Numbers & the Number System

Integers, factors, multiples and primes

The whole numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ are called integers. A factor of a number divides into it exactly, while a multiple is the result of multiplying it by an integer. A prime number has exactly two factors: $1$ and itself. Note that $1$ is not prime, and $2$ is the only even prime.

Key terms Factor — a number that divides exactly into another.

Multiple — the result of multiplying by an integer.

Prime — a number with exactly two distinct factors.

Prime factor — a factor that is also prime.

Every integer greater than $1$ can be written as a product of primes in exactly one way (the Fundamental Theorem of Arithmetic). We find this using a factor tree.

HCF and LCM by prime factorisation

The Highest Common Factor (HCF) is the largest number dividing into two numbers; the Lowest Common Multiple (LCM) is the smallest number both divide into. Write each as a product of primes, then for the HCF take the lower power of each shared prime, and for the LCM take the higher power of every prime present.

Worked example Find the HCF and LCM of $360$ and $84$.

$360 = 2^3 \times 3^2 \times 5$ and $84 = 2^2 \times 3 \times 7$.

A Venn diagram makes this visual: shared primes go in the overlap (their product is the HCF), and multiplying everything gives the LCM.

Fractions and decimals

To add or subtract fractions, use a common denominator; to multiply, multiply across (cancelling first); to divide, multiply by the reciprocal. Always change mixed numbers to improper fractions first.

Worked example Evaluate $2\tfrac{1}{4} + 1\tfrac{2}{3}$.

$\frac{9}{4} + \frac{5}{3} = \frac{27}{12} + \frac{20}{12} = \frac{47}{12} = 3\tfrac{11}{12}$.

Worked example Evaluate $\frac{3}{8} \div \frac{9}{10}$.

$\frac{3}{8} \times \frac{10}{9} = \frac{30}{72} = \frac{5}{12}$.

Terminating decimals convert directly to fractions over powers of $10$, e.g. $0.36 = \frac{36}{100} = \frac{9}{25}$.

Rounding, significant figures and estimation

Decimal places (d.p.) count digits after the point; significant figures (s.f.) count from the first non-zero digit. To estimate, round each number to $1$ s.f. and compute.

Watch out When rounding to s.f., keep place-value zeros: $3962$ to $2$ s.f. is $4000$, not $40$. The trailing zeros hold the size of the number.

Worked example Estimate $\dfrac{38.6 \times 5.1}{0.197}$.

$\approx \dfrac{40 \times 5}{0.2} = \dfrac{200}{0.2} = 1000$.

Upper and lower bounds

A value rounded to a given accuracy lies within half a unit of that accuracy. So a length of $12.4$ cm (to $1$ d.p.) has lower bound $12.35$ and upper bound $12.45$.

Key terms Lower bound — the smallest value that rounds to the stated figure.

Upper bound — the largest value (exclusive) that rounds to it; use the half-unit point in calculations.

For calculations, choose bounds to make the result largest or smallest:

Worked example $d = 84$ m (to $2$ s.f.), $t = 6.2$ s (to $1$ d.p.). Find the upper bound of speed $v = \frac{d}{t}$.

Standard form

Standard form writes a number as $a \times 10^n$ where $1 \le a < 10$ and $n$ is an integer. Large numbers have positive $n$; small numbers have negative $n$.

Worked example Write $0.0000408$ in standard form: $4.08 \times 10^{-5}$.

Worked example Evaluate $(3 \times 10^7) \times (8 \times 10^{-3})$.

$(3 \times 8) \times 10^{7+(-3)} = 24 \times 10^4 = 2.4 \times 10^5$.

Exam tip After multiplying or dividing, always re-adjust so that $a$ is between $1$ and $10$ — $24 \times 10^4$ is not in standard form.

Real world The mass of an electron is about $9.11 \times 10^{-31}$ kg, and the distance to the Sun is roughly $1.5 \times 10^{11}$ m — standard form lets us handle both without a screen full of zeros.

Laws of indices

Worked example Evaluate $27^{-\frac{2}{3}}$.

$= \dfrac{1}{27^{\frac{2}{3}}} = \dfrac{1}{\left(\sqrt[3]{27}\right)^2} = \dfrac{1}{3^2} = \dfrac{1}{9}$.

Worked example Simplify $\dfrac{12x^5}{3x^2}$.

$= 4x^{5-2} = 4x^3$.

Watch out A negative index does not make a number negative — it means a reciprocal: $5^{-2} = \frac{1}{25}$, not $-25$.

Surds

A surd is an irrational root left in exact form, such as $\sqrt{2}$. The key rules are $\sqrt{a}\times\sqrt{b} = \sqrt{ab}$ and $\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}$.

To simplify, take out the largest square factor:

Worked example $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$.

To add or subtract, simplify first, then collect like surds:

Worked example $\sqrt{50} + \sqrt{18} = 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$.

To rationalise the denominator, multiply top and bottom by the surd (or its conjugate):

Worked example $\dfrac{6}{\sqrt{3}} = \dfrac{6}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$.

Worked example Rationalise $\dfrac{4}{3+\sqrt{5}}$ using the conjugate $3-\sqrt{5}$.

Exam tip The conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$. Their product $a^2 - b$ is always rational — this is what clears the surd from the denominator.