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Integer Arithmetic: An Outline

by David Klein The set of integers is the collection of numbers {..., ­3, ­2, ­1, 0, 1, 2, 3, ...}. An integer is any number in this set. Therefore, an integer is either a counting number (1, 2, 3, etc.), or zero, or the opposite (the negative) of a counting number (­1, ­2, ­3, etc.). The number line below is a standard way to exhibit the integers. | ­4 | ­3 | ­2 | ­1 | 0 | 1 | 2 | 3 | 4 | 5

Remark: The set of integers does not include fractions. The positive integers lie to the right of zero on a number line, and the negative integers lie to the left of zero. Like whole numbers, integers on the number line are ordered from left to right with the smaller numbers to the left and larger numbers to the right. Integers can be compared using the symbols for "less than" or "greater than":

­4 < ­ 3 < ­ 2 < ­1 < 0 < 1 < 2 < 3 etc.

These inequalities tell us that ­4 is less than ­3, which is less than ­ 2, and so forth. The number zero is greater than any negative number and less than any positive number. Any positive number is greater than any negative number. An intuitive way to understand the ordering of integers is to think of temperature. For example, ­2 degrees is colder than 0 degrees and ­2 < 0 (or 0 > ­2). A thermometer can be thought of as a number line that includes negative numbers. Usually the number line on a thermometer is vertical instead of horizontal. Number lines with different scales can be used to display larger or smaller numbers than those displayed on the number line above. | ­40 | ­30 | ­20 | ­10 | 0 | 10 | 20 | 30 | 40 | 50 | 60

Money amounts can also be used to help students understand negative numbers. Positive integers can represent the number of dollars you have, and negative integers can represent the number of dollars you owe to someone else. Accountants use negative numbers this way. Remark: Negative numbers are important in science and engineering. For example, in the study of electricity, negative numbers are associated with negative charges and positive numbers are associated with positive charges. Small particles of matter called

atoms consist of electrons, protons, and neutrons. Electrons have a negative charge, protons have a positive charge, and neutrons have no charge, or zero charge. Fifth graders learn how to add and subtract integers.

Addition

Addition of integers may be explained with a number line. We begin with the addition of two positive integers. This builds on earlier grade experience. The addition 2 +3 = 5 can be represented this way:

3 2 | ­4 | ­3 | ­2 | ­1 | 0 | 1 2+3=5 The first arrow has length 2 and the second has length 3. Both arrows point to the right because both 2 and 3 are positive. The same idea applies when one or both of the addends (the numbers being added) are negative. Arrows for negative addends point to the left, as the arrow for ­3 in this example: ­3 + 2 = ­1. | 2 | 3 | 4 | 5

2 ­3 | ­4 | ­3 | ­2 | ­1 | 0 | 1 ­3 + 2 = ­1 | 2 | 3 | 4 | 5

When both addends are negative, both arrows point to the left, as for ­3 + (­2) = ­5.

­2 ­3 | ­5 | ­4 | ­3 | ­2 | ­1 | 0 | 1 | 2 | 3 | 4

­3 + (­2) = ­5

The addition problem 3 + (­2) = 1 gives the same result as the subtraction problem, 3 ­ 2 = 1. Both can be illustrated this way: ­2 3 | ­4 | ­3 | ­2 | ­1 | 0 | 1 | 2 | 3 | 4 | 5

3 + (­2) = 1 and 3 ­ 2 = 1 In order to understand how to subtract integers, we need a new concept: the opposite of an integer. Every integer has an opposite. The opposite of any positive integer is a negative integer, and the opposite of any negative integer is a positive integer. For example, the opposite of 1 is ­ 1, and the opposite of 2 is ­2. The opposite of ­3 is 3, and the opposite of ­17 is 17. The opposite of the number 0 is 0; zero is its own opposite.

| ­4

| ­3

| ­2

| ­1

| 0

| 1

| 2

| 3

| 4

| 5

Opposites identified by arrows

Additive Inverse

When an integer is added to its opposite the result is zero. For example, 4 +(­4) = 0 and ­7 + 7 = 0. For this reason, the opposite of an integer A is sometimes called the additive inverse of A. Definition For any integer A, the symbol ­A means the opposite of A. For example, if A = 2, then ­2 is the opposite of 2. We already knew that. But suppose that A = ­2. The definition tells us that ­A is the opposite of ­2. In other words, ­A = ­(­2) is the opposite of ­2. Since the opposite of ­2 is 2, the definition tells us that: ­(­2) = 2 This statement just says that the opposite of the opposite of 2 is 2. Remark: The parentheses in ­(­2) help the reader to see two minus signs, but it is also correct to write ­ ­2. Both are equal to 2. What does ­0 mean? Since 0 is its own opposite, ­0 = 0 For any integer A, it is true that ­(­A) = A. For example, ­(­25) = 25. What is the meaning of three minus signs in front of a whole number? Think of ­(­(­5)) as ­(­A) when A = ­5, that is, ­(­(­5)) = ­(­A). Since the opposite of the opposite of ­5 is ­5, we see that ­(­(­5)) = ­5 Remark: Sometimes a plus sign (+) is placed in front of a number, purely for emphasis, as in the statement ­(­2) = +2. The meaning of +2 is just 2.

