Signed Number Arithmetic.

Why this matters

Every math topic above this — algebra, calculus, linear algebra, neural network gradients — relies on getting signs right. A sign error in step 2 propagates to a wrong answer in step 10. We slow down here so the rest is built on a clean foundation.

A signed number is a number with a direction:

• Positive ($+3$, or just $3$) — to the right of zero on a number line.
• Negative ($-3$) — to the left of zero.

The four rules

1. Adding numbers with the same sign

Add the values, keep the sign. $(-5) + (-3) = -(5+3) = -8 \qquad 7 + 4 = 11$

2. Adding numbers with different signs

Subtract the smaller value from the larger value, keep the sign of the larger. $(-7) + 4 = -(7-4) = -3$ $(-3) + 8 = 8 - 3 = 5$

3. Subtraction = adding the opposite

$a - b \;=\; a + (-b)$ $7 - (-4)$ becomes $7 + 4 = 11$. $-5 - 3$ becomes $-5 + (-3) = -8$.

4. Multiplication and division — same rule

• Same signs → positive: $(-5) \times (-2) = 10$, $\;-12 \div -3 = 4$.
• Different signs → negative: $(-5) \times 3 = -15$, $\;20 \div -5 = -4$.

A common trap

When you see $-3^2$, the exponent applies only to the $3$: $-3^2 = -(3 \cdot 3) = -9 \qquad (-3)^2 = (-3)(-3) = 9$ Parentheses change the meaning.