LGCSE Mathematics (0178) key terms

A counting number from 1 upwards, such as 1, 2, 3 and so on. It has no decimal or fractional part and is never negative. Some courses include zero in this set.

A whole number that may be positive, negative or zero, such as -7, 0 or 24. It has no fractional or decimal part. Integers are closed under addition, subtraction and multiplication.

A number that can be written as one integer divided by another non-zero integer. Its decimal form either stops or repeats a block of digits forever, as with 3/8 = 0.375 or 2/11 = 0.181818...

A decimal in which a digit or block of digits repeats without end, shown by dots or a bar over the repeating part. For example 0.3 recurring equals one third and can be converted back to a fraction.

A whole number greater than 1 whose only positive factors are 1 and itself, such as 2, 3, 5, 7 and 11. The number 1 is not prime because it has only one factor.

A whole number that divides another exactly, leaving no remainder. For instance 4 is a factor of 12 because 12 divided by 4 equals 3. Every number has itself and 1 as factors.

The result of multiplying a number by a whole number. The multiples of 6 are 6, 12, 18, 24 and so on. Every number has infinitely many multiples but only finitely many factors.

The largest number that divides two or more given numbers exactly. For 18 and 24 the common factors are 1, 2, 3 and 6, so the highest common factor is 6. Found by listing factors or comparing prime factorisations.

The smallest number that is a multiple of two or more given numbers. For 4 and 6 the multiples overlap first at 12. Often used to find a common denominator or to solve problems about events repeating together.

A way of writing very large or very small numbers as a number between 1 and 10 multiplied by a power of 10. For example 45 000 is written 4.5 x 10^4 and 0.00062 is 6.2 x 10^-4.

The digits of a number that carry meaning for its accuracy, counted from the first non-zero digit. Rounding 3.4728 to three significant figures gives 3.47. Leading zeros are never counted as significant.

Replacing a number by a nearby value with fewer digits, chosen to a stated accuracy such as a decimal place or significant figure. If the next digit is 5 or more, round up; otherwise keep the digit unchanged.

Finding an approximate answer by rounding each value to one significant figure or another convenient number before calculating. Used to check whether a calculator answer is sensible, and shown by the symbol approximately equals.

The largest and smallest values a rounded measurement could really have had. A length given as 7.4 cm to one decimal place has a lower bound of 7.35 cm and an upper bound of 7.45 cm.

A fraction expressed out of 100, so 35% means 35/100 or 0.35. Used to compare proportions and to calculate increases, decreases, profit, loss and interest by multiplying by the decimal equivalent.

A comparison of two or more quantities of the same kind, written with a colon such as 3 : 5. To share an amount in a ratio, divide by the total number of parts and multiply each share by its number of parts.

A statement that two ratios are equal, or a relationship where quantities change at the same relative rate. Direct proportion means both grow together; inverse proportion means one grows as the other shrinks.

Interest calculated on both the original amount and on interest already added, so the balance grows by a fixed multiplier each period. After n years at rate r, the amount is P(1 + r/100)^n.

Rules for combining powers of the same base: add indices when multiplying, subtract when dividing, and multiply when raising a power to a power. Also cover zero, negative and fractional indices.

A root of a number that cannot be written exactly as a fraction or terminating decimal, such as the square root of 2. Surds are often left in simplified form rather than rounded.

A combination of numbers, variables and operations without an equals sign, such as 4x - 7 or 3a^2 + 2ab. It can be simplified, expanded or factorised but not solved.

An equation in which each variable appears to the first power only, so its graph is a straight line. Solved by doing the same operation to both sides until the variable stands alone.

Two or more equations that must be true at the same time. Solved by elimination, where one variable is removed by adding or subtracting the equations, or by substitution. The solution satisfies every equation.

An equation containing a squared variable as its highest power, usually written ax^2 + bx + c = 0 with a not zero. Solved by factorising, completing the square or the quadratic formula.

Writing an expression as a product of simpler expressions. Common factors are taken outside brackets, and a quadratic such as x^2 + 5x + 6 becomes (x + 2)(x + 3). The reverse process of expanding.

A statement comparing two expressions that are not necessarily equal, using the symbols <, >, <= or >=. Multiplying or dividing both sides by a negative number reverses the direction of the sign.

A rule that assigns exactly one output value to each allowed input value. Often written f(x), and can be represented by a formula, a table, a graph or a mapping diagram.

A measure of how steep a straight line is, found by dividing the vertical change by the horizontal change between two points. Horizontal lines have gradient zero; vertical lines have undefined gradient.

