United States of America

Grade 8

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.

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:

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.

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

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

Idea summary

The real numbers include rational numbers, irrational numbers, integers, and whole numbers as shown in the diagram below:

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.

Describe the decimal expansion of \dfrac{7}{8}.

Worked Solution

Is \sqrt{35} rational or irrational?

Worked Solution

Idea summary

Rational numbers are numbers with finite or terminating decimal expansions, and numbers that are infinite with repeating decimals.

Irrational numbers are numbers with decimal expansions that are infinite or non terminating and that do not have a repeating decimal.