the 5 out of the 35. The factorization is now 140 = 2 2 × 5 × 7 . Step 4: Look to the remaining factor, 7. Since 7 is prime, the process is complete. The prime factorization of 140 is 2 2 × 5 × 7 . Factor Trees A useful tool for helping with prime factorization is a factor tree . To create a factor tree for the natural number n (where n is not 1), perform the following steps: Step 1: If n is prime, you're done. If n is composite, continue to the next step. Step 2: Identify two divisors of n , call them a and b . Step 3: Write the number n down, and draw two branches extending down (or to the right) of the number n . Step 4: Label the end of one branch a , the other as b . See . Step 5: If a and b are prime, the tree is complete. When a number at this step is a prime number, we refer to it as a leaf of the tree diagram. Step 6: If either a or b are composite, repeat Steps 2 through 4 for a and b . Step 7: The process stops when the leaves are all prime. Step 8: The prime factorization is then the product of all the leaves. This is best seen in an example. Finding the Prime Factorization Find the prime factorization of 66. Since 66 is even, 2 is a factor. Step 1: Factor out the 2. The factorization is 66 = 2 × 33 . The factor tree is shown in . Since 2 is a factor, that branch is done, and 2 is a leaf. Step 2: The 33, though, is divisible by 3, and is the product of 3 and 11. We attach that to the factor tree ( ). Since the 2, 3 and 11 are all prime, the factor tree is done. The prime factorization of 66 is the product of the leaves, so 66 = 2 × 3 × 11 . The factorization is complete. Using a Factor Tree to Find the Prime Factorization Finding the Prime Factorization Find the prime factorization of 135. The number 135 is divisible by 3, and so 3 is a factor of 135. Step 1: Factor out the 3. The factorization is 135 = 3 × 45 . Using the factor tree ( ), 45 is also divisible by 3. Step 2: Factor out a 3 from 45. The other factor is 15. The factor tree is shown in . 15 is also divisible by 3. Step 3: The factors of 15 are 3 and 5. The factor tree is shown in . All the leaves are prime, so the process is complete. The prime factorization of 135 is 3 3 × 5 . Identifying Prime Factors How many different prime factors does 10,241 have? To know how many different prime factors 10,241 has, we need the prime factorization of 10,241. Step 1: Use divisibility rules, to see that the number 10,241 is not divisible by 2 or by 3 (the sum of the digits is 8), or by 5. However, it is divisible by 7. Step 2: After factoring the 7, the factorization is 10 , 241 = 7 × 1463 ; 1,463 is also divisible by 7. Step 3: After factoring the 7, the factorization is 10 , 241 = 7 × 7 × 209 = 7 2 × 209 . The number 209 is not divisible by 7. Step 4: Check the next prime number: 11; 11 does divide 1,463. Step 5: After factoring the 11, the factorization is 10 , 241 = 7 2 × 11 × 19 . Since 19 is prime, the prime factorization of 10,241 is complete. We see that 10,241 has three different prime factors: 7, 11, and 19. Finding the Prime Factorization of 168 Using Wolfram Alpha to Find Prime Factorizations The Wolfram Alpha website is a powerful resource available for free to use. It is designed using AI so that it understands natural language requests. For instance, typing the question “What is the prime factorization of 543,390?” gets a rather quick answer of 2 × 3 × 5 × 59 × 307 . So, if you want to find the prime factorization of a number, you can simply ask Wolfram Alpha to find the prime factorization of the number. Finding the Greatest Common Divisor Two numbers often have more than one divisor in common (all pairs of natural numbers have the common divisor 1). When listing the common divisors, it’s often the case that the largest is of interest. This divisor is called the greatest common divisor and is denoted GCD . It is also sometimes referred to as the greatest common factor (GCF). For instance, 6 is the greatest common divisor of 12 and 18. We can see this by listing all the divisors of each number and, by inspection, select the largest value that shows up in each list. The divisors of 12 1, 2, 3, 4, 6, 12 The divisors of 18 1, 2, 3, 6, 9 It is easy to see that 6 is the largest value that appears in both lists. Finding the Greatest Common Divisor Using Lists Find the greatest common divisor of 1,400 and 250 by listing all divisors of each number. We create a list of all the divisors of 1,400 and of 250, and choose the largest one. The divisors of 1,400 are 1, 2, 4, 5, 7, 8, 10, 14, 20, 25, 28, 35, 40, 50, 56, 70, 100, 140, 175, 200, 280, 350, 700, 1,400. The divisors of 250 are 1, 2, 5, 10, 25, 50, 125, 250. The largest value that appears on both lists is 50, so the greatest common divisor of 1,400 and 250 is 50. Listing all the divisors of the numbers in the set will always work, but for some relatively small numbers, the set of all divisors can become pretty big, and finding them all can be a chore. Another approach to finding the greatest common divisor is to use the prime factorization of the numbers. To do so, use the following steps: Step 1: Find the prime factorization of the numbers. Step 2: Identify the prime factors that appear in every number’s prime factorization. These are called the common prime factors. Step 3: Identify the smallest exponent of each prime factor identified in Step 2 in the prime factorizations. Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. The result is the greatest common divisor. Finding the Greatest Common Divisor Using Prime Factorization Find the greatest common divisor of 1,400 and 250 by using their prime factorizations. Step 1: Find the prime factorizations of the numbers. The prime factorization of 1,400 is 2 3 × 5 2 × 7 . The prime factorization of 250 is 250 = 2 × 5 3 . Step 2: Identify the prime factors that appear in every number’s prime factorization. The common prime factors are 2 and 5. Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. The exponent of common prime factor 2 in the prime factorization of 1,400 is 3, and in the prime factorization of 250 is 1. The smallest of those exponents is 1. The exponent of the common prime factor 5 in the prime factorization of 1,400 is 2 and is in the prime factorization of 250 is 3. The smallest of these exponents is 2. Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. This gives 2 1 × 5 2 = 50 . The greatest common divisor of 1,400 and 250 is 2 × 5 2 = 50 . Notice that the answer matches the one we found in Example 3.15. Finding the Greatest Common Divisor Using Prime Factorization Find the greatest common divisor of 600 and 784 by using their prime factorizations. Step 1: Find the prime factorizations of the numbers. The prime factorization of 600 is 2 3 × 3 × 5 2 . The prime factorization of 784 is 2 4 × 7 2 . Step 2: Identify the prime factors that appear in every number’s prime factorization. There is only one common prime factor, 2. Step 3: Identify the smallest exponent of each prime from identified in Step 2 in the prime factorizations. The exponent of 2 in the prime factorization of 600 is 3. The exponent of 2 in the prime factorization of 784 is 4. So, the smallest exponent of 2 is 3. Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. This gives 2 3 = 8 . The greatest common divisor of 600 and 784 is 8. Srinivasa Ramanujan Srinivasa Ramanujan (credit: Srinivasa Ramanujan, photo by Konrad Jacobs/Oberwolfach Photo Collection/public domain) Ramanujan was born in southern India in 1887, during British rule. He was a self-taught mathematician, who, while in high school, began working through a two-volume text of mathematical theorems and results. His work included examination of the distribution of primes. He eventually came to the attention of British mathematician, G.H. Hardy. During one visit, Hardy mentioned to Ramanujan that his taxicab number was 1,729, remarking that 1,729 appeared to be a rather dull number. To which Ramanujan responded, “It is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” Using Desmos to Find the GCD To find the GCD of a set of numbers in Desmos, type gcd(first_number,second_number…) and Desmos will display the GCD of the numbers, as the numbers are typed! Using Desmos to Find the GCD Applications of the Greatest Common Divisor The greatest common divisor has uses that are related to other mathematics (reducing fractions) but also in everyday applications. We’ll look at two such applications, which have very similar underlying structures. In each case, something must divide the groups or measurements equally. Calculating Floor Tile Size Suppose you have a rectangular room that is 570 cm wide and 450 cm long. You wish to cover the floor of the room with square tiles. What’s the largest size square tile that can be used to cover this floor? Using squares means that the length and width of the tiles are equal. To ensure we are using full tiles, the side length of the square tiles must divide the length of the room and the width of the room. Since we want the largest square tiles, we need the GCD of the width and length of the room or the GCD of 570 and 450. Step 1: Find the prime factorizations of the numbers. The prime factorization of 570 is 2 × 3 × 5 × 19 . The prime factorization of 450 is 2 × 3 2 × 5 2 . Step 2: Identify the prime factors that appear in every number’s prime factorization. The common prime factors are 2, 3, and 5. Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. The exponents of 2 in the prime factorizations of 570 and 450 are 1 and 1. So the smallest exponent for 2 is 1. The exponents of 3 in the prime factorizations of 570 and 450 are 1 and 2, so the smallest exponent for 3 is 1. The exponents of 5 in the prime factorizations of 570 and 450 are 1 and 2, so the smallest exponent for 5 is 1. Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. This gives 2 1 × 3 1 × 5 1 = 30 . The GCD of 450 and 570 is 30, so the largest size square tile that can be used to cover the floor is 30 cm by 30 cm. Organizing Books Per Shelf Suppose you want to organize books onto shelves, and you want the shelves to hold the same number of books. Each shelf will only contain one genre of book. You have 24 sci-fi, 42 fantasy, and 30 horror books. How many books can go on each shelf? Since we want shelves that hold an equal number of books, and a shelf can only hold one genre of book, we need a number that will equally divide 24, 42, and 30. So, we need the GCD of the number of books of each genre or the GCD of 24, 42, and 30. Step 1: Find the prime factorizations of the numbers. The prime factorization of 24 is 2 3 × 3 . The prime factorization of 42 is 2 × 3 × 7 . The prime factorization of 30 is 2 × 3 × 5 . Step 2: Identify the prime factors that appear in every number’s prime factorization. The common prime factors are 2 and 3. Step 3: Identify the smallest exponent of each prime identified in Step 2 in the prime factorizations. The smallest exponent of 2 and 3 in the factorizations is 1. Step 4: Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. This gives 2 1 × 3 1 = 6 . The GCD of 24, 42, and 30 is 6, so the largest number of books that can go on a shelf is 6. Applying the GCD Finding the Least Common Multiple The flip side to a divisor of a number is a multiple of a number. For example, 5 is a divisor of 45 and so 45 is a multiple of 5. More generally, if the number a divides the number b , then b is a multiple of a . This drives the idea of the least common multiple of a set of numbers. A common multiple of a set of numbers is a multiple of each of those numbers. For instance, 45 is a common multiple of 9 and 5, because 45 is a multiple of 9 (9 divides 45) and 45 is also a multiple of 5 (5 divides 45). The least common multiple (LCM) of a set of number is the smallest positive common multiple of that set of numbers. There are (at least) three ways to find the LCM of a set of numbers, and we will explore two of them. One way is to create a list of multiples of each number in the set, and then identify the smallest multiple that appears in those lists. Finding the Least Common Multiple Using Lists Find the LCM of 24 and 90 by listing multiples and choosing the smallest common multiple. Create a list of the multiples of each number. Step 1: The first 15 multiples for 24: 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360. Step 2: The first 15 multiples for 90: 90, 180, 270, 360, 450, 540, 630, 720, 810, 900, 990, 1,080, 1,170, 1,260, 1,350. There is one multiple common to these lists, which is 360. So, 360 is the LCM of 24 and 90. The second method we can use is to find the prime factorizations of the number in the set to build the LCM of the numbers based on the prime divisors of the numbers. The LCM can be built from the prime factorization of the numbers in the set in a similar way as when finding the greatest common divisor. Here are the steps for using the prime factorization process for finding the LCM. Step 1: Find the prime factorization of each number. Step 2: Identify each prime that is present in any of the prime factorizations. Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. Step 4: . Multiply the prime factors identified in Step 2 raised to the powers identified in Step 3. Finding the Least Common Multiple Using Prime Factorization Use the prime factorizations of 24 and 90 to identify their LCM. Step 1: Find the prime factorization of each number. 24 = 2 3 × 3 90 = 2 × 3 2 × 5 Step 2: Identify each prime that is present in any of the prime factorizations. The prime numbers present in the prime factorizations are 2, 3, and 5. Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. Prime 2 3 5 Largest exponent 3 2 1 Step 4: Compute the LCM by multiplying the prime factors identified in Step 2 raised to the powers identified in Step 3. The LCM for 24 and 90 is 2 × 2 × 2 × 3 × 3 × 5 = 2 3 × 3 2 × 5 1 = 360 . Finding the Least Common Multiple Using Prime Factorization Use the prime factorizations of 36, 66, and 250 to identify the LCM. Step 1: Find the prime factorization of each number. 