Topic 1: Number

Almost every question in IGCSE Mathematics depends on secure number skills, whether you are calculating a discount, simplifying an algebraic surd or interpreting a Venn diagram. Topic 1 brings together the arithmetic and number-theory tools that you will reuse in algebra, geometry, statistics and probability. The good news is that these are predictable, rule-based skills: once you know the procedures and practise them, the marks are reliable. This article works through each part of the topic with clear methods and worked examples so you can build the fluency examiners reward.

Types of Number, HCF and LCM

Numbers are classified into sets you must recognise. Natural numbers are the counting numbers $1, 2, 3, \\ldots$ Integers include zero and negatives: $\\ldots, -2, -1, 0, 1, 2, \\ldots$ A rational number can be written as a fraction $\\frac{a}{b}$ where $a$ and $b$ are integers and $b \\neq 0$; an irrational number, such as $\\pi$ or $\\sqrt{2}$, cannot. A prime number has exactly two factors, $1$ and itself, so $2$ is the smallest prime and the only even one. A square number is the result of multiplying an integer by itself, like $49 = 7^{2}$, and a cube number like $27 = 3^{3}$. Every integer greater than $1$ can be written as a product of prime factors. Use a factor tree, then collect repeats as powers. For example $360 = 2^{3} \\times 3^{2} \\times 5$. Prime factorisation is the key to finding the highest common factor (HCF) and lowest common multiple (LCM). Write each number in prime-power form: $60 = 2^{2} \\times 3 \\times 5$ and $84 = 2^{2} \\times 3 \\times 7$. For the HCF, take the lowest power of each shared prime: $2^{2} \\times 3 = 12$. For the LCM, take the highest power of every prime that appears: $2^{2} \\times 3 \\times 5 \\times 7 = 420$. A useful check is the identity $\\text{HCF} \\times \\text{LCM} = a \\times b$, since $12 \\times 420 = 5040 = 60 \\times 84$.

Fractions, Decimals and Percentages

These three are different ways of writing the same value, and you must convert fluently between them. To turn a fraction into a decimal, divide: $\\frac{3}{8} = 0.375$. To turn a decimal into a percentage, multiply by $100$: $0.375 = 37.5\\%$. To turn a percentage into a fraction, write it over $100$ and simplify: $37.5\\% = \\frac{375}{1000} = \\frac{3}{8}$. When adding or subtracting fractions, use a common denominator: $\\frac{2}{3} + \\frac{1}{4} = \\frac{8}{12} + \\frac{3}{12} = \\frac{11}{12}$. To multiply, multiply tops and bottoms and cancel: $\\frac{4}{9} \\times \\frac{3}{8} = \\frac{12}{72} = \\frac{1}{6}$. To divide, multiply by the reciprocal: $\\frac{5}{6} \\div \\frac{2}{3} = \\frac{5}{6} \\times \\frac{3}{2} = \\frac{15}{12} = \\frac{5}{4}$. With mixed numbers, convert to improper fractions first, so $2\\frac{1}{2} = \\frac{5}{2}$. Decimals that stop are terminating, like $0.25$; those that repeat are recurring, written with a dot over the repeating digit, for example $0.\\dot{3} = \\frac{1}{3}$. A fraction in lowest terms gives a terminating decimal exactly when its denominator has only the prime factors $2$ and $5$. To find a fraction of a quantity, multiply: $\\frac{3}{5}$ of $\\$40$ is $\\frac{3}{5} \\times 40 = 24$.

Percentage Change, Interest and Reverse Percentages

A percentage of an amount is found by multiplying by a decimal: $15\\%$ of $80$ is $0.15 \\times 80 = 12$. To increase by $15\\%$, multiply by $1.15$; to decrease by $15\\%$, multiply by $0.85$. These multipliers make percentage change quick and reliable. Percentage change uses the formula $\\text{percentage change} = \\frac{\\text{change}}{\\text{original}} \\times 100$. If a price rises from $\\$40$ to $\\$50$, the change is $10$, so the increase is $\\frac{10}{40} \\times 100 = 25\\%$. Profit and loss percentages work the same way, always dividing by the original (cost) value. Simple interest is calculated only on the original amount: $I = \\frac{P r t}{100}$, where $P$ is the principal, $r$ the rate per year and $t$ the time in years. Compound interest is calculated on the growing balance, using $A = P\\left(1 + \\frac{r}{100}\\right)^{t}$. For example, $\\$500$ at $4\\%$ compound for $3$ years gives $A = 500 \\times 1.04^{3} = \\$562.43$ to the nearest cent. Reverse percentage problems give you the value after a change and ask for the original. Here you must divide by the multiplier, not subtract. If a sale price of $\\$68$ is after a $15\\%$ discount, the original was $\\frac{68}{0.85} = \\$80$. A common error is taking $15\\%$ of $\\$68$ instead of dividing.

