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

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 number.

Real Numbers and Classification of Real Numbers

Real Numbers and Its Classifications

Classification of Real Numbers

  • Natural Numbers:

All the natural numbers are classified as real numbers. It includes all the counting numbers such as 1, 2, 3, 4,…

  • Whole Numbers:

Even whole numbers are also real numbers. To recall, numbers starting with zero are called whole numbers, like 0, 1, 2, 3, 4,…

  • Integers:

Whole numbers and negative of all natural numbers are collectively known as integers which are also real numbers. Examples of real numbers include -3, -2, -1, 0, 1, 2, etc.

  • Rational Numbers:

All the numbers that can be written in the form of p/q, where q≠0 are known as Rational numbers. Examples of rational numbers are ½, 3/4., etc.

  • Irrational Numbers:

The numbers which cannot be written in the form of p ⁄ q (simple fraction) are known as irrational numbers. Irrational numbers are non-terminating and non-repeating in nature.

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Properties of Real Numbers

There are four main real numbers properties which include commutative property, associative property, distributive property, and identity property. These properties are not only important for real number but are also important in algebra topic as well.

  • 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. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2, etc.

  • 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. An example of associative property is 5 + (2 + 3) = (5 + 2) + 3. In this case, both LHS and RHS are 10.

  • Distributive Property:

Take 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. For example, 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.

  • Identity Property:

There are additive and multiplicative identities. For addition: m + (- m) = 0. Similarly, for multiplication, a × 1 = 1 × a = a.

Further Reading:

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 integer)?

More Real Numbers Related Articles

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Practise This Question

When three lines intersect at three points, then how many angles will be formed?