36 = 2 2 × 3 2 66 = 2 × 3 × 11 250 = 2 × 5 3 Step 2: Identify each prime that is present in any of the prime factorizations. The prime numbers present in the prime factorizations are 2, 3, 5, and 11. Step 3: Identify the largest exponent of each prime identified in Step 2 in the prime factorizations. Prime 2 3 5 11 Largest exponent 2 2 3 1 Step 4: Compute the LCM by multiplying the prime factors identified in Step 2 raised to the powers identified in Step 3. The LCM for 36, 66, and 250 is 2 × 2 × 3 × 3 × 5 × 5 × 5 × 11 = 2 2 × 3 2 × 5 3 × 11 1 = 49,500 . Using lists for three or more numbers, particularly larger numbers, could take quite a bit of time. Frequently, as in this example, the prime factorization process is much quicker. In practice, you can use either listing or prime factorization to find the LCM. Finding the Least Common Multiple Using Both Methods Find the LCM of 20, 36, and 45 using lists and prime factorization. Step 1: Use listing. List the multiples: 20 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220 36 36, 72, 108, 144, 180, 216, 252, 288, 324, 360 45 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, 495, 540 The smallest value that appears on all the lists is 180, so 180 is the LCM of 20, 36, and 45. Step 2: Find the prime factorization of each number. 20 = 2 2 × 5 36 = 2 2 × 3 2 45 = 3 2 × 5 Step 3: Identify each prime that is present in any of the prime factorizations. The prime numbers present in the prime factorizations are 2, 3, 5. Step 4: Identify the largest exponent of each prime identified in Step 3 in the prime factorizations. Prime 2 3 5 Largest exponent 2 2 1 Step 5: Compute the LCM by multiplying the prime factors identified in Step 3 raised to the powers identified in Step 4. The LCM for 20, 36, and 45 is 2 2 × 3 2 × 5 1 = 180 . Both listing and prime factorization produced the same result: the LCM is 180. Finding the LCM Using Desmos to Find the LCM To find the LCM of a set of numbers in Desmos, type lcm(first_number,second_number…) and Desmos will display the LCM of the numbers, as the numbers are typed! Using Desmos to find the LCM Applications of the Least Common Multiple Some applications of LCM involve events that repeat at fixed intervals, such as visits to a location. Other applications involve getting things to be of equal magnitude when using parts that are different sizes (see , for instance). In each case, we may be looking to determine when two processes “line up.” Determining Scheduling Overlap Using the Least Common Multiple Two students, João, and Amelia, meet one day at an assisted living facility where they volunteer. João volunteers every 6 days. Amelia volunteers every 10 days. How many days will it be until they are both volunteering on the same day again? If we list the days that each student will volunteer, it becomes clear that we could solve this problem using the LCM of 6 and 10. João will be at the assisted living facility 6, 12, 18, 24, 30, 36, 42, and 48 days later. Amelia will be at the assisted living facility 10, 20, 30, 40, 50, and 60 days later. The smallest number appearing on both lists is 30. João and Amelia will once more be volunteering together 30 days later. Determining the Minimum Height Using the LCM A team-building exercise has teams build a house of cards as high as possible. However, the cards for different teams are of different sizes. Team 1 uses 10 cm × 10 cm cards, while Team 2 uses 8 cm × 8 cm cards. What is the minimum height when the two teams will be tied? Ignore the width of the cards. This is an example where we want to put together objects with different sizes. We want to know the minimum height when they are tied, or when the houses of cards line up the first time. The heights of the towers built using the 10 cm × 10 cm cards will be 10, 20, 30, 40, 50, and 60 cm tall. When the 8 cm × 8 cm cards are used, the tower will be 8, 16, 24, 32, 40, 48, and 56 cm tall. The smallest number appearing on both lists is 40. The first time they are tied is when the two towers are 40 cm tall. Application of LCM Prime Number Life Cycles—Cicadas Cicadas are known to have life cycles of 13 or 17 years, which are prime numbers. Why would a prime number life cycle be an evolutionary advantage? To figure this out, we have to explore how common multiples work with prime numbers. Make a conjecture regarding the LCM of a prime number and another number. Test this conjecture with a few examples of your own making. Check Your Understanding Key Terms natural numbers factor of a number multiple of a number prime number composite number prime factorization greatest common divisor (GCD) least common multiple (LCM) Key Concepts The natural numbers can be categorized as 1, prime numbers, and composite numbers. Prime numbers have as their only factors 1 and themselves. Composite numbers have at least three distinct factors. Composite numbers can be written in their prime factorization form, which is found by repeatedly factoring prime factors from the number. The greatest common divisor (GCD) of a set of numbers is the largest integer that divides all of the numbers in the set. The prime factorizations of the numbers can be used to identify the greatest common divisor. The least common multiple (LCM) of a set of numbers is the smallest integer that is divisible by all of the numbers in the set. The prime factorizations of the numbers can be used to identify the least common multiple. There are various ways that the GCD and LCM are applied. Videos Divisibility Rules Illegal Prime Number Using a Factor Tree to Find the Prime Factorization Finding the Prime Factorization of 168 Using Desmos to find the GCD Applying the GCD Finding the LCM Using Desmos to find the LCM Application of LCM

