The Sets of Real Numbers

Numbers are arranged into different sets. A set is a collection of objects, and in this lesson the objects are numbers. Each number set has special rules about the types of numbers it contains.

In CSEC Mathematics, it is important to recognise the common number sets because they are used in algebra, graphs, equations, inequalities, functions, and number theory.

What you will learn

• Identify natural numbers, whole numbers, integers, rational numbers, irrational numbers and real numbers
• Classify numbers into the correct number sets
• Recognise that some number sets are contained inside larger number sets
• Distinguish between rational and irrational numbers

Key Vocabulary

Set	Symbol	Meaning	Examples
Natural numbers	ℕ	Counting numbers	1, 2, 3, 4, ...
Whole numbers	W	Natural numbers including zero	0, 1, 2, 3, ...
Integers	ℤ	Positive numbers, negative numbers and zero, with no fractions or decimals	..., −3, −2, −1, 0, 1, 2, 3, ...
Rational numbers	ℚ	Numbers that can be written as a fraction	5, 34, −2, 0.25, 0.333...
Irrational numbers	No single basic CSEC symbol	Numbers that cannot be written exactly as a fraction	π, √2, √3
Real numbers	ℝ	All rational and irrational numbers	All numbers on the number line

Key Ideas

How the Number Sets Are Connected

The number sets build on one another. This means that a smaller set is often part of a bigger set.

This means:

• Every natural number is a whole number.
• Every whole number is an integer.
• Every integer is a rational number.
• Every rational number is a real number.

Natural Numbers

Natural numbers are the numbers we use for counting.

Examples of natural numbers:

• 7
• 15
• 100

Non-examples:

• 0 is not usually counted as a natural number in many CSEC contexts.
• −4 is not natural because it is negative.
• 12 is not natural because it is a fraction.

Whole Numbers

Whole numbers are like natural numbers, but they also include zero.

Examples of whole numbers:

• 0
• 6
• 42

Non-examples:

• −1 is not whole because it is negative.
• 2.5 is not whole because it has a decimal part.

Integers

Integers include positive whole numbers, negative whole numbers and zero.

Examples of integers:

• −8
• 0
• 19

Non-examples:

• 35 is not an integer.
• 4.7 is not an integer.

Rational Numbers

A rational number is any number that can be written as a fraction.

Examples of rational numbers include:

• 34
• −52
• 7, because it can be written as 71
• 0.25, because it can be written as 14
• 0.333..., because it can be written as 13

Terminating and Recurring Decimals

Decimals can also be rational.

Type of Decimal	Meaning	Example	Ra­tional?
Termi­nating decimal	The decimal ends	0.75	Yes
Recur­ring decimal	The decimal repeats a pattern forever	0.333...	Yes
Non-termi­nating, non-recur­ring decimal	The decimal goes on forever without repeat­ing	π = 3.14159...	No

Irrational Numbers

Irrational numbers cannot be written exactly as a fraction. Their decimal forms go on forever without ending and without repeating a pattern.

Examples include:

• π
• √2
• √3
• √5

Real Numbers

The real numbers include all rational and irrational numbers. These are the numbers that can be placed on the number line.

Examples of real numbers:

• −6
• 0
• 23
• 4.25
• √2
• π

Worked Examples

Example 1: Classify −5.

1. −5 is negative, so it is not natural or whole.
2. It is an integer because it is a negative whole number.
3. It is rational because it can be written as −51.
4. It is real because all rational numbers are real.

Answer: −5 is an integer, rational number and real number.

Example 2: Classify 34.

1. 34 is a fraction.
2. It is not natural, whole or an integer.
3. It is rational because it is already written as a fraction.
4. It is also real.

Answer: 34 is rational and real.

Example 3: Classify √2.

1. √2 cannot be written exactly as a fraction.
2. Its decimal form goes on without ending and without repeating.
3. Therefore, it is irrational.
4. All irrational numbers are real.

Answer: √2 is irrational and real.

Example 4: Classify 0.

1. 0 is a whole number.
2. It is also an integer.
3. It is rational because it can be written as 01.
4. It is also real.

Answer: 0 is whole, integer, rational and real.

Example 5: Is 0.333... rational or irrational?

1. The decimal 0.333... repeats the digit 3 forever.
2. A recurring decimal is rational.
3. Also, 0.333... can be written as 13.

Answer: 0.333... is rational.

Quick Classification Table

Type of number	8	0	−4	56	√7	π
Natural	Yes	No	No	No	No	No
Whole	Yes	Yes	No	No	No	No
Integer	Yes	Yes	Yes	No	No	No
Rational	Yes	Yes	Yes	Yes	No	No
Irrational	No	No	No	No	Yes	Yes
Real	Yes	Yes	Yes	Yes	Yes	Yes

Common Mistakes

• Mistake 1: Thinking that zero is a natural number in all cases. In many school contexts, natural numbers start at 1.
• Mistake 2: Thinking that all decimals are irrational. Terminating and recurring decimals are rational.
• Mistake 3: Thinking that every square root is irrational. Square roots such as √4, √9 and √16 are rational because they simplify to whole numbers.
• Mistake 4: Forgetting that integers are also rational numbers because they can be written over 1.

Multiple Choice Questions

Practice Questions

1. State whether 7 is a natural number, an integer and a real number.
2. Is π rational or irrational? Explain.
3. To which number sets does −3 belong?
4. Give an example of a number that is real but not rational.
5. Classify 25 as rational or irrational.
6. Explain why 0.6 is rational.
7. State whether √16 is rational or irrational.
8. Write −9 as a fraction.

Answers

1. 7 is natural, integer and real. It is also whole and rational.
2. π is irrational because it cannot be written exactly as a fraction and its decimal does not terminate or recur.
3. −3 is an integer, rational number and real number.
4. Example: √2 or π.
5. 25 is rational because it is a fraction.
6. 0.6 is rational because it can be written as 610 or 35.
7. √16 = 4, so it is rational.
8. −9 can be written as −91.

Lesson Summary