ORT Mathematics key terms

A number that can be written as a fraction p/q, where p and q are integers and q is not zero. Its decimal expansion either stops or repeats a fixed block of digits forever.

A real number that cannot be expressed as a fraction of two integers. In decimal form its digits never end and never settle into a repeating pattern, as with pi or the square root of two.

Writing a whole number greater than one as a product of prime numbers only. Apart from the order of the factors, every such number has exactly one prime factorisation.

The largest positive integer that divides each of two or more given integers without a remainder. It can be found from prime factorisations by taking the lowest power of every shared prime.

The smallest positive integer that is divisible by each of two or more given integers. From prime factorisations, take the highest power of every prime that appears in any of them.

A compact way of writing very large or very small numbers as a number between one and ten multiplied by a power of ten, such as 3.2 times 10 to the fifth.

A ratio compares two quantities by division, while a proportion states that two ratios are equal. Proportions are used to scale maps, recipes and models while keeping the same relative sizes.

The digits in a measured value that carry meaningful information about its precision. Rules decide which zeros count, and a calculated result should not claim more precision than the measurements feeding it.

A rule that assigns exactly one output value to each input value it accepts. The set of accepted inputs is the domain, and the set of resulting outputs is the range.

The domain of a function is the full set of input values for which it is defined. The range is the set of all output values the function actually produces.

A measure of how steep a straight line is, equal to the change in the vertical coordinate divided by the change in the horizontal coordinate between two points. It is also the constant rate of change of a linear function.

A function whose graph is a straight line, commonly written f(x) = mx + c. The constant m is the slope and c is the value where the line crosses the vertical axis.

A function of the form f(x) = ax squared plus bx plus c, where a is not zero. Its graph is a parabola whose turning point, intercepts and number of real roots depend on the coefficients.

A pattern in which a quantity is multiplied by the same factor during each equal time step, giving a curve that rises or falls ever more steeply. The factor is greater than one for growth.

The exponent to which a chosen base must be raised to produce a given number; log base b of x equals y means b to the power y equals x. Logarithms reverse exponentiation.

A function that undoes the effect of another, swapping inputs and outputs. If a function sends a to b, its inverse sends b back to a; only one-to-one functions have inverses that are also functions.

Two or more linear equations that must all be true at the same time. Solving means finding values that satisfy every equation; on a graph this is where the lines meet.

A line that a curve approaches more and more closely as it runs far out or near a forbidden point, without necessarily touching it. Exponential and rational graphs often have asymptotes.

The relationship between figures that have exactly the same shape and size. One can be moved onto the other using translations, rotations and reflections, so corresponding sides and angles are equal.

The relationship between figures that have the same shape but possibly different sizes. Corresponding angles are equal and corresponding lengths are in a fixed ratio called the scale factor.

In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It applies only when one angle is exactly ninety degrees.

A unit of angle measure based on arc length. One radian is the angle at a circle's centre subtended by an arc as long as the radius; a full turn equals two pi radians.

For an acute angle in a right triangle, sine, cosine or tangent compares two side lengths: opposite over hypotenuse, adjacent over hypotenuse, or opposite over adjacent respectively.

A movement of a figure that preserves every distance and angle, namely a translation, rotation or reflection. The resulting image is therefore congruent to the original figure.

The straight-line distance between points with coordinates x1, y1 and x2, y2 equals the square root of the sum of the squared horizontal and vertical differences. It follows from the Pythagorean theorem.

The amount of three-dimensional space a solid occupies, measured in cubic units. For a prism or cylinder it equals the cross-sectional area multiplied by the length or height.

Changing a measurement from one unit to another by multiplying by a conversion factor that equals one, such as one hundred centimetres per metre. Units cancel to show the conversion is correct.

The arithmetic average, found by adding all the values and dividing by how many there are. It uses every data point, so extreme values can pull it away from the typical case.

The middle value when the data are arranged in order; with an even number of values it is the mean of the two middle ones. Unlike the mean, it is little affected by outliers.

A measure of how spread out the values are around their mean. It is the square root of the average squared distance from the mean and is expressed in the same units as the data.

The spread of the middle half of ordered data, equal to the upper quartile minus the lower quartile. It ignores the extreme quarters and so resists the influence of outliers.

A number between minus one and plus one that measures how closely two variables follow a straight-line pattern together. A value near zero means little linear relationship; causation is not implied.

The probability that an event happens given that another event is already known to have happened. It equals the probability of both events divided by the probability of the known event.

Two events are independent when the occurrence of one does not change the probability of the other. For such events, the probability of both happening equals the product of their probabilities.

A symmetric bell-shaped probability distribution described by its mean and standard deviation. About sixty-eight percent of values lie within one standard deviation of the mean and about ninety-five percent within two.

A simplified representation of a real situation using variables, equations, graphs or assumptions. It is judged by how well its predictions match reality for the intended purpose, not by being exact.

A simplifying condition accepted as true when building a model, such as ignoring air resistance or treating a population as constant. Stating assumptions shows when the model's conclusions can reasonably be trusted.

A single case that satisfies the conditions of a general statement but makes its conclusion false. One counterexample is enough to disprove a claim that was thought to hold always.

How much one quantity changes for each unit change in another quantity. On a graph it is the gradient, and its units are output units per input unit.

Checking or deriving relationships by tracking the units of the quantities involved. Both sides of a valid equation must have the same dimensions, which catches many errors in modelling and calculation.