Math


The Integers A ledger comparing assets to debts, resulting in net wealth. (credit: modification of work “Reviewing Financial Statements” by Mary Cullen/Flickr, CC BY 2.0) After completing this section, you should be able to: Define and identify numbers that are integers. Graph integers on a number line. Compare integers. Compute the absolute value of an integer. Add and subtract integers. Multiply and divide integers. Positive net wealth is when the total value of a person’s assets, such as their home, their 401(k), their car, and savings account balance, exceed that of their debts, such as car loans, mortgages, or credit card debt. However, when the total value of debt exceeds the total value of assets, then the person has negative net wealth. Expressing the negative net wealth as a negative number allows people to work with the positive net values and negative net values with the same mathematical processes, and in the same applications. This section introduces the integers and operations with integers. Defining and Identifying Integers Extending the counting numbers to include negative numbers and zero forms the integers . Any other number that cannot be written as { … − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } is not an integer. Identifying Integers Which of the following are integers and which are not? −3 Is an integer, as it is the negative of a counting number 24 This is not written as an integer. Entering the square root of 24 in a calculator, such as desmos, the result is 4.899 (rounded off). Since this is not an integer, then 24 is not an integer. 36/4 Since 36 divided by 4 is 9, and 9 is an integer, then 36/4 is an integer. 45 Is an integer, as it is a counting number 63.9 Is not an integer, because it is not a counting number and not the negative of a counting number. 2/7 Dividing 2 by 7 results in a number less than 1, but greater than 0, so is between two consecutive integers. So, 2/7 is not an integer. −16.0 Is an integer, since the decimal part is 0 Graphing Integers on a Number Line Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the negative direction, while going to the right is going forward, or in the positive direction. A number line ( ) helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number line. Graphing Integers on the Number Line Graphing Numbers on the Number Line Graph the following on the number line: 1 −4 3 Comparing Integers When determining if one quantity or size is larger than another, we know it means there is more of whatever is being discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. This idea applies to negative integers also. This means that a is greater than b when a is to the right of b on the number line. We write a > b . When a is greater than b , we can also say that b is less than a . On the number line, b would be to the left of a . We write b < a . We need to recognize that a > b means the same thing as b < a . This can be seen on the number line in . On this number line, a is to the right of b , so a > b . But this means b is to the left of a , so b < a . Comparing Integers Using the Number Line Comparing Integers Using a Number Line Determine which of −6 and 4 is larger using a number line, and express that using both the greater than and the less than notations. To illustrate this, we use a number line ( ). Since −6 is to the left of 4, then −6 is less than 4. We can write this as −6 < 4. Another way of expressing this is that 4 is greater than −6. So we can also write 4 > − 6 . Comparing Negative Integers Determine which of −6 and −2 is larger, and express that using both the greater than and the less than notations. To illustrate this, we use a number line ( ). Since −6 is to the left of −2, then −6 is less than −2. We can write this as − 6 < − 2 . Another way of expressing this is that −2 is greater than −6. So we can also write − 2 > − 6 . Warning: People often ignore the negative signs, and think than since 6 is greater than 2, −6 is greater than −2. To avoid that error, remember that the greater number is to the right on the number line. Comparing Integers by Quantity Determine which of 27 and 410 is larger, and express that using both the greater than and the less than notations. When thinking about quantity, 410 is more than 27. So, 410 is greater than 27 and 27 is less than 410. We can write this as 410 > 27 