第3章整数

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 Figure 3.10.

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 Figure 3.11.

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 $\mathrm{-3}<-1.$

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 Figure 3.13(a). Similarly, $3$ and $\mathrm{-3}$ are opposites as shown in Figure 3.13(b).

Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol $\text{“−”,}$ in three different ways.