SACE Stage 2 Specialist Mathematics key terms

A number written in the form a + bi, where a and b are real and i satisfies i squared equals negative one. It can be plotted as a point in the complex plane.

The distance from the origin to the point representing a complex number in the complex plane, written |z|. For z = a + bi it equals the square root of a squared plus b squared.

The angle measured anticlockwise from the positive real axis to the line joining the origin to the point representing the complex number. Values are usually given in radians, in the interval from negative pi to pi.

The complex number formed by changing 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 value.

A way of writing a complex number using its modulus r and argument theta, as r times the quantity cosine theta plus i sine theta. It makes multiplication and division of complex numbers simpler.

A rule stating that raising a complex number in polar form to a power n raises the modulus to that power and multiplies the argument by n. It also underlies finding roots of complex numbers.

The complex solutions of the equation z to the power n equals one. There are exactly n of them, equally spaced around the unit circle in the complex plane.

The set of all points in the plane that satisfy a given geometric condition. In the complex plane, conditions on z often produce circles, lines or rays.

A quantity with both magnitude and direction, often represented by a directed line segment or by components in a coordinate system. Vectors can be added and multiplied by scalars.

A vector whose magnitude equals one. Dividing any non-zero vector by its own magnitude produces a unit vector pointing in the same direction.

The length of a vector, written with vertical bars around the vector symbol. In three dimensions it is found using the square root of the sum of the squares of the components.

A scalar obtained from two vectors by multiplying corresponding components and adding the results. It equals the product of the magnitudes and the cosine of the angle between them.

A vector operation defined for two vectors in three dimensions. The result is perpendicular to both original vectors, with magnitude equal to the area of the parallelogram they span.

A way of writing the equation of a plane as the dot product of a normal vector with a position vector equal to a constant. It follows from expanding the normal-and-point form.

A vector that is perpendicular to a line, plane or surface. For a plane it defines the orientation, and it is central to writing the plane's equation.

A set of equations that express coordinates in terms of one or more independent parameters. They are commonly used to describe lines, curves and the path of a moving particle.

An equation relating x and y coordinates directly, without a parameter. It is often obtained by eliminating the parameter from a pair of parametric equations.

A proof technique used for statements about positive integers. It establishes a base case, assumes the statement for some value k, and then proves it for k plus one.

The starting value in a proof by mathematical induction, usually the smallest integer for which the statement is claimed. It must be verified directly before the inductive step is used.

The part of an induction proof that shows: if the statement holds for an arbitrary value k, then it also holds for k plus one. This links the base case to all later cases.

The statement formed by negating and swapping the parts of a conditional statement. If the original conditional is true, its contrapositive is logically equivalent and also true.

A proof method that assumes the statement to be proved is false and shows this assumption leads to an impossible conclusion. The impossibility means the original statement must be true.

A single case that shows a general statement is false. Finding one counterexample is enough to disprove a claim that something is always true.

The statement obtained by swapping the hypothesis and conclusion of a conditional statement. A true statement does not guarantee that its converse is also true.

The derivative found by taking the limit of the difference quotient as the change in x approaches zero. It uses the formal definition of the derivative rather than shortcut rules.

The value that a function or sequence approaches as the input or index approaches a particular point. Limits underpin derivatives, integrals and tests for convergence.

The process of finding the derivative of a function, which measures its instantaneous rate of change. Standard rules cover powers, products, quotients and composite functions.

A rule for differentiating a composite function. The derivative of f of g of x equals f dash of g of x multiplied by g dash of x.

A technique for finding the derivative when a relationship between x and y is given in a form that is not solved for y. Each term is differentiated with respect to x, applying the chain rule to y terms.

The derivative of the derivative of a function, describing how the rate of change itself changes. Its sign helps identify whether a stationary point is a maximum or a minimum.

A point on a curve where the derivative equals zero, so the tangent is horizontal. It may be a local maximum, a local minimum or a point of inflection.

A point where the concavity of a curve changes sign. It is often associated with the second derivative being zero, though that alone does not guarantee an inflection.

A technique that reverses the chain rule by replacing part of an integrand with a new variable. The differential and the limits of integration must also be converted.

A technique derived from the product rule that rewrites an integral of a product as another integral. It is useful when the new integral is simpler than the original.

The integral of a function between two limits, giving a number rather than a family of functions. It represents signed area under a curve and is evaluated using the fundamental theorem of calculus.

The region enclosed between two functions over an interval, found by integrating the difference of the upper and lower functions. Intersection points often set the limits of integration.

An equation involving a function and one or more of its derivatives. Solving it means finding a function that satisfies the relationship, often subject to initial conditions.

A differential equation that can be rearranged so all terms in one variable are on one side and all terms in the other variable on the other. Both sides are then integrated separately.

A known value of a function at a particular point, used to determine the constant of integration in the solution of a differential equation. It selects one particular solution from a family.