3.01 Identify and represent integers

The integers are made up of positive and negative whole numbers, as well as the number 0.

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

The integers to the right of 0 are positive integers and the integers to the left of 0 are negative integers.

0 is neutral. It has no opposite, so it is neither negative or positive.

The arrows at each end indicate that the line extends infinitely in the positive and negative directions.

We can identify an integer from a number line based on its distance from 0. Integers to the left of 0 are negative and integers to the right of 0 are positive.

\,\\\,\\\,

If we move the point at 0 to the right 4 units we get positive 4.

\,\\\,\\\,

If we move the point at 0 to the left 4 places we get -4.

There are other ways we can represent integers. Including vertical number lines, integer chips, and pictures.

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

The point plotted on the number line represents positive 1.

Integer chips can help us model integer expressions. Each chip represents a single integer unit. Integer chips come in all kinds of colors, in this case the green chips represent +1 and the red chips represent -1.

This grouping has 5 green chips representing +1 each. The total grouping represents +5.

This grouping has 7 red chips representing -1 each. The total grouping represents -7.

Examples

Example 1

Is each number an integer? -2, \, \dfrac{1}{2}, \, 0.4, 6, 3\dfrac{2}{5}

Worked Solution

Create a strategy

For each number, we can ask ourselves: is it a positive or negative whole number, or 0?

Apply the idea

• -2 is a negative whole number. So it is an integer.

• \dfrac{1}{2} is a positive fraction and is not a whole number. So it is not an integer.

• 0.4 is a positive decimal and is not a whole number. So it is not an integer.

• 6 is a positive whole number. So it is an integer.

• 3\dfrac{2}{5} is a positive mixed number. So it is not an integer.

Therefore, -2 and 6 are the only integers from the list of numbers.

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Example 2

Consider the number -10.

a

Represent -10 on a number line.

Worked Solution

Create a strategy

We can ask ourselves: is it a positive or negative whole number, or 0? This will help determine where the number belongs on the number lines.

Apply the idea

Since -10 is negative, place it 10 units left of 0 on the number line.

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b

Represent -10 with integer chips.

Worked Solution

Create a strategy

Decide on colors for your positive and negative integer chips. Determine whether you need positive or negative chips to represent -10.

Apply the idea

We need 10 negative integer chips to represent the number -10.

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Example 3

For each number line, determine where the point plotted is.

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

The point is 2 units to the left of -4. so it is at -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.

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.

Apply the idea

The point is located above zero on the vertical number line so it is positive.

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

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Integers are extremely useful numbers because they can be used to describe many things in the real world.

The positive sign \left(+\right) and negative sign \left(-\right) tell us important information about a real world situation.

For example, a positive temperature represents a temperature above zero, while a negative temperature represents a temperature below zero.

In some situations, the positive sign indicates an increase, while the negative sign indicates a decrease.

For example, let's say that a local coffee shop is increasing the prices of some of their menu items by \$ 1. We can quickly describe this situation by writing +1. The positive sign indicates an increase, and the 1 indicates going up by 1.

Similarly, a decrease in the price of a menu item by \$ 1 can be represented by the number -1.

Examples

Example 4

Write an integer to represent the statement:

A loss of \$ 52.

Worked Solution

Create a strategy

We will start by identifying whether our integer should be positive or negative based on the situation. Since we are representing a loss of money, we will have a negative integer.

Apply the idea

We know that our integer will be negative since we are representing a loss. Since it is a loss of \$ 52, the integer that represents the statement is -52.

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Example 5

Let the location of a city be represented by the integer 0, and let a point 7\text{ km} to the east of the city be represented by the integer 7. What integer represents the point 4\text{ km} to the west of the city?

Worked Solution

Create a strategy

Use a number line to represent the information.

Apply the idea

-4 represents the point 4\text{ km} to the west of the city.

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