Topic 1: Number

Number is the foundation of GCSE Mathematics. Nearly every question in algebra, geometry, statistics and ratio quietly relies on confident arithmetic, fluent handling of fractions and percentages, and a clear understanding of how numbers are built from primes. This article works through the whole Number strand of the AQA 8300 specification in order, with worked reasoning at each step. Read it with a pen in hand: stop at each example, predict the next line, then check it. The non-calculator skills here (exact fractions, surds, standard form by hand, estimation) are exactly the ones examiners use to separate grades, so give them extra attention.

Integers and Place Value

The integers are the whole numbers and their negatives: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$ Our number system is base ten (decimal), which means each digit's value depends on its position, or place. In the number $4\,072.65$ the digit $4$ means four thousands, the $7$ means seven tens, the $6$ means six tenths and the $5$ means five hundredths. Reading from the decimal point, columns to the left are units, tens, hundreds, thousands; columns to the right are tenths, hundredths, thousandths. Understanding place value lets you order numbers and multiply or divide by powers of ten quickly. Multiplying by $10$ moves every digit one column to the left (the value gets larger), so $3.6 \times 10 = 36$ and $3.6 \times 100 = 360$. Dividing by $10$ moves digits one column to the right, so $3.6 \div 10 = 0.36$. It is the digits that move, not the decimal point; thinking of it this way avoids errors with trailing zeros. Negative numbers obey careful sign rules. Adding a negative is the same as subtracting: $5 + (-3) = 5 - 3 = 2$. Subtracting a negative is the same as adding: $5 - (-3) = 5 + 3 = 8$. For multiplication and division, two like signs give a positive and two unlike signs give a negative, so $(-4) \times (-3) = 12$ but $(-4) \times 3 = -12$. When ordering negatives remember that $-8 \lt -3$ because $-8$ lies further to the left on the number line, even though $8 \gt 3$.

Factors, Multiples, Primes, HCF and LCM

A factor of a number divides into it exactly with no remainder; for example the factors of $12$ are $1, 2, 3, 4, 6$ and $12$. A multiple is the result of multiplying a number by an integer, so the multiples of $4$ are $4, 8, 12, 16, \ldots$ A prime number has exactly two factors, $1$ and itself: $2, 3, 5, 7, 11, 13, \ldots$ Note that $1$ is not prime (it has only one factor) and $2$ is the only even prime. Every integer greater than $1$ can be written as a product of primes in exactly one way; this is prime factorisation, usually found with a factor tree. For example $360 = 2^{3} \times 3^{2} \times 5$. Writing numbers in this form is the key to finding the highest common factor (HCF) and lowest common multiple (LCM). To find the HCF, take the lowest power of each prime appearing in both numbers; to find the LCM, take the highest power of every prime appearing in either number. Suppose $a = 2^{3} \times 3^{2} \times 5 = 360$ and $b = 2^{2} \times 3 \times 7 = 84$. The HCF uses the shared primes at their lowest powers: $2^{2} \times 3 = 12$. The LCM uses every prime at its highest power: $2^{3} \times 3^{2} \times 5 \times 7 = 2520$. A useful check is that for any two numbers, $\text{HCF} \times \text{LCM} = a \times b$. A Venn diagram, with shared prime factors in the overlap, gives the same answer: the overlap multiplies to the HCF and the whole diagram multiplies to the LCM.

