# Negative numbers: add, subtract, multiply, divide

## In a nutshell

A negative number is any number whose value is less than $0$. You show that a number is negative by placing a minus ($-$) sign before the number. Traditionally, the lack of a minus sign before a number implies it is positive, but to be specific you can place a plus ($+$) sign in front as well.

## Addition and subtraction

There are two rules for adding and subtracting positive and negative numbers. If two signs that are next to each other are the same, then the result is addition; if the two signs next to each other are different, then the result is subtraction.

**SIGN** | **OVERALL** |

$+ +$ | $+$ |

$+-$ | $-$ |

$-+$ | $-$ |

$--$ | $+$ |

##### Example 1

*Calculate *$2+(-2)-(-3)$*.*

*Change the double signs to the correct single sign and calculate the result.*

*$2-2+3=\underline3$*

##### Example 2

*Simplify the expression* $-a + -b +c - -d -e$.

*Change the double signs to the correct single sign.*

*$\begin{aligned}-a + (-b) +c-(-d) &= -a (+-)b+c(--)d \\ &=-a(-)b+c(+)d\\&=\underline{-a-b+c+d}\end{aligned}$*

## Multiplication and division

### Multiplying and dividing two numbers

When multiplying and dividing two positive or negative numbers, there are also rules to follow. If both are positive or negative (the signs are the same), then answer is positive and if one is positive and the other negative (the signs are different), then the answer is negative.

##### Example 3

*Calculate *$-3\times-7$.

*The signs are the same so the answer must be positive.*

$-3\times-7=\underline{21}$

##### Example 4

*Calculate *$-72\div9$*.*

*The signs are different so the answer must be negative.*

$-72\div9=\underline{-8}$

### Multiplying and dividing more than two numbers

When there are more than two numbers involved, you need to count the signs.

#### procedure

- Rewrite any negative numbers as a positive number $\times (-1)$.
- Multiply and/or divide the positive numbers.

- If there are an even number of $(-1)$s, then the answer remains positive. Otherwise, the answer becomes negative.

##### Example 5

*Calculate the following expression: *$3 \times (-33) \times (-24) \div (-12)$*.*

*Rewrite any negative numbers as a positive number *$\times \text{ }(-1)$*.*

$-39=39\times(-1)\\ \ \\-24=24\times(-1)\\ \ \\-12=12\times(-1)$**

*Multiply and divide using whole numbers.*

$\begin{aligned}(3\times33)\times(24\div12)&=99\times2\\&=198\end{aligned}$**

*There are three *$(-1)$*s, which is odd so the answer becomes negative.*

$3 \times (-33) \times (-24) \div (-12)=\underline{-198}$