The Real Number System

Real numbers can be classified into two main categories: Rational Numbers and Irrational Numbers. These two categories together encompass all numbers that can be represented on a number line.

The following diagram shows the real number system and how real numbers can be classified. Scroll down the page for more examples and solutions on classifying real numbers.

 

How to classify real numbers?

1. Rational Numbers (Q)
   

• Real numbers that can be expressed as a fraction a/b, where a,b are integers and b ≠ 0.
  Subsets of Rational Numbers:
  
• Natural Numbers (N) or Counting Numbers: These are the positive integers starting from 1: {1, 2, 3, 4, …}.
  
• Whole Numbers (W): These include all natural numbers plus zero: {0, 1, 2, 3, …}.
  
• Integers (Z): These include all whole numbers and their negatives: {…, -3, -2, -1, 0, 1, 2, 3, …}.
  
• Terminating decimals: Decimals that end after a finite number of digits (e.g., 0.5, 3.75). These can be converted into fractions.
  
• Repeating decimals: Decimals that have a pattern of digits that repeats infinitely (e.g., 0.333…, 1.272727…). These can also be converted into fractions.
  

2. Irrational Numbers (Q’):
   

• Real numbers that cannot be expressed as a fraction a/b, where a,b are integers and b ≠ 0.
  Examples:
  Non-terminating and non-repeating decimals: Their decimal representations go on forever without any repeating pattern (e.g., π (pi) ≈ 3.14159…, √2 (square root of 2) ≈ 1.41421…).
  Square roots of non-perfect squares: Like √2, √3, √5, √6, etc.
  Certain transcendental numbers: Like π and e (Euler’s number).
  

Classifying Real Numbers
This video explains the different classifications of Real Numbers and has a number of examples.

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Ordering and Classifying Real Numbers

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Classifying Real Numbers
Defines the different subsets of Real Numbers we regularly work with and how to classify them using a Venn Diagram.

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Number Types (Math Song)
Natural numbers are naturally
Numbers you count on your hands and your feet
1, 2, 3, 4, 5, 6, and so on

Whole numbers are natural numbers plus zero
That’s the only difference, oh, oh
0, 1, 2, 3, and so on

Integers are whole numbers plus negatives
They look a little bit like this
-2, -1, 0, 1, and so on

Last but not least, not to be outdone
Rational numbers are integers plus fractions
Like -3, 1/2, 5/8 and 14.2

All natural numbers are whole numbers
All whole numbers are integers
All integers are rational numbers
Now listen to that drummer!

What about 4? It’s a natural number so it’s also whole, integer, and rational
What about -2? Integer and rational
What about 1.5? It’s just rational
What about 0? It’s whole, integer, and rational

All natural numbers are whole numbers
All whole numbers are integers
All integers are rational numbers
Now listen to that drummer!

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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