Adding and Subtracting Integers

Integers and Rational Numbers

By the end of this lesson, you will be able to perform operations with integers, compare and order rational numbers, and solve problems involving absolute value.

Integers

Integers: ${\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ — all whole numbers and their opposites.

Number line: negative numbers are to the left of 0; positive numbers to the right.

Absolute value: the distance from zero; always non-negative.

$|5| = 5 \quad |-7| = 7 \quad |0| = 0$

Opposite: the number the same distance from 0 on the other side.

Opposite of $-4$ is $4$. Opposite of $9$ is $-9$.

Adding Integers

Rule 1: Same signs → add the absolute values; keep the sign.

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

$7 + 4 = 11$

Rule 2: Different signs → subtract the smaller absolute value from the larger; keep the sign of the larger.

$(-7) + 3 = -(7-3) = -4$ (negative wins — $|-7| > |3|$)

$8 + (-5) = +(8-5) = 3$ (positive wins — $|8| > |-5|$)

Subtracting Integers

Rule: Subtracting is the same as adding the opposite.

$$a - b = a + (-b)$$

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

$-4 - (-7) = -4 + 7 = 3$

$-6 - 3 = -6 + (-3) = -9$

Multiplying and Dividing Integers

Sign rules:

Signs	Result
Positive × Positive	Positive
Negative × Negative	Positive
Positive × Negative	Negative
Negative × Positive	Negative

Same signs → positive product. Different signs → negative product.

Examples: $(-4) \times (-5) = 20$ $(-6) \times 3 = -18$ $(-20) \div (-4) = 5$ $15 \div (-3) = -5$

Pattern with multiple negatives:

• Even number of negative factors → positive product
• Odd number of negative factors → negative product