3.1 Introduction to integers  (Page 2/9)

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number    .

Order positive and negative numbers

We can use the number line    to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See [link] .

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers . 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. We write $a>b$ (read $a$ is greater than $b$ ) when $a$ is to the right of $b$ on the number line. This is shown for the numbers $3$ and $5$ in [link] .

The numbers lines to follow show a few more examples.

ⓐ

$4$ is to the right of $1$ on the number line, so $4>1.$

$1$ is to the left of $4$ on the number line, so $1<4.$

ⓑ

$\mathrm{-2}$ is to the left of $1$ on the number line, so $\mathrm{-2}<1.$

$1$ is to the right of $\mathrm{-2}$ on the number line, so $1>\mathrm{-2}.$

ⓒ

$\mathrm{-1}$ is to the right of $\mathrm{-3}$ on the number line, so $\mathrm{-1}>\mathrm{-3}.$

$\mathrm{-3}$ is to the left of $\mathrm{-1}$ on the number line, so $\mathrm{-3}<-1.$

Order each of the following pairs of numbers using $<$ or $\text{>:}$

1. ⓐ $14\_\_\_6$
2. ⓑ $\mathrm{-1}\_\_\_9$
3. ⓒ $\mathrm{-1}\_\_\_\mathrm{-4}$
4. ⓓ $2\_\_\_\mathrm{-20}$

Solution

Begin by plotting the numbers on a number line as shown in [link] .

ⓐ Compare 14 and 6.	$14\_\_\_6$
14 is to the right of 6 on the number line.	$14>6$

ⓑ Compare −1 and 9.	$\mathrm{-1}\_\_\_9$
−1 is to the left of 9 on the number line.	$\mathrm{-1}<9$

ⓒ Compare −1 and −4.	$\mathrm{-1}\_\_\_\mathrm{-4}$
−1 is to the right of −4 on the number line.	$\mathrm{-1}>\mathrm{-4}$

ⓓ Compare 2 and −20.	$\mathrm{-2}\_\_\_\mathrm{-20}$
2 is to the right of −20 on the number line.	$2>\mathrm{-20}$

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Find opposites

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 $\mathrm{-2}$ are the same distance from zero, they are called opposites    . The opposite of $2$ is $\mathrm{-2},$ and the opposite of $\mathrm{-2}$ is $2$ as shown in [link] (a). Similarly, $3$ and $\mathrm{-3}$ are opposites as shown in [link] (b).