﻿ Subtraction of Signed Numbers

# Subtraction of Signed Numbers

In this section, we study problems like $\,-3 - (-5)\,$; that is, problems of the form $\,x - y\,.$

The good news is that every subtraction problem is an addition problem in disguise! In one easy step, every subtraction problem is changed to an addition problem, which you already know how to solve.

It's important that you can recognize subtraction problems, and read them aloud correctly. There are several things that you should notice as you study the examples below:

• when a negative number is being subtracted, it goes inside parentheses
• the subtraction sign is read as minus
• a negative number like ‘$\,-3\,$’ is read as negative three
• even though the same symbol ‘$\,-\,$’ is used both for subtraction and for finding the opposite of a number, it is read in different ways

## Examples

 $3 - 5$ the number being subtracted is $\,5\,$; read aloud as three minus five $2 - (-3)$ the number being subtracted is $\,-3\,$; read aloud as two minus negative three $-1 - 6$ the number being subtracted is $\,6\,$; read aloud as negative one minus six $-2 - (-7)$ the number being subtracted is $\,-7\,$; read aloud as negative two minus negative seven

## How to Subtract a Number

To subtract a number, you add its opposite. To subtract $\,3\,,$ you add $\,-3\,.$ To subtract $\,-3\,,$ you add $\,3\,.$

That is, $\,x - y = x + (-y)\,,$ for all real numbers $\,x\,$ and $\,y\,.$ (A good way to read this is:   $\,x\,$ minus $\,y\,$   equals   $\,x\,$ plus the opposite of $\,y\,$)

## Three Steps in a Subtraction Problem

There are three steps in a subtraction problem. These steps are illustrated using this example: $\,-3 - (-5)$

1. Identify the number being subtracted.
Answer: $\,-5\,$
2. Find the opposite of the number being subtracted.
Answer: the opposite of $\,-5\,$ is $\,5\,$
3. Rewrite the subtraction problem as addition of the opposite.
Answer: $-3 - (-5) = -3 \ {\bf\color{green}{+}}\ \color{red}{5} = 2$

Here is a problem with more than two numbers. Notice that every subtraction is turned into an addition in the first step.

$-3 - 5 + (-2) - (-7) + 4$
$= -3 + (-5) + (-2) + 7 + 4$
$= -10 + 11$
$= 1$

## Practice

Here, you will practice subtraction problems of the form ‘$\,x - y\,$’ where $\,x\,$ and $\,y\,$ can be any of these numbers: $\,-10, -9, -8, \ldots, -1, 0, 1, \ldots, 8, 9, 10\,.$ About half of the problems will involve variables!