or as 27 < 410 . The Absolute Value of an Integer When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away from 0. The absolute value of an integer n is the distance from n to 0, regardless of the direction. The notation for absolute value of the integer n is | n | . If we think of an integer as both direction and magnitude (size), absolute value is the magnitude part. Calculating the absolute value of an integer is very straightforward. If the integer is positive, then the absolute value of the integer is just the integer itself. If the integer is negative, then to compute the absolute value of the integer, simply remove the negative sign. Keeping in mind the number line as a path, when you’ve gone 10 steps to the left of 0, you have still taken 10 steps, and the direction does not matter. Evaluating the Absolute Value of an Integer Calculating the Absolute Value of a Positive Integer Calculate |19|. Since the number inside the absolute value symbol is positive, the absolute value is just the number itself. So |19| = 19. Calculating the Absolute Value of a Negative Integer Calculate |−435|. Since the number inside the absolute value is negative, the absolute value removes the negative sign. So |−435| = 435. Adding and Subtracting Integers You may recall having approached adding and subtracting integers using the number line from earlier in your academic life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a negative integer results in a move to the right. This leads to a few adding and subtracting rules, such as: Rule 1: Subtracting a negative is the same as adding a positive. Rule 2: Adding two negative integers always results in a negative integer. Rule 3: Adding two positive integers always results in a positive integer. Rule 4: The sign when adding integers with opposite signs is the same as the integer with the larger absolute value. These rules are good to keep in the back of your mind, as they can serve as a quick error check when you use a calculator. Adding Integers Use your calculator to calculate 4 + (−7). Explain how the answer agrees with what was expected. Using a calculator, we find that 4 + (−7) = −3. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value which |−7|=7. Subtracting Positive Integers Use your calculator to calculate 18 − 9. Explain how the answer agrees with what was expected. Using a calculator, we find that 18 − 9 = 9. Since 18 was larger than 9, we expected the difference to be positive. Subtracting with Negative Integers Use your calculator to calculate 27 − (−13). Explain how the answer agrees with what was expected. Using a calculator, we find that 27 – (−13) = 40. Since we’re subtracting a negative number, it is the same as adding a positive, so this is the same as 27 + 13 = 40. Adding Integers with Opposite Signs Use your calculator to calculate (−13) + 90. Explain how the answer agrees with what was expected. Using a calculator, we find that (−13) + 90 = 77. Since we are adding integers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value, which is positive since 90 is positive. One use of negative numbers is determining net worth , which is all the weath someone owns less all that someone owes. Sometimes net worth is positive (which is good), and sometimes net worth is negative (which can be stressful). Calculating Net Worth Jennifer is owed $50 from her friend Janice, but owes her friend Pat $87. What is Jennifer’s net worth? Net worth is the amount that one is owed minus the amount one owes. Jennifer is owed $50 but owes $87. So, her net worth is $50 – $87 = −$37. The negative indicates that Jennifer owes more than she is owed. Multiplying and Dividing Integers Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules. Rule 1: When multiplying or dividing two integers with the same sign, the result is positive. Rule 2: When multiplying or dividing two integers with opposite signs, the result is negative. Just as before, these rules can serve as a quick error check when using a calculator. Multiplying Positive Integers Use your calculator to calculate 4 × 8. Explain how the answer agrees with what was expected. Entering 4 × 8 into your calculator, the result is 32. This agrees with our expectation. The numbers have the same signs, so the result is positive. Multiplying Integers with Different Signs Use your calculator to calculate 9 × (−10). Explain how the answer agrees with what was expected. Entering 9 × (−10) into your calculator, the result is −90. This agrees with our expectation. The numbers have opposite signs, so the result is negative. Dividing Integers with Different Signs Use your calculator to calculate 400/(−25). Explain how the answer agrees with what was expected. Entering 400/(−25) into your calculator, the result is −16. This agrees with our expectation. The numbers have opposite