GCSE / Maths
Number
Master GCSE Number topics including ordering numbers, place value, four operations, factors and multiples, powers and roots, fractions, decimals, percentages, standard form, units, estimation, rounding, error intervals and bounds.
Numbers can be classified as natural numbers, integers, fractions, decimals, rational numbers, irrational numbers and real numbers.
Positive numbers are greater than zero and negative numbers are less than zero.
A number line can be used to compare and order positive and negative numbers.
Numbers further to the right on a number line are greater than numbers further to the left.
The and ≠ are used to compare numbers and express inequalities.
The .
The to to.
Place value gives the value of a digit according to its position in a number.
Place value is used with integers, decimals, very large numbers and very small numbers.
The four mathematical operations are addition, subtraction, multiplication and division.
Formal written methods can be used to calculate with integers, decimals, fractions and mixed numbers.
Negative numbers follow sign rules when multiplying and dividing.
A positive multiplied by a positive gives a positive result.
A negative multiplied by a negative gives a positive result.
A positive multiplied by a negative gives a negative result.
Inverse operations undo each other.
Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
Inverse operations can be used to check calculations.
The order of mathematical operations determines which operation should be completed first.
BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition and Subtraction.
Brackets should be completed before powers or roots.
Multiplication and division are completed before addition and subtraction.
A reciprocal is found by dividing 1 by a number.
The reciprocal of 5 is 1/5.
A factor is a whole number that divides another number exactly.
A multiple is the result of multiplying a number by an integer.
A prime number has exactly two positive factors: 1 and itself.
A composite number has more than two positive factors.
Common factors are factors shared by two or more numbers.
Common multiples are multiples shared by two or more numbers.
The highest common factor is the greatest factor shared by two or more numbers.
The lowest common multiple is the smallest positive multiple shared by two or more numbers.
Prime factorisation expresses a number as a product of prime numbers.
A factor tree is a method used to find the prime factorisation of a number.
Prime factorisation can be written using index notation.
Every integer greater than 1 has a unique prime factorisation apart from the order of the factors.
Systematic listing is used to organise possible outcomes without missing or repeating any.
Lists, tables and diagrams can be used in systematic counting.
The product rule for counting multiplies the number of choices at each stage.
A power or exponent shows how many times a base is multiplied by itself.
For example, 2⁴ = 2 × 2 × 2 × 2 = 16.
A square number is produced by multiplying a number by itself.
A cube number is produced by multiplying a number by itself three times.
A square root is a number that gives the original number when multiplied by itself.
A cube root is a number that gives the original number when cubed.
Integer indices include positive, zero and negative whole-number powers.
Fractional indices represent roots and powers.
For positive a, .
For positive a, .
When multiplying powers with the same base, add the indices.
When dividing powers with the same base, subtract the indices.
When raising a power to another power, multiply the indices.
An exact answer is written without rounding.
Fractions, π and surds can be used to represent exact values.
A surd is an irrational root written in exact form.
For example, √12 can be simplified to 2√3.
Rationalising the denominator removes a surd from the denominator of a fraction.
Standard form is written as A × 10ⁿ where 1 ≤ .
Standard form is used to represent very large and very small numbers.
Positive powers of 10 are normally used for large numbers.
Negative powers of 10 are normally used for very small numbers.
Calculator displays may show numbers using scientific notation.
A proper fraction has a numerator smaller than its denominator.
An improper fraction has a numerator greater than or equal to its denominator.
A mixed number contains a whole number and a proper fraction.
Fractions must have a common denominator before they can be added or subtracted.
To multiply fractions, multiply the numerators and multiply the denominators.
To divide by a fraction, multiply by its reciprocal.
Fractions should be simplified by dividing the numerator and denominator by a common factor.
A terminating decimal ends after a finite number of decimal places.
A recurring decimal contains digits that repeat indefinitely.
Terminating decimals can be converted into fractions using place value.
Fractions can be converted into decimals by dividing the numerator by the denominator.
Recurring decimals can be converted exactly into fractions using algebra.
Fractions, decimals and percentages can represent the same proportion.
A percentage means a number of parts out of 100.
To convert a decimal to a percentage, multiply by 100.
To convert a percentage to a decimal, divide by 100.
To convert a fraction to a percentage, divide the numerator by the denominator and multiply by 100.
A ratio compares the relative size of two or more quantities.
Ratios can be simplified by dividing every part by a common factor.
Fractions can be used to represent parts of a ratio.
A fraction can be used as an operator to find part of a quantity.
A percentage can also be used as an operator.
To find p% of an amount, multiply the amount by p/100.
A percentage multiplier can be used for percentage increases and decreases.
A percentage increase of p% uses the multiplier 1 + p/100.
A percentage decrease of p% uses the multiplier 1 - p/100.
Percentage change is calculated using change ÷ original value × 100%.
Standard units are used to measure quantities such as length, mass, time, money, area, volume and capacity.
Common metric length units are millimetres, centimetres, metres and kilometres.
10 .
100 .
1000 .
1000 .
1000 .
Metric unit conversions use suitable multiplication or division factors.
Area conversions require the linear conversion factor to be squared.
Volume conversions require the linear conversion factor to be cubed.
Imperial units include inches, feet, yards, miles, ounces, pounds and stones.
Imperial-to-metric conversion factors are normally provided in GCSE questions when needed.
Compound measures combine two or more different units.
Speed is calculated using .
Density is calculated using .
Estimation gives an approximate answer.
Estimation can be used to check whether a calculator answer is reasonable.
Numbers are usually rounded before performing an estimation calculation.
Rounding to decimal places depends on the required number of digits after the decimal point.
Rounding to significant figures begins at the first non-zero digit.
The next digit determines whether the final retained digit stays the same or rounds up.
Values should normally not be rounded during intermediate stages of a calculation.
Truncation removes digits after a specified position without rounding.
An error interval gives the possible range of original values represented by a rounded or truncated value.
Error intervals can be written using inequality notation.
Limits of accuracy describe the range in which the true value lies.
The lower bound is the smallest possible value represented by a rounded measurement.
The upper bound is the greatest limiting value represented by a rounded measurement.
For a number rounded to the nearest whole number, the limits are 0.5 below and 0.5 above the rounded value.
Bounds can affect calculations involving multiplication, division, area, volume and other measures.
Accuracy describes closeness to the true value.
Precision describes the level of detail or consistency in a measurement.