Negative Numbers & Integer Operations

The number line

Negative numbers extend the number line to the left of zero. $-3$ is the same distance from zero as $3$, just on the other side.

The absolute value $|x|$ is the distance from zero — always non-negative.

$|5| = 5 \qquad |-5| = 5$

Adding and subtracting

Same signs: add the absolute values, keep the sign.

$-3 + (-4) = -7 \qquad 5 + 8 = 13$

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

$-7 + 3 = -4 \qquad 9 + (-4) = 5$

Subtraction is just adding the opposite: $a - b = a + (-b)$.

$5 - 8 = 5 + (-8) = -3$ $-3 - (-7) = -3 + 7 = 4$

Multiplying and dividing

The sign rule is symmetric: same signs → positive, different signs → negative.

$(-3) \times (-4) = 12$ $(-3) \times 4 = -12$ $\dfrac{-15}{-3} = 5$ $\dfrac{-15}{3} = -5$

Multiplying or dividing by a negative flips the sign of the result.

Worked example

$-6 + 3 \times (-2) - (-8)$

Order of operations: multiplication first.

1. $3 \times (-2) = -6$. Expression: $-6 + (-6) - (-8)$.
2. Left to right: $-6 + (-6) = -12$. Then $-12 - (-8) = -12 + 8 = -4$.

Key takeaways

• $|x|$ is the distance from zero — never negative.
• Subtracting is adding the opposite: $a - b = a + (-b)$.
• Two negatives multiply/divide to a positive; mixed signs give a negative.