9.1.2: Integers
- Page ID
- 64037
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Locate integers on a number line.
- Find the absolute value of a given number.
- Find the opposite of a given number.
Introduction
You've worked with numbers on a number line. You know how to graph numbers like 0, 1, 2, 3, etc. on the number line. There are other kinds of numbers that can be graphed on the number line, too. Let's see what they look like and where they are located on the number line.
Natural Numbers and Whole Numbers
In mathematics, it's sometimes helpful to talk about groups of things, which are called sets. Numbers can be grouped into sets, and a particular number can belong to more than one set.
You probably are familiar with the set of natural numbers, which are also called the counting numbers. These are the numbers 1, 2, 3, and so on—the numbers we use when counting.
The following illustration shows the natural numbers graphed on a number line.
The number line continues in both directions. The set of natural numbers only continues to the right, so you can include 6, 7, and so on, all the way up into the hundreds, thousands, and beyond. You can only show so much on one picture!
When 0 is added to the set of 1, 2, 3, and so on, it forms the set of whole numbers. These are called “whole” because they have no fractional parts. (A trick to help you remember which are natural numbers and which are whole numbers is to think of a “hole,” which can be represented by 0. The whole ("hole") numbers include 0, the natural numbers do not.)
The following illustration shows the whole numbers graphed on the number line.
Integers
When you work with something like temperature, you sometimes want to use numbers that are less than zero, which are called negative numbers. Negative numbers are written using a negative sign in front, such as -1, -5, and -30. These are read negative one, negative five, and negative thirty. (The negative sign should not be read as minus; minus means subtraction.)
The numbers greater than 0 are called positive numbers and can be written with or without the “+” sign. Notice that 0 is neither positive nor negative!
Integers are the numbers: …, -3, -2, -1, 0, 1, 2, 3, ... Notice that all of the whole numbers are also integers. The following illustration shows the integers graphed on the number line. The integers include zero and continue to the right and to the left.
The number 0 belongs to which of the following sets of numbers?
natural numbers
whole numbers
integers
- natural numbers only
- whole numbers only
- natural and whole numbers
- integers only
- whole numbers and integers
- Answer
-
- Incorrect. The natural numbers are 1, 2, 3, and so on. They don’t include 0. The correct answer is whole numbers and integers.
- Incorrect. While the whole numbers include 0, so do the integers. The correct answer is whole numbers and integers.
- Incorrect. The natural numbers are 1, 2, 3, and so on. They don’t include 0. The correct answer is whole numbers and integers.
- Incorrect. While the integers include 0, so do the whole numbers. The correct answer is whole numbers and integers.
- Correct. Both whole numbers and integers include 0, but the natural numbers do not.
Absolute Value and the Number Line
The number line below shows all the integers between and including -5 and 5. Notice that the positive integers go to the right: 1, 2, 3, and so on. The negative integers go to the left: -1, -2, -3, and so on.
The distance between a number’s place on the number line and 0 is called the number’s absolute value. To write the absolute value of a number, use short vertical lines (|) on either side of the number. For example, the absolute value of -3 is written as bar negative three bar.
Notice that distance is always positive or 0.
|-3| = 3, as -3 is 3 units away from 0 and |3|=3, as 3 is 3 units away from 0.
Here are some other examples.
\(\ \begin{array}{c}
|0|=0 \\
|-23|=23 \\
|6|=6 \\
|817|=817 \\
|-3,000|=3,000
\end{array}\)
Find \(\ |x|\) when \(\ x=-7\).
Solution
|
\(\ |x|\) \(\ |-7|\) |
Substitute -7 for \(\ x\) in the expression. |
| \(\ |-7|=7\) | Since -7 is 7 units from 0, the absolute value is 7. |
To locate an integer on the number line, imagine standing on the number line at 0. If the number is 0, you’re there. If the number is positive, face to the right toward numbers greater than 0. If the number is negative, face to the left toward numbers less than 0. Then, move forward the number of units equal to the absolute value of the number.
Find -4 on the number line. Then determine |-4|.
Solution
 |
Imagine standing at 0. Since -4 is negative, face to the left. Move 4 units from 0 in the negative direction. |
 |
Draw a dot on the number line at that location, which is -4. |
| \(\ |-4|=4\) | Direction moved does not affect absolute value, only the distance moved. |
Which point represents -2 on this number line?
- Answer
-
- Incorrect. This point is 4 units to the left of 0, so it represents -4. The point that represents -2 is 2 units to the left of 0. Point B is correct.
- Correct. Point B is 2 units to the left of 0, so it represents -2.
- Incorrect. This point is 1 unit to the left of 0, so it represents -1. The point that represents -2 is 2 units to the left of 0. Point B is correct.
- Incorrect. This point is 1 unit to the right of 0, so it represents 1. A negative number must be to the left of 0. The point that represents -2 is 2 units to the left of 0. Point B is correct.
- Incorrect. This point is 2 units to the right of 0. A negative number must be to the left of 0. The point that represents -2 is 2 units to the left of 0. Point B is correct.
Opposites
You may have noticed that, except for 0, the integers come in pairs of positive and negative numbers: 1 and -1, 3 and -3, 72 and -3, 72 and -72, and so on. Each number is the opposite of the other number in the pair: 72 is the opposite of -72, and -72 is the opposite of 72.
A number and its opposite are the same distance from 0, so they have the same absolute value.
\(\ |72|=72, \text { and }|-72|=72\)
The set of integers are all the whole numbers and their opposites.
What is the opposite of -29?
- Answer
-
The opposite of -29 is 29.
Summary
Some numbers are natural numbers (1, 2, 3, ...) or whole numbers (0, 1, 2, 3, ...). Whole numbers are also integers. There are other integers which are the opposites of the whole numbers (-1, -2, -3, ...). These negative numbers lie to the left of 0 on the number line. Integers are the whole numbers and their opposites. The absolute value of a number is its distance to 0 on the number line. Absolute values are always positive or 0.