UK Secondary (7-11)

Classification of Numbers

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 whole numbers (also called natural 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 all the negatives we now have a set of numbers called the integers.

Integers

Whole numbers together with negative numbers 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 recurring 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.

Using the diagram, classify the number $-42$−42.

Select all that apply.

$-42$−42 is an irrational number.

A$-42$−42 is a whole number.

B$-42$−42 is an integer but not a whole number.

C$-42$−42 is rational but not a whole number.

D$-42$−42 is an irrational number.

A$-42$−42 is a whole number.

B$-42$−42 is an integer but not a whole number.

C$-42$−42 is rational but not a whole number.

D

Roxanne determines that $\sqrt[3]{8}$^{3}√8 is an integer, and $\sqrt{30}$√30 is an irrational number.

Using the diagram, classify the number resulting from the addition $\sqrt[3]{8}+\sqrt{30}$^{3}√8+√30. Select ALL that apply.

The result will be a rational number.

AThe result will be a real number.

BThe result will be a whole number.

CThe result will be an integer.

DThe result will be an irrational number.

EThe result will be a rational number.

AThe result will be a real number.

BThe result will be a whole number.

CThe result will be an integer.

DThe result will be an irrational number.

E