Fractions, Decimals, Percentages and Conversions

A fraction $\frac{a}{b}$ represents $a$ parts out of $b$. To add or subtract fractions you need a common denominator: $\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$. To multiply, multiply tops and bottoms: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$. To divide, multiply by the reciprocal (flip the second fraction): $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$. Always cancel to lowest terms by dividing top and bottom by their HCF. With mixed numbers, convert to improper fractions first: $2\frac{1}{2} = \frac{5}{2}$. Percentages are fractions out of $100$, and decimals are another way of writing the same value. To convert a percentage to a decimal, divide by $100$, so $37\% = 0.37$; to convert a decimal to a percentage, multiply by $100$. A fraction becomes a decimal by dividing the numerator by the denominator: $\frac{3}{8} = 3 \div 8 = 0.375$, which is $37.5\%$. Fractions either terminate or recur as decimals. A fraction in lowest terms terminates exactly when its denominator's only prime factors are $2$ and $5$; otherwise it recurs, like $\frac{1}{3} = 0.\dot{3}$. To convert a recurring decimal to a fraction, set $x$ equal to it and multiply by a power of ten so the recurring parts line up. For $x = 0.\dot{4}\dot{5} = 0.454545\ldots$, multiply by $100$ to get $100x = 45.4545\ldots$; subtracting gives $99x = 45$, so $x = \frac{45}{99} = \frac{5}{11}$. Knowing the common conversions ($\frac{1}{2} = 50\%$, $\frac{1}{4} = 25\%$, $\frac{1}{5} = 20\%$, $\frac{3}{4} = 75\%$) saves time.

Percentage Change, Reverse Percentages and Interest

To find a percentage of an amount, use a multiplier. To increase by $15\%$, multiply by $1.15$; to decrease by $15\%$, multiply by $0.85$. So $\pounds 240$ increased by $15\%$ is $240 \times 1.15 = \pounds 276$. Percentage change is found with $\text{percentage change} = \frac{\text{change}}{\text{original}} \times 100$. If a value rises from $50$ to $62$, the change is $12$ and the percentage increase is $\frac{12}{50} \times 100 = 24\%$. Always divide by the original value, not the new one. Reverse percentage problems give you the value after a change and ask for the original. Because the new value is the original multiplied by the multiplier, you divide to undo it. If a price is $\pounds 84$ after a $20\%$ reduction, then $\pounds 84$ represents $80\%$ of the original, so the original is $84 \div 0.8 = \pounds 105$. A common mistake is to add $20\%$ back onto $\pounds 84$, which is wrong because the $20\%$ was of the larger original amount. Simple interest is paid only on the original amount each period: interest of $\pounds I = P \times r \times t$ where $P$ is the principal, $r$ the rate as a decimal and $t$ the time. Compound interest is paid on the growing balance, so it uses repeated multiplication: the amount after $n$ years is $A = P(1 + r)^{n}$. For $\pounds 2000$ at $3\%$ compound interest over $4$ years, $A = 2000 \times 1.03^{4} = \pounds 2251.02$. Depreciation works the same way with a multiplier below $1$, so a car losing $12\%$ per year is multiplied by $0.88^{n}$.

Powers, Roots and the Laws of Indices

A power (or index) tells you how many times to multiply a number by itself, so $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$. The square root reverses squaring: $\sqrt{49} = 7$, and the cube root reverses cubing: $\sqrt[3]{27} = 3$. You should know the squares up to $15^{2} = 225$ and the cubes of $2, 3, 4, 5$ and $10$ by heart. The laws of indices let you simplify expressions with the same base. To multiply, add the powers: $a^{m} \times a^{n} = a^{m+n}$, so $3^{4} \times 3^{2} = 3^{6}$. To divide, subtract the powers: $a^{m} \div a^{n} = a^{m-n}$, so $5^{7} \div 5^{3} = 5^{4}$. To raise a power to a power, multiply: $(a^{m})^{n} = a^{mn}$, so $(2^{3})^{2} = 2^{6} = 64$. Three special cases follow from these rules. Any nonzero number to the power zero is one: $a^{0} = 1$, because $a^{m} \div a^{m} = a^{0}$ and also equals $1$. A negative power means a reciprocal: $a^{-n} = \frac{1}{a^{n}}$, so $2^{-3} = \frac{1}{8}$. A fractional power means a root: $a^{\frac{1}{n}} = \sqrt[n]{a}$, and more generally $a^{\frac{m}{n}} = (\sqrt[n]{a})^{m}$. For example $27^{\frac{2}{3}} = (\sqrt[3]{27})^{2} = 3^{2} = 9$. Combining the rules, $16^{-\frac{1}{2}} = \frac{1}{\sqrt{16}} = \frac{1}{4}$.

