Opposite

Same distance, other side

$$\text{opposite of } 4 = -4 \qquad \text{opposite of } -7 = 7$$

The minus sign means “the opposite of”, so taking it twice returns you to the start:

$$-(-7) = 7$$

Zero is its own opposite. It is the only number that is.

They add to zero

A number and its opposite are additive inverses:

$$6 + (-6) = 0 \qquad -14 + 14 = 0$$

Six steps right then six steps left ends where it began. A deposit of $50$ and a withdrawal of $50$ leave a balance unchanged.

Why subtraction depends on it

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

Every subtraction can be rewritten as adding the opposite, which means only one set of sign rules is ever needed:

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

The second line is the one worth remembering. Taking away a debt leaves you better off, so subtracting a negative increases the result. See adding and subtracting integers.

Not the same as a reciprocal

	Combines to	Example
opposite (additive inverse)	$0$ by adding	$5$ and $-5$
reciprocal (multiplicative inverse)	$1$ by multiplying	$5$ and $\tfrac{1}{5}$

Both are called inverses, and they undo different operations.