Why does negative times negative equal positive?

When you multiply two negatives, think of it as reversing direction twice. If you go backward (negative) and then reverse course (another negative), you end up going forward (positive). In multiplication, two negative factors produce a positive product because the negatives cancel each other out.

Is negative times positive always negative?

Yes. By Rules of Signs, any product of one negative factor and one positive factor is always negative. It doesn't matter which factor is which or how large they are—negative times positive always gives negative, and positive times negative also always gives negative.

Does Rules of Signs apply to addition and subtraction?

No, Rules of Signs applies only to multiplication (and division). Addition and subtraction of negative numbers follow different rules. For example, (-3) + (-2) = -5, not 6 (which is what Rules of Signs would give for multiplication).

Can I use Rules of Signs for exponents like (-2)^3?

Yes. Exponents like (-2)^3 mean (-2) \times (-2) \times (-2), so you apply Rules of Signs to the repeated multiplication: (-2)(-2) = 4, then 4(-2) = -8. A useful shortcut is to count the negatives—an odd number gives negative, an even number gives positive.

Rules of Signs

Rules of Signs tell you whether the product of two numbers is positive or negative, which you need for solving equations with negative numbers.

(-a)(-b) = ab \qquad (-a)(b) = -ab

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What each symbol means

What Rules of Signs takes

Symbol	Meaning
$a$	A number or variable representing one of the factors in the multiplication; it has a sign (positive or negative) that matters—misreading the sign as opposite will flip the sign of your entire answer.
$b$	A number or variable representing the other factor in the multiplication; its sign is just as important as $a$ 's. If you treat it as having the opposite sign, your product's sign will be inverted.

When to use it

Use Rules of Signs when you multiply two numbers and need to determine the sign of the product.

Level

Usually taught in: Pre-Algebra

Worked examples

1. Multiply two negative integers

Problem

Simplify

(-3)(-4)

$(-3)(-4)$
We have two negative factors to multiply.
$= 3 \cdot 4$
By Rules of Signs, negative times negative becomes positive, so we can remove both negative signs.
$= 12$
Multiply the positive values: $3 \times 4 = 12$ .

Answer: $12$

When you multiply two negative numbers, the result is always positive. This is the first part of Rules of Signs: two negatives make a positive because the negatives cancel each other out.

2. Multiply a negative fraction by a positive integer

Problem

Simplify

\left(-\frac{2}{3}\right) \cdot 9

$-\frac{2}{3} \cdot 9$
We have a negative fraction multiplied by a positive integer.
$-\frac{2 \cdot 9}{3}$
By Rules of Signs, negative times positive equals negative, so we multiply the numerator by the integer and keep the negative sign.
$-\frac{18}{3}$
Calculate the numerator: $2 \times 9 = 18$ .
$-6$
Simplify the fraction: $18 \div 3 = 6$ , and the negative sign remains.

Answer: $-6$

This example combines Rules of Signs with fraction multiplication. The key insight is that negative times positive always gives negative, even when fractions are involved, and then you simplify the resulting fraction as usual.

3. Calculate total cost savings for a fence section

Problem

A rectangular garden is being fenced. One section of fencing has a cost of

-8

dollars per meter (representing a bulk discount). The section is 15 meters long. Find the total cost:

(-8) \cdot 15

$(-8) \cdot 15$
We multiply the cost per meter ( $-8$ , a savings) by the number of meters (15).
$= -8 \times 15$
This is negative times positive, so by Rules of Signs the result will be negative.
$= -120$
Calculate: $8 \times 15 = 120$ , and the negative sign remains.

Answer: $-120 \text{ dollars}$

In this real-world context, a negative cost (savings) multiplied by a positive length gives a negative total (net savings). Rules of Signs tells us that negative times positive always produces negative, which makes intuitive sense.

Common mistakes

Where Rules of Signs usually goes wrong

The mistake	Why it is wrong	The fix
(-2) + (-3) = 6	The student confused addition with multiplication—Rules of Signs applies only to multiplication, not addition.	Rules of Signs applies to multiplication only. For addition: $(-2) + (-3) = -5$ . For multiplication: $(-2) \times (-3) = 6$ .
(-4)(3) = 12	The student correctly applied Rules of Signs but forgot to include the negative sign in the final answer.	Negative times positive equals negative: $(-4)(3) = -12$ , not $12$ .
$(-6)(2) = -12$ but $(2)(-6) = 12$	The student thinks the order of the factors determines the sign, but multiplication is commutative—the sign never depends on order.	$(-6)(2) = (2)(-6) = -12$ always. By Rules of Signs, negative times positive always equals negative regardless of the order of the factors.

Tips and when to use something else

Memorize the pattern: like signs (negative × negative or positive × positive) give positive, and unlike signs (negative × positive or positive × negative) give negative.
Rules of Signs works for division too—negative divided by positive equals negative, just like the multiplication rules ( $-10 ÷ 2 = -5$ ).
With multiple negatives, apply Rules of Signs step by step: $(-2)(-3)(-4) = 6(-4) = -24$ . Or use the shortcut: count the negatives—an even count gives positive, odd gives negative.
Don't confuse this with the Distributive Property, which is a separate rule for expanding expressions like $-3(x + 2)$ .

Related formulas

Order of OperationsOrder of Operations is the set of rules that tells you which calculation to do first when you have multiple operations in one expression.Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.Absolute ValueAbsolute value measures the distance a number is from zero without regard to direction, so it is always positive or zero.Commutative PropertyThe Commutative Property says you can switch the order of numbers when adding or multiplying without changing the result.

Frequently asked questions

Why does negative times negative equal positive?: When you multiply two negatives, think of it as reversing direction twice. If you go backward (negative) and then reverse course (another negative), you end up going forward (positive). In multiplication, two negative factors produce a positive product because the negatives cancel each other out.
Is negative times positive always negative?: Yes. By Rules of Signs, any product of one negative factor and one positive factor is always negative. It doesn't matter which factor is which or how large they are—negative times positive always gives negative, and positive times negative also always gives negative.
Does Rules of Signs apply to addition and subtraction?: No, Rules of Signs applies only to multiplication (and division). Addition and subtraction of negative numbers follow different rules. For example, $(-3) + (-2) = -5$ , not $6$ (which is what Rules of Signs would give for multiplication).
Can I use Rules of Signs for exponents like $(-2)^3$ ?: Yes. Exponents like $(-2)^3$ mean $(-2) \times (-2) \times (-2)$ , so you apply Rules of Signs to the repeated multiplication: $(-2)(-2) = 4$ , then $4(-2) = -8$ . A useful shortcut is to count the negatives—an odd number gives negative, an even number gives positive.

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Reviewed 2026-09-18