Operations of Integers

Integers: Addition and Multiplication

In this section we shall develop the definitions for adding and multiplying integers in such a way that all of the basic properties of natural numbers are also valid for the integers. Let us summarize those properties now, so that it will be clear just what our objective is. We want each of these to be true for every integer replacement of the variables.

For the positive integers, these objectives may be readily met by making an obvious agreement. Since a positive integer is also a natural number, we shall agree that the sum or product of two positive integers is found by adding or multiplying these natural numbers according to the rules already established for them.

Now consider the sum of two negative integers. How shall we define -3+-4? We may approach this problem indirectly by first considering the sum of 7 and (-3 + -4). The associative, inverse and identity properties will require that 7+ (-3+-4) = (4+3) + (-3 + -4) = (4+ -4) + (3 + -3) = 0 + 0 = 0. But this shows that the sum of 7 and (-3 + -4) is 0, and hence that 7 and (-3 + -4) are additive inverses. Then -3+-4 = -7, the additive inverse of 7.

Example 1 Use the above method to show that -5 + -8 = -13.

Solution 13 + (-5+-8) = (8+5) + (-5 + -8) = (8 + -8) + (5 + -5) = 0+0 =0. Since 13 + (-5 + -8) = 0, then -5 + -8 = -13, the additive inverse of 13.

Example 2 Find the simplest form of these sums.
(a) -10 + -2 (b) -5 + -5 (c) -73 + -215

Solution (a) -12 (b) -10 (c) -288

Next we shall consider several sums of positive and negative integers.

8 + -5 = (3 +5) + -5 = 3 + (5 + -5) = 3+0 = 3

Example 3 Use a similar method to find the simplest form.
(a) 8 + -10 (b) 8 + -11

Solution
(a) 8 + -10 = 8 + (-8 + -2) = (8 + -8) + -2 = 0 + -2 = -2
(b) 8 + -11 = 8 + (-8 + -3) = (8 + -8) + -3 = 0 + -3 = -3

Let us now take a close look at the sums we have so far considered.

We see that in every case the sum of two negative integers is a negative integer, while the sum of a positive integer and a negative integer is sometimes positive and sometimes negative. However, the absolute value of an integer can be used to state some simple rules which will make it easier to find sums of integers.

We first note that when adding two negative integers the absolute value of the sum is the sum of the absolute values of the terms. For example, -3 + -4 = -7. The absolute values of the terms are 3 and 4, while the absolute value of the sum is 7. Then we note that when adding a positive and a negative integer, the absolute value of the sum is the difference of the absolute values of the terms! For example, 8 + -5 = 3. The absolute values of the terms are 8 and 5, while the absolute value of the sum is 3, the difference of 8 and 5. Again with 8 + -10 = -2, the absolute values of the terms are 8 and 10, and the absolute value of the sum is 2, the difference 10-8. Finally we note that for the sum of a positive and a negative integer, whenever the positive term has the larger absolute value, the sum is positive, but whenever the negative term has the larger absolute value, the sum is negative. For example, 8 + -7 = 1 ; the positive term, 8, has the larger absolute value, and the sum is positive. But 8 + -11 = -3; the negative term, -11, has the larger absolute value, and the sum is negative.

These observations are actually quite general and can be collected into a simple two-part rule for adding integers:

Example 4 For each sum indicate: (a) whether the absolute values of the terms should be added or subtracted; (b) whether the sum is a positive integer or a negative integer; (c) the simplest form of the integer.

(i) -4 + 9 (ii) -8 + -2 (iii) 3 + -5

Solution

(i) (a) Subtract (b) Positive (c) 5
(ii) (a) Add (b) Negative (c) -10
(iii) (a) Subtract (b) Negative (c) -2

Now let us turn our attention to multiplication of integers. We shall first consider the product of a positive and a negative integer, (3) (-4). We may approach this product indirectly by first finding the sum of 12 and (3)(-4). 12 + (3)(-4) = (3)(4) + (3)(-4) = (3)(4 + -4) = 3∙0 = 0. But since the sum of 12 and (3)(-4) is zero, then (3) (-4) must be the additive inverse of 12, or (3)(-4) = -12.

Example 5 Show that (-5)(6) = -30.

Solution 30 + (-5)(6) = (5)(6) + (-5) (6) = (5 + -5)(6) = 0∙6 = 0.
Hence (-5)(6) = -30, the additive inverse of 30.

