01 · DECIMAL EXPANSIONS
Types of Decimal Expansions
Every real number has a decimal expansion. They fall into three categories:
DECIMAL EXPANSION TYPES
Terminating vs Non-Terminating
1. Terminating Decimals (Rational): Remainder becomes zero after a finite number of steps (e.g. 7/8 = 0.875, 1/2 = 0.5).
2. Non-Terminating Recurring/Repeating (Rational): Remainder repeats periodically (e.g. 1/3 = 0.333... = 0.3̄, 1/7 = 0.142857142857...).
3. Non-Terminating Non-Recurring (Irrational): Remainder never becomes zero and never repeats (e.g. √2 = 1.41421356..., π = 3.14159265...).
02 · CONVERTING BAR DECIMALS TO P/Q
Converting 0.q̄ or 0.p q̄ to p/q Form
SOLVED EXAMPLE
Show that 0.333... = 0.3̄ can be expressed in the form p/q
Let x = 0.333... (Equation 1)
Since 1 digit repeats, multiply both sides by 10:
10x = 3.333... (Equation 2)
Subtract (1) from (2):
10x - x = 3.333... - 0.333...
9x = 3 ⟹ x = 3/9 = 1/3
Subtract (1) from (2):
10x - x = 3.333... - 0.333...
9x = 3 ⟹ x = 3/9 = 1/3
03 · RATIONALISATION
Rationalising the Denominator
When the denominator of an expression contains a square root, we multiply the numerator and denominator by its conjugate to make the denominator rational.
RATIONALISATION EXAMPLE
Rationalise the denominator of 1 / (2 + √3)
Conjugate of (2 + √3) is (2 - √3).
1 / (2 + √3) × (2 - √3) / (2 - √3)
= (2 - √3) / [2² - (√3)²]
= (2 - √3) / (4 - 3) = 2 - √3
= (2 - √3) / [2² - (√3)²]
= (2 - √3) / (4 - 3) = 2 - √3