Numbers and Their Operations covers prime factorisation, HCF and LCM, the real number system and the number line, the laws of indices, standard form, and approximation and estimation. Prime factorisation is the toolkit for the whole chapter: HCF and LCM, square and cube roots without a calculator, and later work on indices all depend on writing a number as a product of prime powers.
Almost every later topic in E-Math assumes you can handle negative numbers, fractions and surds fluently. In this topic, marks are rarely lost to genuinely hard questions; they are lost to a small set of recurring slips, such as swapped HCF and LCM powers, negative indices, standard form outside the allowed range, and rounding too early.
What these notes cover
- •Primes and prime factorisation
- •HCF and LCM from prime factorisation
- •The HCF–LCM relationship
- •Perfect squares, perfect cubes and their roots
- •The real number system
- •Sign rules and the number line
- •The five laws of indices
- •Zero, negative and fractional indices
- •Standard form
- •Decimal places and significant figures
- •Estimation and premature rounding
- •Common mistakes and exam technique
Key definitions
- Prime number
- A whole number greater than 1 that has exactly two different factors: 1 and itself. 1 is not prime, and 2 is the only even prime.
- Rational number
- A number that can be written as a fraction of two integers, a ÷ b with b ≠ 0. Every integer, terminating decimal and recurring decimal is rational, e.g. 0.6 = 3 ÷ 5.
- Irrational number
- A number that cannot be written as a fraction of integers; its decimal expansion goes on forever without repeating, e.g. √2 = 1.41421… and π. Rationals and irrationals together make up the real numbers.
- Standard form
- A number written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. n is positive for large numbers, negative for numbers between 0 and 1, and 0 for numbers from 1 up to (but not including) 10.
- Significant figures
- Digits counted from the first non-zero digit, reading left to right. Zeros between non-zero digits are significant, leading zeros are not, and trailing zeros after the decimal point are: 0.0030 has 2 s.f.
Key formulas
HCF–LCM relationship
Holds for two numbers only; for 4, 6 and 8 the HCF is 2 and the LCM is 24, but 2 × 24 ≠ 4 × 6 × 8. Use it to check working or to find the LCM once the HCF is known.
Product and quotient laws
Same base a only, with a ≠ 0 where division is involved. Add the indices when multiplying; subtract (top minus bottom) when dividing. If the bases differ, e.g. 2³ × 5², no law applies.
Power laws
A power of a power means multiply the indices; do not confuse this with the product law, where you add. The power distributes over every factor: (3x)² = 9x², not 3x². b ≠ 0 in the quotient.
Zero and negative indices
a ≠ 0. A negative index means reciprocal, never a negative answer: 2⁻³ = 1 ÷ 8. The index applies only to what it touches: 5x⁰ = 5, but (5x)⁰ = 1.
Fractional index
The denominator n is the root and the numerator m is the power; in numerical work take the root first. For a negative fractional index: minus sign → reciprocal, denominator → root, numerator → power.
Multiplying in standard form
For division use (A ÷ B) × 10ᵐ⁻ⁿ. Re-normalise if the front number leaves 1 ≤ A < 10, e.g. 14.4 × 10⁷ = 1.44 × 10⁸. To add or subtract, match the powers first; never add the indices.
Worked example
Without using a calculator, evaluate (2⁵ × 8⁻¹)² ÷ 4⁻³.
- 1Express everything in base 2: since 8 = 2³ and 4 = 2², the expression becomes (2⁵ × (2³)⁻¹)² ÷ (2²)⁻³.
- 2Apply the power law to each bracket-of-a-power: (2³)⁻¹ = 2⁻³ and (2²)⁻³ = 2⁻⁶, giving (2⁵ × 2⁻³)² ÷ 2⁻⁶.
- 3Apply the product law inside the bracket: 5 + (−3) = 2, so 2⁵ × 2⁻³ = 2², giving (2²)² ÷ 2⁻⁶.
- 4Apply the power law again: (2²)² = 2⁴, giving 2⁴ ÷ 2⁻⁶.
- 5Apply the quotient law, subtracting carefully: 4 − (−6) = 10, so 2⁴ ÷ 2⁻⁶ = 2¹⁰ = 1024.
Watch out: Dividing by 2⁻⁶ means subtracting −6, which adds 6 to the index. Working out 4 − 6 = −2 (giving 2⁻²) instead of 4 − (−6) = 10 is the most common slip in this question type, so always write the subtraction with brackets before simplifying the signs.
Common mistakes
HCF and LCM powers swapped
Wrong: HCF takes the highest powers and LCM takes the lowest.
Right: HCF = lowest power of each common prime; LCM = highest power of every prime that appears in either number. Check that HCF ≤ both numbers ≤ LCM and that HCF × LCM = a × b.
A negative index gives a negative answer
Wrong: 2⁻³ = −8.
Right: A negative index means reciprocal: 2⁻³ = 1 ÷ 2³ = 1 ÷ 8, which is positive. A positive base raised to any index stays positive.
Premature rounding
Wrong: Rounding √2 to 1.41 and then squaring gives 1.41² = 1.9881 ≈ 1.99 (3 s.f.).
Right: The true value is exactly 2, so the correct 3 s.f. answer is 2.00. Keep full calculator accuracy and round once, at the very end; if you must write an intermediate value down, keep at least 5 significant figures.
Questions students ask
How do you find the HCF and LCM using prime factorisation?
Write both numbers in index notation first, lining up the primes. The HCF uses only the primes common to both numbers, each to its lowest power; the LCM uses every prime that appears in either number, each to its highest power. For 540 = 2² × 3³ × 5 and 252 = 2² × 3² × 7, HCF = 2² × 3² = 36 and LCM = 2² × 3³ × 5 × 7 = 3780.
How do you find the smallest k so that 720k is a perfect square?
Prime factorise in index form and top up each odd index to the next even number; k is the product of the missing factors only. Since 720 = 2⁴ × 3² × 5, only the index of 5 is odd, so k = 5 and 720 × 5 = 3600 = 60². Students often lose marks by making k too big.
How do you add numbers in standard form?
Rewrite both numbers with the same power of 10, usually by matching the smaller power to the larger, then add the front numbers. For example, 7.4 × 10⁶ + 8.9 × 10⁵ = 7.4 × 10⁶ + 0.89 × 10⁶ = 8.29 × 10⁶. Never add the indices, because index laws apply to multiplication and division only.
What is the difference between significant figures and decimal places?
Decimal places count digits after the decimal point, zeros included; significant figures count digits from the first non-zero digit, reading left to right. The rounding decision is the same in both: if the first digit dropped is 5 or more, round up. For 0.03046, 3 s.f. gives 0.0305 while 3 d.p. gives 0.030.
In the full PDF
- •Factor-tree and repeated-division diagrams, plus the ladder method for finding HCF and LCM in one working
- •Number-line diagrams for inequalities and sign diagrams with critical points
- •Further worked examples on classifying numbers, mixed operations with negatives and fractions, ordering, standard form conversion and arithmetic, and 1 s.f. estimation
- •All eight classic traps, including the −3² calculator trap and standard form outside the allowed range
- •Calculator skills for the fraction, (−) sign, power and root, and × 10ˣ keys
- •Four exam-technique habits and a one-page rule bank summary