1.06 Identify and represent integers

Integers are made up of positive and negative whole numbers, as well as the number 0. The positive sign (+) indicates a quantity that is greater than zero, while the negative sign (-) indicates a quantity that is less than zero. These signs are important in mathematics for determining the direction and value of quantities. Here is a partial list of the set of integers, which continues up toward positive infinity and down toward negative infinity.

\left\{\ldots ,-3,\,-2,\,-1,\,0,\,1,\,2,\,3,\,\ldots\right\}

We know that 1 is less than 2, that 2 is less than 3, and so on. We can apply that to negatives as well, for example -3 is less than -2.

When we are asked to identify if a number is an integer or not, we can ask ourselves, is it a positive or negative whole number, or 0? If the answer is yes, then the number is an integer.

Examples

We can identify and represent integers using a number line. A number line is a visual representation of numbers on a straight line. The straight line can be horizontal or vertical and can be extended to both directions. Number lines can be used to compare numbers and perform operations.

In this vertical number line, the numbers above zero are positive integers while the numbers below zero are negative integers.

A point plotted on the number line represents positive 1.

We can identify an integer on the number line by plotting a point at that integer. On the number line below we can see that the point is at the mark labeled with the integer -3. This means that the point on the number line is at -3.

However, not every number line has labels for all the marks. On the number line below, the location of the point on the number line has not been labeled.

In order to find the missing label we need to find the distance between each mark. We can do this by comparing the number of gaps between the existing labels with the distance between the integer labels.

In this case, we can see that there are three gaps between the labels of 0 and 3. This means that there is a distance of 3 units shared between three gaps, so each gap will be equal to 1 unit. Since the point is located one mark to the right of 3, the integer we are looking for is 1 unit greater than 3. So the location of the point is 4.

Examples

Example 3

Determine if the following number is an integer:

Worked Solution

Create a strategy

We can ask ourselves: is it a positive or negative whole number, or 0?

Apply the idea

We can see that the number that has been plotted is -10. Because -10 is a negative whole number, we know that it is an integer.

Example 4

Where is the point plotted on the number line?

a

Worked Solution

Create a strategy

Every second tick is labeled with a multiple of 4. Since there are 4 units between every second tick, each tick is 2 units apart.

Apply the idea

\text{Number} = -6

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b

Worked Solution

Create a strategy

Remember that numbers above zero are positive numbers and numbers below zero are negative numbers on a vertical number line.

Apply the idea

The point is located above zero on the vertical number line. The marked numbers are 0 and 5. The segment between 0 and 5 is divided into 5 parts so each part is equivalent to one unit.

The point is on the third mark. The integer is positive 3.