Similar treatment would give (-7)(4) = -28 and (12)(-3) = -36. Next consider the product of two negative integers, (-3)(-4). Again we will make an indirect approach by first considering the sum of this product and -12. We will use the above result that (3)(-4) = -12. Then (-3)(-4) + -12 = (-3)(-4) + (3)(-4) = (3 + 3)(-4) = (0)(-4) = 0. But since the sum of (-3)(-4) and (-12) is 0, then this product must be the additive inverse of -12. Hence (-3)(-4) = 12.

Example 6 Show that (-5)(-6) = 30.

Solution (-5)(-6) + -30 = (-5)(-6) + (-5)(6) = (-5)(-6 + 6) = (-5)(0) = 0. Therefore (-5)(-6) = 30, the additive inverse of -30.

Similarly we have (-12)(-3) = 36 and (-7)(-4) = 28.

Let us examine the products we have found so far.

(3)(-4) = -12 (-3)(-4) = 12
(-5)(6) = -30 (-5)(-6) = 30
(-7)(4) = -28 (-7)(-4) = 28
(12)(-3) = -36 (-12)(-3) = 36

Note that whenever one factor is positive and the other factor is negative, the product is negative. On the other hand we see that when both factors are negative the product is positive. Also note that in all cases the absolute value of the product can be obtained by multiplying the absolute values of the factors.

These observations are also quite general and we collect them into another two-part rule for multiplying integers:

Neither the rules for adding integers nor the rules for multiplying them have considered zero as a term or ag a factor. We do not need to do so, for our assumptions about the integers already yield. “For every x ∈ I , x+0= x and x∙0=0.”

Integers: Subtraction and Division

Recall that with natural numbers the operation of subtraction is defined as the inverse of addition. For example, we say that the difference of 17 and 9 is 8, since the sum of 8 and 9 is 17. In the statement 17-9 = 8 we call 17 the minuend, 9 the subtrahend, and 8 the difference. Then, in general, the difference of two natural numbers is a number whose sum with the subtrahend equals the minuend. We will adopt exactly this same definition for subtraction with integers.

Example 1 Use the definition to find the simplest form of these differences.
(a) 3 — -8 (b) -12 — 4 (c) -15 — -7

Solution
(a) 3 — -8 = 11, since 11 + -8 = 3
(b) -12 — 4 = -16, since -16 + 4 = -12
(c) -15 — -7 = -8, since -8 + -7 = -15

As the example shows, the definition can be easily used to “check’’ the difference, but it is not too useful for actually finding the difference. Consider again Example 1(c), -15 — -7 = -8. Let us form the sum of the minuend and the additive inverse of the subtrahend. Remember the minuend is -15 and the subtrahend is -7, so the sum we want is -15 +7 = -8. The fact that the difference of two integers is also equal to the sum of the minuend and the additive inverse of the subtrahend is generally true as is shown in the following theorem.

Theorem 1

Proof We shall use the definition, which requires that the sum of the difference and the subtrahend equals the minuend:

(a + -b) + b = a + (-b + b ) [Associative property of addition]
= a + 0 [Additive inverse]
= a [Additive identity]

Then we have shown that the sum of the difference (a + -b) and the subtrahend b, equals the minuend a.

Example 2 Use the theorem to find the simplest form.
(a) 14 — -9 (b) -8-7 (c) 0 — -19

Solution
(a) 14 — -9 = 14 + 9 = 23 (b) -8 -7 = -8 + -7 = -15
(c) 0 — -19 = 0 + 19 = 19

The expression 4 — -3 illustrates the two uses we have made of the ‘minus sign.’ The first minus sign indicates subtraction, while the second elevated minus sign indicates the additive inverse. Then 4 — -3 means the difference of 4 and the additive inverse of 3. And we have shown that 4 — -3 = 4 + 3 = 7. Often we will find it more convenient to indicate the additive inverse by using a minus sign which is not elevated. The context of its use will make it clear whether this symbol means to subtract or to find the additive inverse.

Example 3 Find the simplest form of each of the following:
(a) 4-2 (b) 7 + (-2) (c) (-5)(4) (d) -15 — (-1)

Solution

(a) Here — indicates subtraction and 4-2= 2.

(b) Here — indicates additive inverse, as the parentheses make clear, and 7+ (-2) = 5.

(c) Again, the parentheses used with the minus sign indicate the first factor is the additive inverse of 5. Thus (-5)(4) = -20.

(d) The first — indicates additive inverse since no minuend precedes 15. The second — is a subtraction symbol. And the third — again illustrates additive inverse. Thus -15 — (-1) = -15 + 1 = -14.