Standard Form

Standard form (scientific notation) writes a number as $A \times 10^{n}$, where $A$ is at least $1$ but less than $10$, and $n$ is an integer. It is ideal for very large or very small numbers. For instance $4\,500\,000 = 4.5 \times 10^{6}$ and $0.000\,32 = 3.2 \times 10^{-4}$. A positive power of ten means a large number; a negative power means a small number between $0$ and $1$. The power counts how many places the digits move relative to the decimal point. To multiply or divide in standard form, deal with the number parts and the powers of ten separately, using the index laws. For example $(3 \times 10^{5}) \times (2 \times 10^{4}) = 6 \times 10^{9}$, and $(8 \times 10^{7}) \div (2 \times 10^{3}) = 4 \times 10^{4}$. Sometimes the front number falls outside the allowed range and must be adjusted: $(5 \times 10^{6}) \times (4 \times 10^{3}) = 20 \times 10^{9} = 2 \times 10^{10}$. To add or subtract, the powers of ten must match first. To compute $3.2 \times 10^{5} + 4 \times 10^{4}$, rewrite the second term as $0.4 \times 10^{5}$, then add to get $3.6 \times 10^{5}$. On a calculator the $\times 10^{x}$ or EXP button enters standard form directly; never type the digits $10$ as well, or you will introduce an extra factor of ten.

Surds

A surd is a root that cannot be simplified to a rational number, such as $\sqrt{2}$ or $\sqrt{3}$; these are irrational, with decimal expansions that never terminate or repeat. Working with surds keeps answers exact rather than rounded, which is why non-calculator questions often demand surd form. Two rules drive all simplification: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ and $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. To simplify a surd, factor out the largest perfect square. For example $\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$. You can add or subtract only like surds, treating them like algebraic terms: $5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}$, while $\sqrt{2} + \sqrt{3}$ cannot be combined. Expanding brackets uses the same care: $(2 + \sqrt{5})(3 - \sqrt{5}) = 6 - 2\sqrt{5} + 3\sqrt{5} - 5 = 1 + \sqrt{5}$. Rationalising the denominator removes a surd from the bottom of a fraction. For a single surd, multiply top and bottom by that surd: $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$. When the denominator is a sum or difference such as $3 + \sqrt{2}$, multiply by its conjugate $3 - \sqrt{2}$, because $(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7$, which is rational. So $\frac{1}{3 + \sqrt{2}} = \frac{3 - \sqrt{2}}{7}$.

Rounding, Significant Figures, Estimation, and Bounds

Rounding makes numbers simpler while keeping them close to the true value. To round to a given decimal place, look at the next digit: if it is $5$ or more, round up; otherwise round down. So $3.4567$ to two decimal places is $3.46$. Significant figures count from the first nonzero digit: in $0.004\,839$ the first significant figure is $4$, so to two significant figures it is $0.0048$, and to three it is $0.004\,84$. For large numbers keep place value with zeros: $48\,620$ to two significant figures is $49\,000$. Estimation rounds every number to one significant figure to get a quick, sensible approximation. To estimate $\frac{31.2 \times 5.8}{0.21}$, round to $\frac{30 \times 6}{0.2} = \frac{180}{0.2} = 900$. The symbol $\approx$ means approximately equal to. Estimation is also a fast check that a calculator answer is reasonable. A rounded value hides a range of possible true values, described by error intervals and bounds. A measurement rounded to the nearest unit could be up to half a unit either side. If a length is $24$ cm to the nearest centimetre, the lower bound is $23.5$ cm and the upper bound is $24.5$ cm, written $23.5 \leq x \lt 24.5$. Note the upper bound takes the half value even though $24.5$ would itself round up; it is the boundary of the interval. When combining bounded quantities, choose bounds to make the result largest or smallest. The maximum of a sum or product uses both upper bounds; for a maximum quotient $\frac{a}{b}$, use the upper bound of $a$ and the lower bound of $b$, since dividing by a smaller number gives a larger result.