Real Numbers

Real numbers are simply the combination of rational and irrational numbers. The concepts related to real numbers are explained here in detail along with examples and practice questions. The key real number concepts that are included in this article are:

  • Real Numbers Definition
  • Classification of Real Numbers
  • Properties of Real Numbers
  • Real Numbers Worksheet

Real Numbers Definition

Real numbers can be defined as the combination of both the rational and irrational numbers. Real numbers can be both positive or negative, and they are denoted by the symbol “R”. Numbers like a natural number, decimals, and fractions come under the real numbers category.

Real Numbers

Real Numbers and its Classifications

Classification of Real Numbers

Real numbers are classified under different categories. Below chart will help you to understand in a better way:

Category Definition Example
Natural Numbers Contain all counting numbers which start from 1.

N = {1,2,3,4,……}

All numbers such as 1, 2, 3, 4,5,6,…..…
Whole Numbers Collection of zero and natural number.

W = {0,1,2,3,…..}

All numbers including 0 such as 0, 1, 2, 3, 4,5,6,…..…
Integers The collective result of whole numbers and negative of all natural numbers. All are real numbers. Includes: -infinity,……..-4, -3, -2, -1, 0, 1, 2, 3, 4, ……+infinity
Rational Numbers Numbers that can be written in the form of p/q, where q≠0. Examples of rational numbers are ½, 5/4 and 12/6 etc.
Irrational Numbers All the numbers which are not rational and cannot be written in the form of p/q. Irrational numbers are non-terminating and non-repeating in nature like √2

Real Numbers Chart

Real Numbers

Real Numbers Chart

Properties of Real Numbers

There are four main real numbers properties which include commutative property, associative property, distributive property, and identity property. Consider m, n and r are real numbers.

Commutative Property:

If we have real numbers m, n then, the general form will be m + n = n + m for addition and m.n = n.m for multiplication.

  • Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2
  • Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2

Associative Property:

If we have real numbers m, n, r. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.

  • Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
  • Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).

Distributive Property:

For three real numbers as m, n, and r. Now, the distributive property of real numbers is in the form of m (n + r) = mn + mr and (m + n) r = mr + nr.

  • Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.

Identity Property:

There are additive and multiplicative identities.

  • For addition: m + 0 = m. (zero is additive identity)
  • For multiplication: a × 1 = 1 × a = a. (1 is multiplicative identity)

Real Numbers Worksheet (Questions)

  1. Which is the smallest composite number?
  2. Prove that any positive odd integer is of the form 6x + 1, 6x + 3, or 6x + 5.
  3. Evaluate 2 + 3 × 6 – 5
  4. What is the product of a non-zero rational number and irrational number?
  5. Can every positive integer be represented as 4x + 2 (where x is an integer)?

More Real Numbers Related Articles

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Practise This Question

A fraction is