Learning Objectives
- Understand the classification of numbers: Natural, Whole, Integers, Rational, and Irrational
- Represent real numbers on the number line
- Learn about decimal expansions of real numbers
- Understand the laws of exponents for real numbers
- Rationalise the denominator of irrational expressions
Key Concepts
Classification of Numbers
Natural Numbers (N): Counting numbers 1, 2, 3, 4, ... (positive integers starting from 1).
Whole Numbers (W): Natural numbers including zero: 0, 1, 2, 3, ...
Integers (Z): All whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational Numbers (Q): Numbers that can be expressed in the form p/q where p and q are integers and q ≠ 0. Examples: 1/2, -3/4, 5, 0.75
Irrational Numbers: Numbers that cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating. Examples: √2, √3, π
Real Numbers (R): The collection of all rational and irrational numbers together form the set of real numbers.
Decimal Expansions
Terminating decimals: Decimal expansion ends after a finite number of digits. Example: 1/4 = 0.25. These are always rational.
Non-terminating repeating decimals: Decimal expansion repeats a pattern forever. Example: 1/3 = 0.333... These are always rational.
Non-terminating non-repeating decimals: Decimal expansion neither terminates nor repeats. Example: √2 = 1.41421356... These are always irrational.
Representing Real Numbers on the Number Line
Every real number corresponds to a unique point on the number line. To represent √n on the number line, use the method of successive construction of right triangles (spiral of Theodorus).
Operations on Real Numbers
The sum, difference, product, and quotient of a rational and an irrational number is irrational (except when multiplying or dividing by zero).
Example: 2 + √3 is irrational; 5 × √2 is irrational.
Rationalisation
To rationalise 1/(a + b√c), multiply numerator and denominator by the conjugate (a - b√c).
Example: 1/(√2 + 1) = (√2 - 1)/((√2 + 1)(√2 - 1)) = (√2 - 1)/(2 - 1) = √2 - 1
Laws of Exponents for Real Numbers
For positive real numbers a and b, and rational numbers p and q:
- a^p × a^q = a^(p+q)
- (a^p)^q = a^(pq)
- a^p × b^p = (ab)^p
- a^p / a^q = a^(p-q)
- a^(1/n) = ⁿ√a
Summary
Real numbers are the union of rational and irrational numbers. Every real number has a unique position on the number line. Rational numbers have terminating or repeating decimal expansions, while irrational numbers have non-terminating non-repeating expansions. Rationalisation removes irrational numbers from the denominator. The laws of exponents extend naturally to all real numbers.
Important Terms
- Rational Number: A number expressible as p/q where q ≠ 0 and p, q are integers
- Irrational Number: A number that cannot be written as p/q; its decimal is non-terminating and non-repeating
- Real Numbers: The set of all rational and irrational numbers
- Rationalisation: The process of eliminating irrational numbers from the denominator
- Conjugate: For a + √b, the conjugate is a - √b
Quick Revision
- N ⊂ W ⊂ Z ⊂ Q ⊂ R; Irrational numbers are also in R
- √2, √3, √5, π are irrational numbers
- Every rational number is a real number but not every real number is rational
- To rationalise, multiply by the conjugate of the denominator
- a^(1/n) means the nth root of a