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.

A number line from -4 to 4 with rational numbers. Ask your teacher for more information.

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

They cannot all 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. We can use a decimal approximation or a decimal expansion to determine if a number is irrational. The decimal expansion of an irrational number will be a decimal that does not terminate or repeat.

A number line from -4 to 4 with irrational numbers. Ask your teacher for more information.

Even with those added examples, we can see that the number line is still not "full". In fact, we can never truly "fill" a number line because between any two real numbers, there is always another real number. This is called the density property.

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?

Integers

Whole numbers

Worked Solution

Create a strategy

If the height above sea level is a positive value, this implies that the sea level can be represented by zero. Anything below sea level can be represented by a negative value.

Apply the idea

The position of a submarine below sea level is negative. Whole numbers are counting number starting from 0, which do not include negative numbers.

The correct option is letter A.

Reflect and check

Other subsets of the real numbers could also be used to represent a submarine's position relative to sea level. For example, a submarine could be 4.5 meters below sea level, or -4.5 meters deep. The number -4.5 is a rational number because it has a terminating decimal.

Example 2

Using the diagram, complete the following statement:

A real number is either:

a whole number or an irrational number.

an integer or an irrational number.

a rational number or an irrational number.

an integer or a rational number.

Worked Solution

Create a strategy

Use the diagram to consider how we can split the real numbers without excluding any numbers.

Apply the idea

A: This is not correct because real numbers also include numbers that are not whole, such as fractions and negative integers.
B: This statement is not correct because it excludes numbers that are not integers, like \dfrac{1}{2} or - \dfrac{3}{4}.
C: This is correct because real numbers are composed of exactly these two sets of numbers; every real number is either rational or irrational, covering all possibilities without overlap.
D: This statement is not correct because it excludes irrational numbers such as \pi, which are also real numbers.

The rational numbers and irrational numbers together include all real numbers.

The correct option is C.

Example 3

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.

\sqrt{49} is an irrational number.

\sqrt{49} is an integer.

\sqrt{49} is an rational number.

\sqrt{49} is a whole number.

Worked Solution

Create a strategy

Check to see if the square root can be simplified, then use the diagram to find the options that best apply.

Apply the idea

Since 49 is a perfect square, \sqrt{49}=\sqrt{7^2}=7.

Irrational numbers cannot be expressed as fractions. However, \sqrt{49} can be written as 7 or \dfrac{7}{1}, so it is not irrational.

Integers are numbers that can be written without a fractional component, such as 7, so \sqrt{49} is an integer.

Rational numbers can be expressed as fractions, such as \dfrac{7}{1}, so \sqrt{49} is rational.

Whole numbers are non-negative integers, such as 7, so \sqrt{49} is a whole number.

The correct options are letters B, C, D.

Example 4

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

Worked Solution

Create a strategy

To find the decimal expansion of \dfrac{7}{8}, divide 7 by 8 using long division or a calculator.

Apply the idea

\displaystyle 7\div 8

\displaystyle =

\displaystyle 0.875

The decimal expansion of \dfrac{7}{8} is finite or terminating, and is therefore a rational number.

Reflect and check

The decimal expansion of \dfrac{7}{8} is a terminating decimal because the denominator is a power of 2. Fractions whose denominators are powers of 2, or products of powers of 2 and 5 (e.g., 2, 4, 5, 8, 10, 16, 20, ...), always yield terminating decimals.

Example 5

Is \sqrt{35} rational or irrational?

Worked Solution

Create a strategy

Use a calculator to find and describe the decimal expansion of \sqrt{35}.

Apply the idea

The number \sqrt{35} when squared equals 35.

This number cannot be written as the quotient of two integers.

Using a calculator, \sqrt{35}= 5.9160797830996160. It has a decimal expansion that is infinite or non-terminating and that does not have a repeating pattern.

\sqrt{35} is an irrational number.

Reflect and check

A rational number can be expressed as a fraction \dfrac{a}{b}, where a and b are integers and b does not equal 0. However, the square root of a non-perfect square, like 35, does not simplify to a fraction of integers and therefore cannot be expresssed as a precise ratio.

Idea summary

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

Outcomes

8.NS.2

The student will investigate and describe the relationship between the subsets of the real number system.

8.NS.2a

Describe and illustrate the relationships among the subsets of the real number system by using representations (e.g., graphic organizers, number lines). Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural numbers.

8.NS.2b

Classify and explain why a given number is a member of a particular subset or subsets of the real number system.

8.NS.2c

Describe each subset of the set of real numbers and include examples and non-examples.

1.01 The real number system

Ideas

The real number system

Examples

Example 1

Example 2

Example 3

Example 4

Example 5

Outcomes

8.NS.2

8.NS.2a

8.NS.2b

8.NS.2c

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