Ratio and Proportion

A ratio compares quantities of the same kind, such as $3 : 5$. Simplify ratios by dividing all parts by their HCF, so $12 : 18 = 2 : 3$. To divide a quantity in a given ratio, add the parts to find the total number of shares, then find the value of one share. To share $\\$60$ in the ratio $2 : 3$, there are $5$ shares, each worth $\\$12$, giving $\\$24$ and $\\$36$. Direct proportion means two quantities increase together at a constant rate, written $y \\propto x$, or $y = kx$ for a constant $k$. If $5$ pens cost $\\$2$, then $\\$1$ buys $2.5$ pens and the unitary method scales any amount. Inverse proportion means one quantity increases as the other decreases, written $y \\propto \\frac{1}{x}$, or $y = \\frac{k}{x}$. If $4$ workers take $6$ hours, then $\\text{workers} \\times \\text{hours} = 24$, so $3$ workers take $8$ hours. More advanced proportion may involve squares or roots, for instance $y \\propto x^{2}$ or $y \\propto \\sqrt{x}$. In every case the method is the same: find $k$ from the given pair of values, write the equation, then substitute. Rates such as speed, density and exchange rates are applied proportion: $\\text{speed} = \\frac{\\text{distance}}{\\text{time}}$ and $\\text{density} = \\frac{\\text{mass}}{\\text{volume}}$.

Indices and Standard Form

An index, or power, shows repeated multiplication: $2^{5} = 32$. The laws of indices let you simplify expressions. When multiplying powers of the same base, add the indices: $a^{m} \\times a^{n} = a^{m+n}$. When dividing, subtract: $a^{m} \\div a^{n} = a^{m-n}$. To raise a power to a power, multiply: $\\left(a^{m}\\right)^{n} = a^{mn}$. Any nonzero number to the power zero is $1$, so $a^{0} = 1$. A negative index means a reciprocal: $a^{-n} = \\frac{1}{a^{n}}$, so $2^{-3} = \\frac{1}{8}$. A fractional index means a root: $a^{\\frac{1}{n}} = \\sqrt[n]{a}$ and $a^{\\frac{m}{n}} = \\sqrt[n]{a^{m}}$, so $8^{\\frac{2}{3}} = \\left(\\sqrt[3]{8}\\right)^{2} = 4$. Standard form writes a number as $a \\times 10^{n}$, where $1 \\leq a \\lt 10$ and $n$ is an integer. Large numbers have positive powers: $4\\,500\\,000 = 4.5 \\times 10^{6}$. Small numbers have negative powers: $0.00032 = 3.2 \\times 10^{-4}$. To multiply or divide numbers in standard form, deal with the $a$ values and the powers of $10$ separately, then adjust so the front number stays between $1$ and $10$. For example $\\left(3 \\times 10^{4}\\right) \\times \\left(2 \\times 10^{5}\\right) = 6 \\times 10^{9}$.

Surds and Exact Values

A surd is an irrational root left in exact form, such as $\\sqrt{2}$ or $\\sqrt{50}$, rather than a rounded decimal. Working in surds keeps answers exact, which the harder (extended) paper often demands. The key rules are $\\sqrt{a} \\times \\sqrt{b} = \\sqrt{ab}$ and $\\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}$. To simplify a surd, factor out the largest perfect square: $\\sqrt{50} = \\sqrt{25 \\times 2} = 5\\sqrt{2}$. You can add or subtract only like surds, just as with algebraic terms: $3\\sqrt{2} + 4\\sqrt{2} = 7\\sqrt{2}$, while $\\sqrt{2} + \\sqrt{3}$ cannot be combined. When multiplying brackets, expand carefully: $\\left(1 + \\sqrt{3}\\right)\\left(2 - \\sqrt{3}\\right) = 2 - \\sqrt{3} + 2\\sqrt{3} - 3 = -1 + \\sqrt{3}$. Rationalising the denominator means removing a surd from the bottom of a fraction. For a single surd, multiply top and bottom by that surd: $\\frac{6}{\\sqrt{3}} = \\frac{6\\sqrt{3}}{3} = 2\\sqrt{3}$. This gives a tidy exact answer that examiners expect.

Bounds, Limits of Accuracy and Set Notation

Any rounded measurement has limits of accuracy. A value given to the nearest unit can be up to half a unit either side. If a length is $24\\,\\text{cm}$ to the nearest centimetre, its lower bound is $23.5\\,\\text{cm}$ and its upper bound is $24.5\\,\\text{cm}$, so the true length $L$ satisfies $23.5 \\leq L \\lt 24.5$. Note the upper bound uses a strict inequality. Bounds combine in calculations. For a maximum sum or product, use the upper bounds; for a minimum, use the lower bounds. Division is the tricky case: the maximum of $\\frac{a}{b}$ uses the largest $a$ with the smallest $b$, and the minimum uses the smallest $a$ with the largest $b$. For speed from distance $\\div$ time, the fastest possible speed pairs the upper-bound distance with the lower-bound time. Set notation describes collections of objects. The universal set is $\\xi$, and a set is listed in braces, for example $A = \\{1, 2, 3\\}$. The number of elements in $A$ is written $n(A)$. Membership is shown with $\\in$ ($3 \\in A$) and non-membership with $\\notin$. Union $A \\cup B$ contains everything in either set; intersection $A \\cap B$ contains only what is in both. The complement $A'$ is everything in $\\xi$ not in $A$, and the empty set is $\\varnothing$. If every element of $A$ lies in $B$, then $A$ is a subset of $B$, written $A \\subseteq B$. These symbols are read directly from Venn diagrams in the exam.