Integers
Integers
The set of integers consists of the counting numbers 1, 2, 3, ₀, their opposites ^{–}1, ^{–}2, ^{–}3, #8320;, and the number 0. The negative integers are needed to find the answer to exercises such as ^{+}5 − ^{+}9 = ^{–}4, as well as to record such quantities as the temperature below zero, the elevation below sea level, how much has been lost, and how much is owed. 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. Positive integers are represented to the right of 0 and negative integers to the left of 0. Integers such as ^{–}3 and ^{+}3 are opposites.
Students can compare integers by using a number line. On a number line, the integer farthest to the left is the least and the integer farthest to the right is the greatest.
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. Examples are |^{–}3| = 3, |^{+}3| = 3. Notice that ^{–}3 and ^{+}3 are both the same distance from 0, so both have the same absolute value.
Addition and Subtraction of Integers
As students learn to add and subtract integers, the use of counters and the number line will help them understand the process. As students become more proficient with addition and subtraction of integers, they should be able to make such computations without aids.
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
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.
Examples:
= (^{–}3 + ^{+}3) + ^{+}2
= 0 + ^{+}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:
5 − 3 = 2
|^{+}5| > |^{–}3|, so the sum is positive.
^{–}3 + ^{+}5 = ^{+}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:
m − ^{–}5 = 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 |
Teaching Model 22.1: Integers and Absolute Value