1.01 The real number system

Our real numbers system hasn't been around in its current state forever. It was developed slowly over time.

The real number system includes rationals, irrationals, integers, whole numbers, and natural numbers.

Classification of numbers is about identifying which set, or sets, a number might belong to. It might be helpful to remember the different types of numbers as a story about filling in the numbers on a number line.

The first numbers we put on the line are the natural numbers. The set of natural numbers are the counting numbers, starting from 1: 1, 2, 3, 4, 5, 6, 7, ...

Next, we will add 0 to our line to show the whole numbers. The set of whole numbers are the counting numbers, starting from 0:0, 1, 2, 3, 4, 5, 6, 7, ...

The left side of the lines above look pretty empty. If we add all the negatives we now have a set of numbers called the integers. Whole numbers together with negative numbers make up the set of integers: ..., -4, \, -3, \, -2, \, -1, \, 0, \, 1, \, 2, \, 3, \, 4, ...

But are there numbers between the ones we already have marked on the above number lines? The answer is yes - an infinite amount of numbers between every little mark.

What sort of numbers are these? Well, rational numbers are all numbers that indicate whole numbers as well as parts of whole numbers. So fractions, decimals, and percentages are added to our number line to create the set of rational numbers.

A rational number is a number that can be expressed as a fraction where both the numerator and the denominator in the fraction are integers, and the denominator is not equal to zero.

Integers together with all fractions (including repeating or terminating decimals) make up the set of rational numbers.

They cannot be listed, but here are some examples:

..., -8, \, -7.4, \, -7, \, -6, \, -5.333 \, 87, \, -4, \, -2, \, 0, \, \dfrac{1}{2}, \, 75\%, \, 1, \, 2, \, 3, \, 3.565 \,6 , \, ...

But wait, our number line is still not quite full, there are still gaps. These gaps are filled with numbers we call irrational numbers. These are numbers like \sqrt{21} and \pi:

Now we can revisit the picture from above of the different sets of numbers in the real number system. Looking at the image, we can see some examples of numbers that belong to each set:

Notice that some number sets are entirely contained within larger number sets. For example, all of the whole numbers like 1,\, 2, \, 3, \, 17, \, 28 \, 736, ... etc. are also integers. But there are some integers, -1, \, -2, \, -56, \, -98\, 324 that are not whole numbers.

Similarly, rational numbers are also real numbers, but the set of real numbers includes all the rational numbers and all the irrational numbers.

Examples

Example 1

Height above sea level is expressed as a positive quantity. Which set of numbers is the most appropriate for describing the position of a submarine relative to sea level?

A

Integers

B

Whole numbers

Worked Solution

Create a strategy

Choose the option that is the best way of representing sea level.

Apply the idea

The position of a submarine below sea level is negative.

The correct option is letter A.

Watch question walkthrough

Example 2

Consider that we can express \sqrt{49} as \sqrt{7^2}.

Using the diagram, classify the number \sqrt{49}.

Select the three options that apply.

A

\sqrt{49} is an irrational number.

B

\sqrt{49} is an integer.

C

\sqrt{49} is an rational number.

D

\sqrt{49} is a whole number.

Worked Solution

Create a strategy

Compare and use the diagram to find the three options that best apply.

Apply the idea

Integers are numbers that can be written without a fractional component, such as -7.

Rational numbers can be expressed as fractions, such as \dfrac{1}{7}.

Whole numbers are non-negative integers, such as 7.

The correct options are letters B, C, D.

Watch question walkthrough

Every number has a decimal expansion. When we are asked to determine the decimal expansion of a number, we are expected to write the number in decimal form.

Numbers with decimal expansions that are infinite or non-terminating and that do not have a repeating decimal are irrational numbers. On the other hand, numbers with finite or terminating decimal expansions, and those numbers that are infinite with repeating decimals, are rational numbers.

Examples