Subtraction

Now we are ready to subtract integers. Here is the definition of subtraction. Definition For any integers A and B, A ­ B means A + (­B). Remark: The minus sign (­) is used in two different ways: to indicate the opposite of a number, and to indicate the operation of subtraction. This can be confusing to students. In some texts, an elevated minus sign, as in ­4, is used to indicate "the opposite of," in this case the opposite of 4. The usual minus sign refers to subtraction, as in 5 ­ 4. With that convention, one would write "3 minus the opposite of 4" as "3 ­ ( ­ 4)." However, most people do not use different symbols, and we follow the usual convention here.

Let's look at some examples. First, let's compare familiar subtraction statements involving only positive integers to this definition. An example discussed above is 3 ­ 2 = 1. According to the definition of subtraction of integers, 3 ­ 2 means 3 + (­2) We saw earlier that 3 + (­2) = 1, so this new definition of subtraction is consistent with students' earlier understanding of subtraction. The value of this new definition of subtraction is that we get the same answers to subtraction problems as before when all numbers were positive, and we can extend the operation of subtraction to negative integers too. The definition allows us to change any subtraction problem to an addition problem. To illustrate, let's calculate ­3 ­ 4: ­3 ­ 4 means ­3 + (­4) A ­ B means A + (­B) (A = ­3 and B = 4) ­3 + (­4) can be found using a number line as described above, or after some practice, students can see directly that ­3 + (­4) = ­7. Therefore ­3 ­ 4 = ­7. As another example, consider the subtraction problem, ­3 ­ (­4), that is ­3 minus ­4. ­3 ­ (­4) means ­3 + (­(­ 4)) A ­ B means A + (­B) (A = ­3 and B = ­ 4) To evaluate ­3 + (­(­ 4)), we use the fact that ­(­ 4) = 4 (the opposite of the opposite of 4 is 4). Then, ­3 ­ (­4) = ­3 + (­(­ 4)) = ­3 + 4 = 1 The key to addition of integers is the number line, and the key to subtraction of integers is to write the subtraction problem as an addition problem.

Absolute Value

Another concept associated with integers is absolute value. The absolute value of an integer is its distance from zero on the number line. The absolute value of a number A is written as |A|. For example, the absolute value of ­3 is written as |­3|, and |­3| = 3 because the distance from ­3 to 0 is three units on a number line.

| ­4

| ­3

| ­2

| ­1

| 0

| 1

| 2

| 3

| 4

| 5

Likewise, |3| = 3 because the distance from 3 to 0 is three units. The absolute value of 0 is 0, that is, |0| = 0. The absolute value of any number except zero is positive.

Multiplication

Why is multiplication of integers defined so that "neg × pos = neg" and "neg × neg = pos"? Consider first a positive integer times a negative integer, for example, 3 × (­ 7). It is natural to require, 3 × (­ 7) = (­ 7) + (­ 7) + (­ 7) = ­21 3 groups of ­ 7 Even if 3 is replaced by a different positive integer, and ­ 7 is replaced by another negative integer, the argument is the same. The conclusion is: "pos × neg = neg". What about switching the order of the factors: "neg × pos"? E.g., what about ­ 7 × 3? If we want the Commutative Property for multiplication to hold for integers ( we do!), then, ­ 7 × 3 = 3 × (­ 7) = ­21 Again, using different integers does not change the basic argument. We conclude that, neg × pos = neg and pos × neg = neg The last case to establish is "neg × neg = pos." As with whole numbers, we assume that 0 multiplied by an integer is always 0. To find ­ 7 × ­ 3, start this way, 0=­7×0 0 = ­ 7 × [ 3 + ­ 3] If we want the Distributive property to hold for integers (and we do!), then 0=­7×3+­7×­3 Using the fact that we already know neg × pos = neg, and therefore ­ 7 × 3 = ­ 21, we get, 0 = ­ 21 + ­ 7 × ­ 3

A California 4th grade math standard is to know that "equals added to equals are equal". In other words, adding the same number to both sides of an equation results in an equation. Add 21 to both sides: 21 = 21 + (­ 21) + ­ 7 × ­ 3 So, 21 = ­ 7 × ­ 3 In other words, ­ 7 × ­ 3 = 21, a positive number. This same argument works to show that for any integers, "neg × neg = pos." These multiplication rules extend to rational numbers, and division rules follow from the multiplication rules. For example, the quotient of two negative numbers is positive. A result that is sometimes useful is the following theorem. It can be deduced from what has already been done, or independently: Theorem For any integer N, ­N = ­ 1 × N. proof. ­1 + 1 = 0 (­1 + 1) × N = 0 × N = 0 ­1×N+ 1×N=0 ­1×N+ N=0 Now add ­N to both sides of the equation to get, ­ 1 × N = ­N. The proof assumes the validity of basic properties, especially the Distributive Property. Corollary ­1 × ­1 = 1 proof. Let N = ­1 in the theorem. Corollary For any integers A and B, ­(A × B) = (­A) × B = A × (­B). In addition, (­A) × (­B) = A × B. The proof uses the theorem and the Associative and Commutative Properties for multiplication.

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