# Arithmetic with Negative Numbers

## Adding and Subtracting

Adding a negative number is the same as subtracting a positive number. For example,

Subtracting a negative number is the same as adding a positive number. For example,

## Multiplying and Dividing

Multiplying a negative number by a positive number, or a positive number by a negative number, makes the answer negative.

For example,

It’s a good habit to write parentheses around negative numbers, as above. Doing so helps reduce preventable errors. Also, writing parentheses allows you to convey multiplication without using the multiplication symbol. It’s a good idea to omit the multiplication symbol, because in a handwritten context, it is too easily confused with the variable *x*. In this book, we will follow printed convention and often use the symbol, but you should make a habit of writing on your scratch paper (and ultimately your noteboard in the testing center) in this fashion:

and

Multiplying a negative number by a negative number makes the answer positive.

You do the same for dividing.

## Exponentiation

Taking a **negative number to an even power** generates a **positive number**.

In this case, writing the -3 in parentheses is required to communicate that we are squaring the negative number, not multiplying the squared number by -1.

Taking a **negative number to an odd power** generates a **negative number** every time.

For example,

or

The -1’s cancel out to become positive 1’s in pairs. When the exponent is odd, there’s always one of them left over, making the overall product negative.

## Order of Operations

As we saw above, parentheses are important when dealing with exponents of negative powers. We get a different result *without* parentheses,

…than we do *with* parentheses:

This difference is explained by “order of operations,” which is covered in an upcoming chapter. In those terms, the **e**xponent (“E” in PEMDAS) comes before **m**ultiplication by -1 (“M” in PEMDAS) unless we make the multiplication by -1 come first by putting it in **p**arentheses (“P” in PEMDAS).

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