signs, so the result is negative. Dividing Negative Integers Use your calculator to calculate −750/(−3). Explain how the answer agrees with what was expected. Entering −750/(−3) into your calculator, the result is 250. This agrees with our expectation. The numbers have the same signs, so the result is positive. At the end of a season, a team may wish to buy their coach an end-of season gift. It makes sense to share the cost equally among the members. To do so, the team would need to find the average (or mean) cost per member. The average (or mean) of a set of numbers is the sum of the numbers divided by the number values that are being averaged. Finding the Average of a Set of Numbers The daily low temperatures in Barrie, Ontario, for the week of February 14, 2021, were −20°, −12°, −15°, −23°, −17°, −13°, and −19° degrees Celsius. What was the average daily temperature for the week of February 14, 2021, in Barrie? Step 1: To find the average daily temperature, we first need to add the temperatures. (−20) + (−12) + (−15) + (−23) + (−17) + (−13) + (−19) = −119 Step 2: That sum will then be divided by 7 since we are averaging over seven days, giving −119/7 = −17. So, the average daily temperature in Barrie, Ontario the week of February 14, 2021, was −17° Celsius. Check Your Understanding Key Terms integer absolute value average of a set of numbers Key Concepts A set of numbers that can be built from the natural numbers are the integers, which consist of the natural numbers, zero (0), and the negatives of the natural numbers. Integers are often graphed on a number line, which helps display the relative positions and values of those numbers. The number line can be used to visualize when one integer is larger than or smaller than another integer. Arithmetic operations with integers are similar to the operations with natural numbers, except that the sign (positive or negative) of the numbers will determine the sign (positive or negative) of the result. Videos Graphing Integers on the Number Line Comparing Integers Using the Number Line Evaluating the Absolute Value of an Integer

Math


Order of Operations Calculators may automatically apply order of operations to calculations. (credit: “Precision” by Leonid Mamchenkov/Flickr, CC BY 2.0) After completing this module, you should be able to: Simplify expressions using order of operations. Simplify expressions using order of operations involving grouping symbols. Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first. You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.” Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations , arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, 4 × 2 2 + 3 + 5 2 can be calculated in many ways. You could get 5,184. Or, you could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find 4 × 2 2 + 3 + 5 2 = 44 . Simplify Expressions Using Order of Operations The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS: E Exponents M/D Multiplication and division A/S Addition and Subtraction So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example. Using Two Order of Operations Calculate 21 − 4 × 13 . There are no exponents in this expression, so the next operations to check are multiplication and division. Step 1: Moving left to right, the first multiplication encountered is 4 multiplied by 13. We perform that operation first. 21 − 4 × 13 = 21 − 52 Step 2: The only operation remaining is the subtraction. 21 − 52 = − 31 . So, 21 − 4 × 13 = − 31 . Using Two Order of Operations Calculate 4 × 8 3 . Step 1: Moving left to right, we see there is an exponent. We calculate the exponent first. 4 × 8 3 = 4 × ( 8 × 8 × 8 ) = 4 × 512 Step 2: The only operation remaining is the multiplication. 4 × 512 = 2,048 So, 4 × 8 3 = 2,048 . Using Three Order of Operations Calculate 2 + 3 2 × 4 . Step 1: To calculate this, move left to right, and compute all the exponents first. The only exponent we see is the squaring of the 3, so that is calculated first. 2 + 3 2 × 4 = 2 + ( 3 × 3 ) × 4 = 2 + 9 × 4 Step 2: Since the exponents are all calculated, now calculate all the multiplications and divisions moving left to right. The only multiplication or division present is 9 times 4. 2 + 9 × 4 = 2 + 36 Step 3: Moving left to right, perform the additions and subtractions. There is only one such operation, 2 plus 36. 2 + 36 = 38 So, 2 + 3 2 × 4 = 38 . Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS. Order of Operations 1 Using Eight Order of Operations Correctly apply the order of operations to compute the following: 4 − 25 × 6 / 10 × 3 2 + 7 × 2 3 . Step 1: To do so, calculate the exponents first, moving left to right. There are two occurrences of exponents in the expression, 3 squared and 2 cubed. 