Example 4 Use the integers 7 and -4 to show that subtraction is not a commutative operation.

Solution (7) — (-4) = 11; (-4) — (7) = -11

Note that when the order of subtraction is reversed, the differences are additive inverses. The next theorem shows that this is generally true.

Theorem 2

Proof We form the sum of a-b and b-a. Then
(a-b) + (b-a)
= [ a+(-b) ] + [ b+(-a) ] [Theorem 1]
= [ a + (-b) + b ] + (-a) [Associative property of addition]
= (a + 0) + (-a) [Additive inverse]
= a+ (-a) [Additive identity]
= 0 [Additive inverse]

Since the sum of a-b and b-a is zero, we see that each is the additive inverse of the other. Hence a-b = -(b-a).

Example 5 Use the integers -4, 7, and -8 to show that subtraction is not an associative operation.

Solution [ (-4) — (7) ] — (-8) = (-11) — (-8) = -3;
(-4) — [ (7) — (-8) ] = (-4) — (15) = -19

We have shown by example that subtraction of integers is neither commutative nor associative. Is the set of integers closed with respect to subtraction? Theorem 2 says that the difference of two integers is also equal to the sum of the minuend and the additive inverse of the subtrahend. We have assumed that the additive inverse of every integer is also an integer, and we have assumed that the sum of any two integers is also an integer. Hence, we see that the difference of two integers is always an integer, and I is closed with respect to subtraction.

Example 6 Are these subsets of I closed with respect to subtraction?
(a) {0} (b) {2, -2, 3, -3, 4, -4, ...} (c) N

Solution
(a) Closed [ 0-0 = 0 ]
(b) Not closed [ 2-2 = 0, and 0 is not a member of the set]
(c) Not closed [ 2-5 = -3, and -3 is not a member of N ]

As with the natural numbers, we define division of integers to be the inverse of multiplication. Then (-15)÷(3)=-5, since (-5)(3)=-15. Similarly, -44/(-4)=11, since (11)(-4) = -44. In the statement -44/(-4)=11, we call -44 the dividend, -4 the divisor, and 11 the quotient.

Note that the product of the quotient and the divisor equals the dividend.

Example 7 Find the simplest form of the quotient.

(a) 45/(-9) (b) -15/(-5) (c) (27)÷(-5)
Solution

(a) 45/(-9)=-5, since (-5)(-9) = 45
(b) -15/(-5)=3, since (3)(-5) = -15
(c) There is no integer whose product with -5 equals 27.
Hence (27)÷(-5) is not an integer.

As is seen in the example, when the quotient of two integers is an integer, there are rules analogous to those for multiplication. That is, the quotient of two positive or two negative integers is positive, but the quotient of one positive and one negative integer is negative. And of course the last example shows that the set of integers is not closed with respect to division.

There is a problem with division when we try to use zero as the divisor. Consider 3÷0, which we may also write 3/0. Our definition requires that the quotient must be a number whose product with 0, the divisor, equals 3, the dividend. Perhaps the first impulse is to say that 3/0 = 0. But then we must have 0∙0=3, a false statement. If we try 3/0 = 3, we are lead to the equally false statement, 3∙0=3. In fact there is no number whose product with zero is 3, and hence we must conclude that 3÷0 is not defined. Thus 3/0 is not a numeral since it does not represent any number at all ! A similar argument will show that 1/0, 2/0, 4/0, 5/0, etc. and -1/0, -2/0, -3/0 etc. are not numerals and do not represent numbers.

The expression 0/0 presents a unique difficulty. We might write 0/0 = 0 since 0∙0=0. But we might also accept 0/0 = 1, since 1∙0=0. Similarly, it is reasonable to conclude 0/0 = 2 since 2∙0=0 and 0/0 = 17 since 17∙0=0 . We see that 0/0 is hopelessly ambiguous so we say that 0/0 is not a numeral since it may represent any number. In conclusion we shall agree that division by zero is undefined and if the denominator of a fraction is a symbol which represents zero, then that fraction does not represent any number at all, but is in fact undefined.

Example 8

Determine which of the following are defined, and find the simplest form for those which are.

(a) -7/(2-2) (b) ((-3)-(-3))/(3+3)

Solution

(a)-7/(2-2)=-7/0, not defined.

(b) ((-3)-(-3))/(3+3)=(-3+3)/6=0/6=0. Note that the numerator of a fraction may be zero.

(d) ((-2)(-3)-6)/(2(-3)+6)=(6-6)/(-6+6)=0/0, not defined.

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