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Grade 9

1.08 The real number system

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 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 repeating 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.

 

Practice questions

QUestion 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?

  1. Integers

    A

    Whole numbers

    B

    Integers

    A

    Whole numbers

    B

QUESTION 2

Using the diagram, complete the following statement.

A real number is either:

  1. a whole number or an irrational number.

    A

    an integer or an irrational number.

    B

    a rational number or an irrational number.

    C

    an integer or a rational number.

    D

    a whole number or an irrational number.

    A

    an integer or an irrational number.

    B

    a rational number or an irrational number.

    C

    an integer or a rational number.

    D

QUESTIOn 3

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

Select all that apply.

  1. $\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
 

Outcomes

9.B1.2

Describe how various subsets of a number system are defined, and describe similarities and differences between these subsets.

9.B1.3

Use patterns and number relationships to explain density, infinity, and limit as they relate to number sets.

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