4 − 25 × 6 / 10 × 3 2 + 7 × 2 3 = 4 − 25 × 6 / 10 × 9 + 7 × 8 Step 2: Now that the exponents are calculated, perform the multiplication and division, moving left to right. The first is the product of 25 and 6. 4 − 25 × 6 / 10 × 9 + 7 × 8 = 4 − 150 / 10 × 9 + 7 × 8 Step 3: Next is the 150 divided by 10. 4 − 150 / 10 × 9 + 7 × 8 = 4 − 15 × 9 + 7 × 8 Step 4: Next is 15 multiplied by 9. 4 − 15 × 9 + 7 × 8 = 4 − 135 + 7 × 8 Step 5: Finally, multiply the 7 and 8. 4 − 135 + 7 × 8 = 4 − 135 + 56 As all the multiplications and divisions have been calculated, the additions and subtractions are performed, moving left to right. 4 − 135 + 56 = − 131 + 56 = − 131 + 56 = − 75 The computed value is −75. Order of Operations 2 Using Six Order of Operations Correctly apply the rules for the order of operations to accurately compute the following: 10 − 3 × 5 3 / 15 + 56 / 4 . Step 1: Calculate exponents first, moving left to right: 10 − 3 × 5 3 / 15 + 56 / 4 = 10 − 3 × 125 / 15 + 56 / 4 Step 2: Multiply and divide, moving left to right: 10 − 3 × 125 / 15 + 56 / 4 = 10 − 375 / 15 + 56 / 4 = 10 − 25 + 56 / 4 = 10 − 25 + 14 Step 3: Add and subtract, moving left to right: 10 − 25 + 14 = − 15 + 14 = − 1 Using Order of Operations Correctly apply the rules for the order of operations to accurately compute the following: ( − 8 ) / 2 × 3 − 9 × 2 4 / 12 + 9 × ( − 4 ) 2 / 2 3 . Step 1: Calculate the exponents first, moving left to right: ( − 8 ) / 2 × 3 − 9 × 2 4 / 12 + 9 × ( − 4 ) 2 / 2 3 = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 12 2 / 2 3 = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 2 3 = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 8 Step 2: Multiply and divide, moving left to right: = ( − 8 ) / 2 × 3 − 9 × 16 / 12 + 9 × 144 / 8 = ( − 4 ) × 3 − 9 × 16 / 12 + 9 × 144 / 8 = ( − 12 ) − 9 × 16 / 12 + 9 × 144 / 8 = ( − 12 ) − 144 / 12 + 9 × 144 / 8 = ( − 12 ) − 12 + 9 × 144 / 8 = ( − 12 ) − 12 + 1,296 / 8 = ( − 12 ) − 12 + 162 Step 3: Add and subtract, moving left to right: = ( − 12 ) − 12 + 162 = ( − 24 ) + 162 = 138 Using the Order of Operations Involving Grouping Symbols We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below. P Parentheses E Exponents M/D Multiplication and division (division is just the multiplication by the reciprocal) A/S Addition and subtraction (subtraction is just the addition of the negative) As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: ( 4 × 7 ) 2 . This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS. Prioritizing Parentheses in the Order of Operations Correctly apply the rules for the order of operations to accurately compute the following: ( 10 − 3 ) × 5 3 . Step 1: Perform all calculations within the parentheses before all other operations. ( 10 − 3 ) × 5 3 = 7 × 5 3 Step 2: Since all parentheses have been cleared, move left to right, and compute all the exponents next. 7 × 5 3 = 7 × 125 Step 3: Perform all multiplications and divisions moving left to right. 7 × 125 = 875 Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward. Order of Operations 3 Working Innermost Parentheses in the Order of Operations Correctly apply the rules for order of operations to accurately compute the following: 4 + 2 × ( 3 2 − ( 2 + 5 ) 2 × 4 ) / ( 3 + 8 ) . Step 1: Perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time. The innermost set of parentheses has the 2 + 5 inside. The 3 + 8 is in a separate set of parentheses, so that addition can occur at the same time as the 2 + 5. 4 + 2 × ( 3 2 − ( 2 + 5 ) 2 × 4 ) / ( 3 + 8 ) = 4 + 2 × ( 3 2 − ( 7 ) 2 × 4 ) / ( 11 ) Step 2: Now that those parentheses have been handled, move on to the next set of parentheses. Applying the order of operation rules inside that set of parentheses, the exponent is evaluated first, then the multiplication, and then the addition. 4 + 2 × ( 3 2 − 7 2 × 4 ) / 11 = 4 + 2 × ( 3 2 − 49 × 4 ) / 11 = 4 + 2 × ( − 187 ) / 11 Step 3: Since all parentheses have been cleared, apply the EMDAS rules to finish the calculation. 4 + 2 × ( − 187 ) / 11 = 4 − 374 / 11 = 4 − 34 = − 30 Order of Operations 4 Check Your Understanding Key Terms order of operations PEMDAS Key Concepts Establishing shared rules on which arithmetic operations are calculated first is necessary. Without them, different people may find different values for the same expression. The highest precedence is with expressions in parentheses. This allows parts of an expression to be calculated in an order different than the basic order of operations. The lowest precedence is addition and subtraction, as they are the basis for all other calculations. Multiplication and division have precedence over addition and subtraction, as they are representations of repeated addition or subtraction. Exponents have precedence over multiplication and division, as they represent repeated multiplication and division. Videos Order of Operations 1 Order of Operations 2 Order of Operations 3 Order of Operations 4

Math