Integers

Integers
The set of integers consists of the counting numbers 1, 2, 3, …, their opposites 1, 2, 3, …, and the number 0. The negative integers are needed to answer problems like 3 − 5 = 2, as well as to record such quantities as the temperature below zero, the depth below sea level, how much has been lost, or how much is owed. In the beginning lessons of this chapter, a raised minus sign is used for negative integers and a raised plus sign is used for positive integers. Whether or not students are required to write the raised plus sign is left to the discretion of the teacher.

The integers can be represented on a number line that extends to the left and to the right. The positive integers are represented to the right of 0, and the negative integers are represented to the left of 0.

Numbers such as +3 and 3 are opposite numbers. It is important that students understand the following concept.

When opposites are added, the result is zero.

Example: +3 +3 =0

The absolute value of an integer m is denoted by |m|. The absolute value of a number is its distance from the number 0 on a number line. For example, |+3| = 3 and |3| = 3. Notice that 3 and +3 are the same distance from 0, so both integers have the same absolute value.

Comparing and Ordering Integers
When comparing and ordering integers, it is easiest for students to think about a number line. On a number line, the number farthest to the left is the least and the number farthest to the right is the greatest. Students need to develop the idea that the greater the absolute value of a negative integer, the smaller the number. For example, 332 < 2 and 332 < 299.

When adding integers with the same sign, the sum is equal to the sum of the absolute values of each addend and has the sign common to the addends.

Examples:
+5 + +2 = +7
5 + 2 = 7
Both addends are positive, so the sum is positive.
Both addends are negative, so the sum is negative.

When adding integers with different signs, the sign of the sum depends upon the sign of the number with the greater absolute value. This can be understood by regrouping and then using addition of opposites.

3 + +5 = 3 + (+3 + +2)
= (3 + +3) + +2
= 0 + +2 = +2
6 + +2 = (4 + 2) + +2
= 4 +(2 + +2)
= 4 + 0 = 4

A rule for adding integers with opposite signs is to subtract the lesser absolute value from the greater absolute value and then use the sign of the integer with the greater absolute value.

Examples:

Find 3 + +5.
5 − 3 = 2
|+5| > |3|, so the sum is positive.
3 + +5 = +2
Find 6 + +2.
6 − 2 = 4
|6| > |+2|, so the sum is negative.
6 + +2 = 4

Subtraction is the inverse operation of addition. Subtracting an integer is the same as adding its opposite.

Examples:

m5 = m + +5

6 − 5 = 6 + +5
Addends have different signs so find the difference. 6 − 5 = 1 Since |6| > |+5|, the sum is negative.
6 − 5 = 1
m+5 = m + 5

+6 − 5 = +6 + +5
Addends have the same signs so find the sum. 6 + 5 = 11 Addends are positive, so the sum is positive.
+6 − 5 = +11
+1 − +5 = +1 + 5
Addends have different signs so find the difference. 5 − 1 = 4 Since |5| > |+1|, the sum is negative.
+1 − +5 = 4
1 − +5 = 1 + 5
Addends have the same signs so find the sum. 1 + 5 = 6 Addends are negative, so the sum is negative.
1 − 5 = 6

Multiplication of Integers
The product of two integers can be found by finding the product of their absolute values and then using this rule.

The product is positive if the factors have the same signs.
The product is negative if the factors have opposite signs.

Examples:

+3 × +2 = +6
+3 × 2 = 6
3 × +2 = 6
3 × 2 = +6

Division of Integers
Since multiplication and division are inverse operations, the sign rules for multiplication also apply to division.

The quotient of two integers with the same signs is positive.
The quotient of two integers with opposite signs is negative.

Examples:

+6 ÷ +3 = +2
6 ÷ +3 = 2
6 ÷ 3 = +2
+6 ÷ 3 = 2

Teaching Model 11.2: Compare and Order Integers