Negative Numbers — Rules for Adding & Multiplying

Negative numbers are numbers less than zero. To add them, decide whether the signs match. To multiply them, look only at whether the signs match or differ.

They show up whenever a value is below a reference point, such as a temperature below $0$, money you owe, or a position to the left of $0$ on a number line.

Rules For Adding And Multiplying Negative Numbers

If you only need the core rules, use these:

When you add numbers with the same sign, add their distances from zero and keep that sign.

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

When you add numbers with different signs, subtract the smaller distance from zero from the larger one, then keep the sign of the number with the larger distance from zero.

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

When you multiply two numbers with the same sign, the product is positive. When the signs are different, the product is negative.

$$(-4)(-2) = 8,\quad (-4)(2) = -8$$

What Negative Numbers Mean

A negative number is not just "a number with a minus sign." The sign tells you the value is on the opposite side of zero from a positive number, and the size tells you how far from zero it is.

That is why negative numbers are useful. They can represent being below sea level, owing money, moving left on a graph, or dropping below a baseline.

How To Add Negative Numbers

Addition becomes easier if you picture movement on a number line.

Adding a positive number means move right. Adding a negative number means move left.

For example,

$$-2 + (-3)$$

starts at $-2$ and moves $3$ units left, so the result is $-5$.

If the signs are different, the moves compete. In

$$-7 + 4$$

you start at $-7$ and move $4$ units right. You do not reach zero, so the answer is $-3$.

A reliable rule is:

1. Same signs: add the distances from zero and keep the common sign.
2. Different signs: subtract the distances from zero and keep the sign of the number farther from zero.

Worked Example: $-6 + 9$

Find:

$$-6 + 9$$

The signs are different, so do not add $6$ and $9$ directly. First compare their distances from zero. Since $9$ is farther from zero than $6$, the final answer will be positive.

Now subtract the distances:

$$9 - 6 = 3$$

So

$$-6 + 9 = 3$$

This is the key idea for adding signed numbers: with opposite signs, the larger distance from zero decides the sign of the answer.

Why A Negative Times A Negative Is Positive

The multiplication rule is different from the addition rule. For multiplication, you multiply the sizes first and then decide the sign from whether the signs match.

If the signs are different, the product is negative:

$$3 \cdot (-2) = -6$$

If the signs are the same, the product is positive:

$$(-3)\cdot(-2) = 6$$

For most school problems, this rule is enough:

1. Same signs give a positive product.
2. Different signs give a negative product.

Common Mistakes With Negative Numbers

One common mistake is using the multiplication sign rule for addition. In addition, the sign depends on which number is farther from zero when the signs are different. In multiplication, it does not.

Another mistake is dropping parentheses. For example,

$$-3^2 = -(3^2) = -9$$

but

$$(-3)^2 = 9$$

The parentheses change what is being squared.

A third mistake is keeping the wrong sign in expressions like $-10 + 6$. Since $10$ is farther from zero than $6$, the result is $-4$, not $4$.

When Negative Numbers Are Used

Negative numbers appear in temperature, elevation, finance, coordinate geometry, algebra, and physics. Once the sign rules feel natural, it becomes much easier to read equations correctly and avoid small sign errors that change the whole answer.

Try A Similar Problem

Try these without a calculator: $-4 + 11$, $-9 + (-2)$, and $(-5)(3)$. Say the rule before you calculate. If you want to check a longer expression step by step, try your own version in a solver and watch where the sign changes.