Think of a horizontal line extending infinitely in both directions. This is our Number Line.
In the middle, we have zero (0).
To the right of zero are the positive integers: 1, 2, 3, 4, and so on.
To the left of zero are the negative integers: -1, -2, -3, and so on.
The complete set of integers, often denoted by the symbol Z, includes all these whole numbers:
Z = { …, -3, -2, -1, 0, 1, 2, 3, 4, … }
Adding integers might seem tricky at first because of the negative signs, but it follows two simple rules based on whether the signs of the numbers are the same or different.
If you are adding two integers that have the same sign:
Find the sum of their absolute numerical values (ignore the signs for a moment).
Assign the sign of the integer that has the greater magnitude (in this case, since they have the same sign, the result will simply keep that sign).
Examples:
$3 + 5 = 8$ (Both are positive, the result is positive)
$-3 – 5 = -(3 + 5) = -8$ (Both are negative, we add 3 and 5 to get 8, then assign the negative sign)
$12 + 13 = 25$
$-12 – 13 = -25$
If you are adding a positive and a negative integer:
Find the difference between their numerical values (subtract the smaller value from the larger value).
Assign the sign of the integer that has the greater magnitude.
Examples:
$-3 + 5 = +(5 – 3) = 2$ (5 is larger than 3, and 5 is positive, so the result is positive 2)
$3 – 5 = -(5 – 3) = -2$ (5 is larger than 3, and 5 is negative, so the result is negative 2)
$-12 + 13 = 1$ (13 is larger than 12 and is positive)
$12 – 13 = -1$ (13 is larger than 12 and is negative)
When you have more than two integers to add or subtract, you can apply these rules step-by-step or group similar terms.
Example 1: Step-by-Step
$12 + 13 – 38 – 56$
First, add the positive numbers: $12 + 13 = 25$
Next, “add” the negative numbers (sum of magnitudes with a negative sign): $-38 – 56 = -94$
Now, find the difference: $25 – 94 = -69$
Example 2: Different Grouping Methods
Expression: $12 + 38 – 11 – 32$
$(12 – 11) + (38 – 32) = 1 + 6 = 7$