Integers: Learn It 2

Integers on a Number Line

Both positive and negative numbers can be represented on a number line. Recall that a number line created with only whole numbers starts at [latex]0[/latex] and shows the counting numbers increasing to the right as shown in the number line below. The counting numbers [latex](1, 2, 3, \ldots )[/latex] on the number line are all positive. We could write a plus sign, [latex]+[/latex], before a positive number such as [latex]+2[/latex] or [latex]+3[/latex], but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

This figure is a number line scaled from 0 to 6.

 

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with [latex]-1[/latex] at the first mark to the left of [latex]0,-2[/latex] at the next mark, and so on. Refer to the number line below for reference.

On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.

A number line with 0 in the middle. The scaling has positive numbers 1 to 4 to the right of 0 and negative numbers, negative 1 to negative 4 to the left of 0.

 

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.

Plot the numbers on a number line:

  1. [latex]3[/latex]
  2. [latex]-3[/latex]
  3. [latex]-2[/latex]

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 the number line below.

A number line. Above it, there is an arrow pointing to the right labeled increasing. Below the number line there is an arrow pointing to the left labeled decreasing.

 

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 [latex]a < b[/latex] (read [latex]a[/latex] is less than [latex]b[/latex] ) when [latex]a[/latex] is to the left of [latex]b[/latex] on the number line. We write [latex]a > b[/latex] (read [latex]a[/latex] is greater than [latex]b[/latex] ) when [latex]a[/latex] is to the right of [latex]b[/latex] on the number line.

The number [latex]3[/latex] is to the left of [latex]5[/latex] on the number line. So [latex]3[/latex] is less than [latex]5[/latex], and [latex]5[/latex] is greater than [latex]3[/latex].

A number line with points 3 and 5 labeled with dots. Below the number line is the statements 3 is less than 5 and 5 is greater than 3.

 

The numbers lines to follow show a few more examples.

A number line with points 1 and 4 labeled with dots.

 

[latex]4[/latex] is to the right of [latex]1[/latex] on the number line, so [latex]4>1[/latex].
[latex]1[/latex] is to the left of [latex]4[/latex] on the number line, so [latex]1<4[/latex].

 

A number line with points negative 2 and 1 labeled with dots.

 

[latex]-2[/latex] is to the left of [latex]1[/latex] on the number line, so [latex]-2<1[/latex].
[latex]1[/latex] is to the right of [latex]-2[/latex] on the number line, so [latex]1>-2[/latex].

 

A number line with points negative 3 and negative 1 labeled with dots.

 

[latex]-1[/latex] is to the right of [latex]-3[/latex] on the number line, so [latex]-1>-3[/latex].
[latex]-3[/latex] is to the left of [latex]-1[/latex] on the number line, so [latex]-3<-1[/latex].