Rational Numbers, Compound Interest, Expansions & Factorisation
1. Rational and Irrational Numbers
Rational Numbers
Numbers that CAN be expressed as p/q where p,q ∈ Z, q ≠ 0. Terminating decimals. Non-terminating RECURRING decimals.
Irrational Numbers
Numbers that CANNOT be expressed as p/q. Non-terminating NON-RECURRING decimals. √2, √3, π, e.
Key Results
- The sum or product of a RATIONAL and an IRRATIONAL is IRRATIONAL.
- √ab = √a × √b. √(a/b) = √a/√b.
- To RATIONALISE a denominator: multiply numerator and denominator by the CONJUGATE.
2. Compound Interest (Without Formula)
Concept
Each year, the interest is ADDED to the principal. The NEXT year's interest is calculated on the INCREASED amount. 'Interest on interest.'
Calculation by Successive Method
Compute interest year-by-year. Add to principal. Repeat. Best for 2-3 years.
Using the Formula
A = P(1 + r/100)ⁿ. CI = A — P. For half-yearly: rate halved, periods doubled.
Growth and Depreciation
- Population: Pₙ = P₀(1 + r/100)ⁿ. Same formula as CI.
- Depreciation: Vₙ = V₀(1 — r/100)ⁿ.
3. Expansions (Algebraic)
Key Identities
| Identity |
|---|
| (a + b)² = a² + 2ab + b² |
| (a — b)² = a² — 2ab + b² |
| (a + b)(a — b) = a² — b² |
| (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| (a — b)³ = a³ — 3a²b + 3ab² — b³ |
| a³ + b³ = (a + b)(a² — ab + b²) |
| a³ — b³ = (a — b)(a² + ab + b²) |
| (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca |
4. Factorisation
Methods
| Method | When to Use |
|---|---|
| Common factor | All terms share a factor |
| Grouping | Group terms. Factor each group. Find common binomial. |
| Difference of squares | a² — b² = (a+b)(a—b) |
| Sum/Difference of cubes | a³ ± b³ |
| Splitting middle term | For quadratics ax² + bx + c: find two numbers that multiply to ac and add to b |
| Perfect squares | a² ± 2ab + b² = (a ± b)² |
Factor Theorem
If f(a) = 0 for polynomial f(x), then (x — a) is a FACTOR of f(x). Useful for factorising cubic and higher-degree polynomials.
