1. Expressions & Operations

Lesson

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.

Natural numbers

The set of natural numbers are the counting numbers, starting from $1$1:

$1,2,3,4,5,6,7,\ldots$1,2,3,4,5,6,7,…

Next, we will add $0$0 to our line to show the whole numbers.

Whole numbers

The set of whole numbers are the counting numbers, starting from $0$0:

$0,1,2,3,4,5,6,7,\ldots$0,1,2,3,4,5,6,7,…

The left side of this line looks pretty empty. If we add the opposites of the whole numbers, we now have a set of numbers called the integers.

Notice that the opposites of the whole numbers are written with a negative sign, and they appear at the same distance from zero but on the other side.

Also notice that $-0$−0, the opposite of $0$0, hasn't been marked on this line. Where should it be? Why haven't we needed to include it?

Integers

Whole numbers together with their opposites make up the set of integers:

$\ldots,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,\ldots$…,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,…

But are there numbers between the ones we already have marked? 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.

Rational numbers

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:

$\ldots,-8,-7.4,-7,-6,-5.33387,-4,-2,0,\frac{1}{2},75%,1,2,3,3.5656,\ldots$…,−8,−7.4,−7,−6,−5.33387,−4,−2,0,12,75%,1,2,3,3.5656,…

But wait! Our number line is still not quite full. There are still gaps in a few places. These gaps are filled with numbers we call irrational numbers. These are numbers like $\sqrt{21}$√21 and $\pi$π:

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

Notice that some number sets are entirely contained within larger number sets. For example, all of the whole numbers like $1,2,3,17,28736,\ldots$1,2,3,17,28736,… etc. are **also** integers. But there are some integers, like $-1,-2,-56,-98324$−1,−2,−56,−98324, 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?

Integers

AWhole numbers

BIntegers

AWhole numbers

B

Using the diagram, complete the following statement.

A real number is either:

a whole number or an irrational number.

Aan integer or an irrational number.

Ba rational number or an irrational number.

Can integer or a rational number.

Da whole number or an irrational number.

Aan integer or an irrational number.

Ba rational number or an irrational number.

Can integer or a rational number.

D

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

Select all that apply.

$\sqrt{49}$√49 is an irrational number.

A$\sqrt{49}$√49 is an integer.

B$\sqrt{49}$√49 is a rational number.

C$\sqrt{49}$√49 is a whole number.

D$\sqrt{49}$√49 is an irrational number.

A$\sqrt{49}$√49 is an integer.

B$\sqrt{49}$√49 is a rational number.

C$\sqrt{49}$√49 is a whole number.

D