Definition

  • Real numbers are all the numbers that can be found on the number line.
  • They include both rational and irrational numbers.
  • The symbol for real numbers is R.
  • Real numbers can be positive, negative, or zero.

Classification

  • Real Numbers are divided into two main types:
    • Rational Numbers
    • Irrational Numbers

Rational Numbers

  • Rational numbers are numbers that can be written in the form p/q, where p and q are integers and q ≠ 0.
  • All integers are rational numbers. (e.g., 3 = 3/1)
  • Terminating decimals are rational numbers. (e.g., 0.5 = 1/2)
  • Repeating or recurring decimals are rational. (e.g., 0.333… = 1/3)
  • Examples: 1/2, 5, -3, 0.25, 7.777…

Irrational Numbers

  • Irrational numbers cannot be expressed as p/q.
  • They have non-terminating and non-repeating decimal representations.
  • Examples include √2, π (pi), e, √3, etc.
  • π is a famous irrational number, approximately 3.14159…
  • The square root of a non-perfect square is always irrational.

Properties of Real Numbers

  • Real numbers are closed under addition, subtraction, multiplication, and division (except division by zero).
  • Commutative property holds for addition and multiplication.
  • Associative property also holds for addition and multiplication.
  • Distributive property of multiplication over addition: a × (b + c) = a×b + a×c
  • 0 is the additive identity, and 1 is the multiplicative identity.

Number Line Representation

  • All real numbers can be placed on a number line.
  • This includes integers, fractions, decimals, roots, etc.
  • Real numbers form a continuous number system without any gaps.

Set Relationships

  • Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
  • Irrational numbers and rational numbers together form the set of real numbers.
  • Real numbers ⊂ Complex numbers (in broader mathematics)

Examples

  • Rational: −3, 0, 1/5, 0.75, 6
  • Irrational: √2, π, √5, e

Applications in Real Life

  • Used in banking, engineering, geometry, and measurements.
  • Rational numbers are used for currency, data, and fractions.
  • Irrational numbers are used in scientific calculations and engineering designs.

Special Notes

  • 0 is a rational number (0 = 0/1).
  • Square roots of perfect squares are rational. (e.g., √9 = 3)
  • Every real number has a unique position on the number line.

Category

Questions