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By the end of this section, you will be able to:
  • Use negatives and opposites
  • Simplify: expressions with absolute value
  • Add integers
  • Subtract integers

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers .

Use negatives and opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than 0 . The negative numbers are to the left of zero on the number line. See [link] .

A number line extends from negative 4 to 4. A bracket is under the values “negative 4” to “0” and is labeled “Negative numbers”. Another bracket is under the values 0 to 4 and labeled “positive numbers”. There is an arrow in between both brackets pointing upward to zero.
The number line shows the location of positive and negative numbers.

The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.

Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.

Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See [link] .

A number line ranges from negative 4 to 4.  An arrow above the number line extends from negative 1 towards 4 and is labeled “larger”. An arrow below the number line extends from 1 towards negative 4 and is labeled “smaller”.
The numbers on a number line increase in value going from left to right and decrease in value going from right to left.
Doing the Manipulative Mathematics activity “Number Line-part 2” will help you develop a better understanding of integers.

Remember that we use the notation:

a < b (read “a is less than b”) when a is to the left of b on the number line.

a > b (read “ a is greater than b ”) when a is to the right of b on the number line.

Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in [link] are called the integers. The integers are the numbers 3 , −2 , −1 , 0 , 1 , 2 , 3

A number line extends from negative four to four. Points are plotted at negative four, negative three, negative two, negative one, zero, one, two, 3, and four.
All the marked numbers are called integers .

Order each of the following pairs of numbers, using<or>: 14 ___ 6 −1 ___ 9 −1 ___ −4 2 ___ −20 .

Solution

It may be helpful to refer to the number line shown.
A number line ranges from negative twenty to fifteen with ticks marks between numbers. Every fifth tick mark is labeled a number. Points are plotted at points negative twenty, negative 4, negative 1, 2, 6, 9 and 14.


  1. 14 ___ 6 14 is to the right of 6 on the number line. 14 > 6


  2. −1 ___ 9 −1 is to the left of 9 on the number line. −1 < 9


  3. −1 ___ −4 −1 is to the right of −4 on the number line. −1 > −4


  4. 2 ___ −20 2 is to the right of −20 on the number line. 2 > −20
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Order each of the following pairs of numbers, using < or > : 15 ___ 7 −2 ___ 5 −3 ___ −7
5 ___ −17 .

> < > >

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Order each of the following pairs of numbers, using < or > : 8 ___ 13 3 ___ −4 −5 ___ −2
9 ___ −21 .

< > < >

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You may have noticed that, on the number line    , the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and −2 are the same distance from zero, they are called opposite     s . The opposite of 2 is −2 , and the opposite of −2 is 2.

Opposite

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

[link] illustrates the definition.

A number line ranges from negative 4 to 4. There are two brackets above the number line. The bracket on the left spans from negative three to 0. The bracket on the right spans from zero to three. Points are plotted on both negative three and three.
The opposite of 3 is −3 .

Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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