The value at which a graph crosses the vertical axis, that is the y-value when x equals zero. In the form y = mx + c it is the constant c, and it marks the starting value of the relationship.

A rule of the form y = mx + c, where m is the gradient and c the y-intercept. Any equation of this shape graphs as a straight line, so equal x-steps always give equal y-steps.

The point exactly halfway between two given points. Its coordinates are the averages of the corresponding coordinates: for (2, 5) and (8, 1) the midpoint is (5, 3).

The length of the straight line joining two coordinate pairs, found with Pythagoras' theorem. For points (x1, y1) and (x2, y2) the distance is the square root of the sum of the squared coordinate differences.

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. Used to find a missing side and to test whether a triangle contains a right angle.

A ratio linking an angle of a right-angled triangle to two of its sides: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. Remembered as SOH-CAH-TOA.

A relationship for any triangle stating that each side divided by the sine of its opposite angle is constant. Used when you know a matching side and angle together with one further side or angle.

A generalisation of Pythagoras' theorem to any triangle, relating one side to the other two sides and the angle between them. Used when two sides and the included angle, or all three sides, are known.

A direction measured clockwise from due north, written as three digits such as 047 degrees. It always lies between 000 and 360 and is the standard way of describing travel direction on a map.

A named geometric fact about angles in circles, for example that the angle in a semicircle is a right angle, or that angles in the same segment are equal. Each needs a named reason in a proof.

A four-sided shape whose vertices all lie on one circle. Its opposite angles add up to 180 degrees, and this property is often used to find unknown angles in circle problems.

A straight line that touches a curve at exactly one point without crossing it. For a circle, a tangent is perpendicular to the radius drawn to the point of contact.

The relationship between shapes that are exactly the same size and shape, though possibly rotated, reflected or moved. Triangles are proved congruent using the conditions SSS, SAS, ASA and RHS.

The relationship between shapes with the same angles but possibly different sizes, where corresponding lengths share a constant scale factor. Areas scale by the square of that factor and volumes by its cube.

A diagram in which every length is a fixed fraction of the real length, stated as a scale such as 1 : 50 000. Measurements taken from the drawing can be converted back to real distances.

The amount of flat surface enclosed by a two-dimensional shape, measured in square units such as cm^2. Formulas differ by shape: length times width for a rectangle, half base times height for a triangle.

The total distance around the outside edge of a flat shape, found by adding all its side lengths. For a circle the matching quantity is the circumference, calculated as pi times the diameter.

The amount of three-dimensional space a solid occupies, measured in cubic units such as cm^3. For a prism it equals the cross-sectional area multiplied by the length, and for a cylinder pi r^2 h.

The total area of all the faces or curved surfaces of a solid, measured in square units. Found by calculating the area of each face separately and adding them, or by using a net.

The distance along part of the circumference of a circle. It equals the fraction of the full circle represented by the central angle, multiplied by pi times the diameter.

The area of the pie-slice region bounded by two radii and an arc. Calculated as the angle divided by 360, multiplied by pi times the radius squared.

The average found by adding all values and dividing by how many there are. It uses every data value, so a single extreme value can pull it noticeably away from the centre of the data.

The middle value when data is arranged in order. With an even number of values it is the mean of the two middle ones. Unlike the mean, it is not affected by extreme values.

The value that occurs most often in a set of data. A set may have one mode, several modes, or none at all if every value appears equally often. Useful for categorical data.

A simple measure of spread found by subtracting the smallest value from the largest. It is quick to calculate but depends only on the two extreme values, ignoring everything in between.

The difference between the upper quartile and the lower quartile, covering the middle half of the data. It ignores extreme values and so gives a more reliable picture of spread than the range.

A running total of frequencies, showing how many values fall at or below each point. Its graph rises in steps or a smooth curve and is used to estimate the median and quartiles.

A chart for continuous data in unequal class intervals, where the area of each bar represents frequency. The vertical axis is frequency density, that is frequency divided by class width.

A measure from 0 to 1 of how likely an event is, where 0 means impossible and 1 means certain. For equally likely outcomes it is the number of favourable outcomes divided by the total number of outcomes.

Events that cannot happen at the same time, such as rolling a 3 and rolling a 5 on one die. Their combined probability is found by adding the individual probabilities together.

Events where the outcome of one does not affect the probability of the other, such as separate coin tosses. For such events the probability of both happening is the product of their probabilities.

A branching diagram showing the possible outcomes of two or more stages of an event. Probabilities are written on the branches and multiplied along a path, then added across paths.