medium difficulty6 min

Number Systems — Natural to Real

The number system is a family of number sets arranged from counting numbers to all real numbers: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

In one line: Counting to complete: natural grows into real.

Practice This Concept

Understand it

Natural numbers are used for counting, integers include negatives and zero, rational numbers can be written as p/q where q is not zero, and irrational numbers cannot be written in that form. Rational and irrational numbers together make the real number system.

Example

5 is natural, 0 is whole, -3 is an integer, 7/4 is rational, and √2 is irrational. All of these are real numbers.

Formula and rule

Hierarchy rule: ℕ ⊂ W ⊂ ℤ ⊂ ℚ ⊂ ℝ, and irrational numbers are also part of ℝ but not part of ℚ.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Classify 0, -5, 3/2, and √7.
2Here 0 is whole, integer, rational, and real; -5 is integer, rational, and real; 3/2 is rational and real; √7 is irrational and real.

Method you can reuse

1First check whether the number is used for counting.
2If not, check whether it is an integer.
3If it is a fraction p/q with q not zero, mark it rational.
4If it is non-terminating and non-repeating, mark it irrational.
5Finally, place it under real numbers.

Sketch it in your answer

Use a nested-set diagram showing natural numbers inside whole numbers, whole numbers inside integers, integers inside rational numbers, and rational and irrational numbers together inside real numbers.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Is -8 a rational number and a real number?

Think first, then show answer

Yes, -8 is a rational number because it can be written as -8/1, and it is also a real number because all rational numbers are real numbers.

Common wrong answer

Calling √9 irrational.

Why it loses marks: √9 = 3, and 3 is a natural, integer, rational, and real number.

Don't mix this up

Students often say every integer is a natural number, but negative integers and zero are not natural numbers in the usual school convention.

Exam tip

When asked to classify a number, place it in the smallest correct set first and then mention the larger sets it belongs to.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to identify the smallest set for a given number or arrange number sets in correct inclusion order.

1
Enough for 1 mark
The number system is a family of number sets arranged from counting numbers to all real numbers: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
2
Add for 2 marks
Example: 5 is natural, 0 is whole, -3 is an integer, 7/4 is rational, and √2 is irrational. All of these are real numbers.
3
Add for 3 marks
Natural numbers are used for counting, integers include negatives and zero, rational numbers can be written as p/q where q is not zero, and irrational numbers cannot be written in that form.

MCQ Quiz

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