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Class 9 The World of Numbers: cheat sheet
CBSE Mathematics. Real numbers are rational (p/q) or irrational. Decimals that end or repeat are rational; surds like √2 are not. Exponent laws work for fractional powers too.
Key ideas
- A rational number can be written p/q with integers p, q and q ≠ 0. Its decimal either ends or repeats.
- In lowest terms, p/q gives an ending decimal only when q has no prime factors other than 2 and 5.
- An irrational number has a decimal that never ends and never repeats, e.g. √2, √3, π.
- Every real number is a point on the number line, and every point is a real number.
- To rationalise 1/(√a + √b), multiply the top and bottom by √a − √b.
- Fractional powers are roots: a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ.
Remember
| Exponent laws | aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ |
| Zero and negative powers | a⁰ = 1; a⁻ⁿ = 1/aⁿ |
| Surds | √a × √b = √(ab); (√a + √b)(√a − √b) = a − b |
| Repeating decimal | x = 0.272727…: 100x − x = 27, so x = 27/99 = 3/11 |
Watch out
- Writing √(a + b) = √a + √b: √(9 + 16) = 5, not 7.
- Forgetting to put the fraction in lowest terms before checking for 2 and 5.
- Multiplying by √a + √b instead of the conjugate √a − √b.
Worked example
Write 73/80 in decimal form.
Working: 80 has only 2 and 5 as prime factors, so it terminates: 73 ÷ 80 = 0.9125
Answer: 0.9125
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