Number Systems
Rational and Irrational Numbers
What is a Number Line?
A number line is a straight line with numbers placed at equal intervals. It helps us visualize the order and relationships between different types of numbers.
Types of Numbers on the Number Line:
| Number Type | Definition | Examples | Position on Number Line |
|---|---|---|---|
| **Whole Numbers** | Natural numbers including 0 | 0, 1, 2, 3, ... | Zero at center, all others right |
| **Integers** | Whole numbers and their negatives | ..., -3, -2, -1, 0, 1, 2, 3, ... | Both left (negative) and right (positive) of zero |
| **Rational Numbers** | Numbers that can be expressed as p/q where q ≠ 0 | 1/2, -3/4, 2.5, 0.333... | Between integers also, infinitely dense |
Important Properties:
- Every point on the number line represents a real number
- Between any two distinct rational numbers, there are infinitely more rational numbers
- The number line extends infinitely in both directions
- Rational = p/q with integers p, q and q ≠ 0.
- Irrational decimals are non-terminating and non-recurring (√2, π).
Decimal Expansions and the Number Line
What are Terminating and Recurring Decimals?
When we convert a rational number (p/q) to decimal form by dividing p by q, we get either:
Terminating Decimals: The division ends after a finite number of steps.
- Example: 1/2 = 0.5, 3/4 = 0.75, 7/8 = 0.875
- Occurs when denominator (in simplest form) has only prime factors 2 and 5
Recurring (Repeating) Decimals: The division never ends; a digit or block of digits repeats forever.
- Example: 1/3 = 0.333..., 2/7 = 0.285714285714...
- Represented with a bar: 0.3̅, 0.285714̅
Why Do Recurring Decimals Occur?
When denominator has prime factors other than 2 and 5 (like 3, 7, 11, etc.), the decimal repeats.
- Terminating or recurring ⇒ rational; non-recurring ⇒ irrational.
- √2, √3, √5 are placed using right-triangle (Pythagoras) constructions.
Exponents and Rationalising Denominators
What is Rationalization?
Rationalization is the process of eliminating irrational numbers (like √2, √3) from the denominator of a fraction. This makes expressions easier to work with and compare.
Why Rationalize?
- To simplify expressions
- To add/subtract fractions with irrational denominators
- To compare numbers easily
- Standard form in mathematics often requires rationalized denominators
Methods for Rationalization:
| Type of Denominator | Multiply Numerator & Denominator by | Result |
|---|---|---|
| a + √b | a - √b (conjugate) | Denominator becomes a² - b |
| √a + √b | √a - √b (conjugate) | Denominator becomes a - b |
The Conjugate Trick:
The conjugate of (a + √b) is (a - √b). Their product:
(a + √b)(a - √b) = a² - (√b)² = a² - b (a rational number)
- Surds follow aᵐ · aⁿ = aᵐ⁺ⁿ and a^(1/n) = ⁿ√a.
- Rationalise by multiplying by the conjugate of the denominator.