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Lesson 1: Multiply with negatives

# Negative numbers: multiplication and division FAQ

## Why do we get a positive number when we multiply two negative numbers?

When we multiply or divide two negative numbers, the result is a positive number. This might seem strange at first, but it's important to remember that a negative sign in math is really just an instruction to change the direction of a number on a number line. So when we multiply or divide two negative numbers, we're reversing the direction twice, which brings us back to a positive number.
For example, we would show $3×4$ as $3$ jumps, each $4$ units long and to the right, because $4$ is positive. We would start at $0$, then reach $4$, $8$, and $12$. So $3×4=12$.
For example, we would show $3×-4$ as $3$ jumps, each $4$ units long and to the left, because $-4$ is negative. We would start at $0$, then reach $-4$, $-8$, and $-12$. So $3×-4=-12$.
When we multiply by a negative number, we go in the opposite direction than we normally would. So, we would show $-3×-4$ as $3$ backwards jumps, each $4$ units long. A regular jump of $-4$ would go to the left, so a backward jump goes to the right. We would start at $0$, then reach $4$, $8$, and $12$. So $-3×-4=12$.
Try it yourself with our Multiplying negative numbers exercise.

## What's the deal with negative signs in fractions?

The same rules apply to fractions as they do to whole numbers. A negative sign before a fraction means the whole fraction is opposite of the value without the negative sign. But we can also have a negative sign in the numerator (top of the fraction) or denominator (bottom of the fraction). So is the fraction negative or positive when there is more than one negative sign?
Each negative sign before the fraction or in the numerator is the same as multiplying the fraction by $-1$. For example, if we have the fraction $-\frac{-3}{8}$, that is the same as $\frac{3}{8}$ multiplied by $-1$ twice.
$\begin{array}{rl}-\frac{-3}{8}& =\frac{3}{8}\cdot -1\cdot -1\\ \\ & =\frac{3}{8}\cdot 1\\ \\ & =\frac{3}{8}\end{array}$
If the negative sign is in the denominator, that is the same as dividing the fraction by $-1$. For example, if we have the fraction $\frac{9}{-7}$, that is the same as $\frac{9}{7}$ divided by $-1$.
$\begin{array}{rl}\frac{9}{-7}& =\frac{9}{7}÷-1\\ \\ & =-\frac{9}{7}\end{array}$
Try it yourself with our Negative signs in fractions exercise.

## Are there any special rules for order of operations with negative numbers?

There are not any special rules, but we do need to know where a couple more operations fit.
The negative sign is the same as multiplying the term by $-1$, so it happens in the same stage as multiplication and division. For example, in the expression $-{3}^{2}$, we apply the exponent first, then the negative sign.
$\begin{array}{rl}-{3}^{2}& =-1\cdot 3\cdot 3\\ \\ {}^{=}-9\end{array}$
However, in the expression $\left(-3{\right)}^{2}$, the negative sign is inside the parentheses. So we apply the negative sign first, then the exponent.
$\begin{array}{rl}\left(-3{\right)}^{2}& =-3\cdot -3\\ \\ & =9\end{array}$
We've also learned about absolute value. So first of all, the absolute value symbols are a grouping symbol. We perform the operations inside the absolute value symbol during the same stage as we evaluate work inside of parentheses.
Then, we actually apply the absolute value during the same stage as we apply exponents and roots. That is because $|x|=\sqrt{{x}^{2}}$.
Let's try it all together. What is the value of $3-7\cdot |5-8|-{4}^{3}$?
$\begin{array}{rlr}3-7\cdot |5-8|-{4}^{3}& =3-7\cdot |-3|+4& \text{Grouping symbols}\\ \\ & =3-7\cdot 3-64& \text{Exponents and absolute value}\\ \\ & =3-21-64& \text{Multiplication}\\ \\ & =-82& \text{Addition and subtraction}\end{array}$