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 โ 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: โ2, โ3, ฯ
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: โ2 = 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 โn 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 + โ3 is irrational; 5 ร โ2 is irrational.
Rationalisation
To rationalise 1/(a + bโc), multiply numerator and denominator by the conjugate (a - bโc).
Example: 1/(โ2 + 1) = (โ2 - 1)/((โ2 + 1)(โ2 - 1)) = (โ2 - 1)/(2 - 1) = โ2 - 1
Laws of Exponents for Real Numbers
For positive real numbers a and b, and rational numbers p and q:
- a^p ร a^q = a^(p+q)
- (a^p)^q = a^(pq)
- a^p ร b^p = (ab)^p
- a^p / a^q = a^(p-q)
- a^(1/n) = โฟโa
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 โ 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 + โb, the conjugate is a - โb
Quick Revision
- N โ W โ Z โ Q โ R; Irrational numbers are also in R
- โ2, โ3, โ5, ฯ 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