QCAA Specialist Mathematics key terms

A proof technique showing a statement holds for a base case, then that whenever it holds for some integer it holds for the next, so it holds for all integers from the base onward.

The initial instance of a statement verified directly at the start of an induction proof, providing the anchor from which all later cases follow.

The part of an induction proof that assumes the statement true for an arbitrary case and deduces it for the next case, linking each case to its successor.

A proof that assumes the negation of the desired statement and derives an impossibility, thereby establishing the original statement must be true.

A single specific case in which a general claim fails, which is sufficient to disprove that claim without any further argument.

A statement believed true on the basis of evidence but not yet proved, which may later be established by proof or refuted by counterexample.

A number written in the form a + bi, where a and b are real and i satisfies i squared equals negative one, allowing solutions to equations with no real roots.

The symbol i defined by the property that its square equals negative one, forming the building block of all complex numbers.

The number obtained by reversing the sign of the imaginary part, so the conjugate of a + bi is a minus bi; multiplying a number by its conjugate gives a real result.

The distance of a complex number from the origin in the complex plane, written as the square root of the sum of the squares of its real and imaginary parts.

The angle measured anticlockwise from the positive real axis to the line joining the origin to the point representing the number in the complex plane.

Writing a complex number as its modulus multiplied by the combination cosine of the argument plus i sine of the argument, convenient for multiplication, division and powers.

The result that raising a complex number in polar form to a power raises the modulus to that power and multiplies the argument by that power.

The values whose given power equals a specified complex number; an nth root problem has exactly n distinct solutions arranged symmetrically around a circle in the complex plane.

The result that every polynomial of degree at least one with complex coefficients has at least one complex root, so a degree n polynomial has exactly n roots counted with multiplicity.

A quantity having both magnitude and direction, represented by a directed line segment and often written as a column of components.

A vector of length one, obtained by dividing any nonzero vector by its own magnitude, used to specify direction independently of size.

The operation combining two vectors to give a number equal to the product of their magnitudes and the cosine of the angle between them; it is zero when the vectors are perpendicular.

The component of one vector along the direction of another, found using the scalar product and the unit vector in that direction.

A description of a line as a fixed point plus a scalar multiple of a direction vector, where the scalar varies over all real values.

A rectangular array of numbers organised in rows and columns, used to represent transformations, systems of equations and other structured data.

A single number calculated from a square matrix that indicates whether the matrix has an inverse and describes how the associated transformation scales areas or volumes.

The matrix that, when multiplied by a given square matrix, produces the identity matrix; it exists only when the determinant is nonzero.

A square matrix whose determinant equals zero, so it has no inverse and its associated transformation collapses area in at least one direction.

A mapping of points in the plane or space defined by multiplying their position vectors by a matrix, producing rotations, reflections, dilations or shears.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of a difference quotient and interpreted as the gradient of the tangent.

A rule for differentiating a composite function, giving the derivative as the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function.

A rule for differentiating a product of two functions, giving the derivative as the first times the derivative of the second plus the second times the derivative of the first.

A point on a curve where the derivative equals zero, which may be a local maximum, a local minimum or a point of inflection.

The derivative of the derivative of a function, measuring how the gradient itself changes and used to classify stationary points through its sign.

An equation involving a function and one or more of its derivatives, whose solution is a family of functions rather than a single number.

A differential equation that can be rearranged so all terms in the dependent variable sit on one side and all terms in the independent variable on the other, allowing both sides to be integrated.

A known value of the solution at a particular point, used to select one specific member from the family of solutions of a differential equation.

A variable that can take any value in an interval, described by a probability density function whose total area under the curve equals one.

A non-negative function whose integral over an interval gives the probability that a continuous random variable lies in that interval.

A symmetric bell-shaped continuous distribution determined by its mean and standard deviation, widely used to model naturally occurring variation.

The normal distribution with mean zero and standard deviation one, to which any normal variable can be converted by subtracting the mean and dividing by the standard deviation.

The number of standard deviations a value lies from the mean, calculated by subtracting the mean and dividing by the standard deviation.

The long-run average of a random variable, computed as a sum or integral of each value weighted by its probability or density.

The angle subtended at the centre of a circle by an arc equal in length to the radius; a full turn equals two pi radians.

The circle of radius one centred at the origin, on which the cosine and sine of an angle appear as the coordinates of the corresponding point.

The relation stating that the square of sine plus the square of cosine of the same angle equals one, following from the equation of the unit circle.

An identity expressing a trigonometric function of a sum or difference of two angles in terms of functions of the individual angles, such as sine of A plus B.

A special case of the compound angle formulas giving trigonometric functions of twice an angle in terms of functions of the original angle.

A pair of equations expressing the coordinates of points on a curve separately in terms of a third variable, called the parameter.

A simplified mathematical description of a real situation, built on stated assumptions and used to make predictions that must be checked against evidence.