ASSEB HSLC General 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 form either stops or repeats a fixed block of digits.

A real number that cannot be expressed as a fraction of two integers. Its decimal expansion never ends and never settles into a repeating block, as with the square root of 2.

Any number on the continuous number line, covering both rational and irrational values. Every point on the line corresponds to exactly one real number.

An irrational root of a rational number that cannot be simplified to a rational value, such as the square root of 3. Surds are often left in root form in exact answers.

Rewriting a fraction so that no surd remains in the denominator, usually by multiplying numerator and denominator by a suitable surd or conjugate. This gives an equivalent exact value in standard form.

The paired expression obtained by changing the sign between two terms, for example a minus root b paired with a plus root b. Multiplying such a pair removes the surd because the result is a difference of squares.

An algebraic expression formed by adding or subtracting terms of the form a times x raised to a whole-number power, where coefficients are real. Powers must be non-negative integers and no variable appears in a denominator.

The largest exponent of the variable that appears with a non-zero coefficient in a polynomial. It fixes the maximum number of roots and the general shape of the graph.

A statement that a polynomial has (x minus a) as a factor exactly when substituting a gives zero. It converts factor questions into simple substitution checks.

A rule saying that dividing a polynomial by (x minus a) leaves a remainder equal to the polynomial evaluated at a. It allows the remainder to be found without long division.

A value of the variable that makes the polynomial equal to zero. These values are the roots of the related equation and the points where the graph crosses the horizontal axis.

An equation such as ax plus by equals c in which each variable appears only to the first power. Its solutions form a straight line, and two such equations may be solved simultaneously.

Two or more equations that must hold true at the same time for the same unknowns. Their solution is the set of values satisfying all equations together, found by elimination or substitution.

A technique for solving simultaneous equations in which one equation is rearranged to express a variable and that expression is put into the other equation. It reduces the system to a single equation in one unknown.

A technique for solving simultaneous equations by adding or subtracting scaled versions of the equations so that one variable cancels. The remaining single-variable equation is then solved.

An equation of the form ax squared plus bx plus c equals zero with a not zero. It has at most two real roots, which may be found by factorising, completing the square or the quadratic formula.

The quantity b squared minus 4ac taken from a quadratic equation. Its sign tells whether the roots are real and distinct, real and equal, or not real.

The expression giving the roots of a quadratic equation as negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. It solves any quadratic once written in standard form.

A sequence in which each term after the first is obtained by adding a fixed number called the common difference. Examples include 3, 7, 11, 15 and sequences with negative differences.

The constant amount added to each term of an arithmetic progression to obtain the next term. It equals any term minus the term just before it and may be positive, negative or zero.

The general formula a plus (n minus 1) times d, where a is the first term and d is the common difference. It gives any term of an arithmetic progression directly from its position number.

The total of the first n terms of an arithmetic progression, given by n over 2 times the sum of the first and last terms, or n over 2 times twice a plus (n minus 1) d.

The value obtained by dividing the sum of a set of quantities by how many there are. In an arithmetic progression of three terms, the middle term equals the average of the other two.

A measure of steepness found as the change in vertical distance divided by the change in horizontal distance between two points. It is often denoted m and is undefined for a vertical line.

An algebraic relation whose graph is a straight line, commonly written as y equals mx plus c or in general form ax plus by plus c equals zero. The constants fix the slope and position of the line.

The coordinate where a line or curve meets an axis. The y-intercept is the value of y when x is zero, and the x-intercept is the value of x when y is zero.

Lines in the same plane that never meet however far they are extended. They have equal slopes and always keep the same perpendicular distance apart.

Lines that meet at a right angle. Their slopes multiply to negative one, so one slope is the negative reciprocal of the other, except for horizontal and vertical lines.

A ratio of two sides of a right-angled triangle linked to one acute angle, namely sine, cosine or tangent. These ratios depend only on the angle, not on the size of the triangle.

The angle measured upward from a horizontal line to the line of sight when an observer looks at an object above eye level. It is used with trigonometric ratios to find heights and distances.

The angle measured downward from a horizontal line to the line of sight when an observer looks at an object below eye level. It equals the angle of elevation from the object back to the observer.

The rule that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides. It underlies distance calculations and many trigonometric identities.

The side of a right-angled triangle opposite the right angle. It is always the longest side and is the side used as the denominator in the sine and cosine ratios.

A four-sided figure whose vertices all lie on one circle. Its opposite angles add up to 180 degrees, a property often used to find unknown angles.

The statement that an angle formed by two chords meeting on a circle equals half the central angle standing on the same arc. Angles in the same segment are therefore equal.

A straight line that touches a circle at exactly one point, called the point of contact. It is perpendicular to the radius drawn to that point.

The amount of three-dimensional space enclosed by a solid figure, measured in cubic units. For a cylinder it equals the base area multiplied by the height.

The sum of the areas of all faces or curved surfaces of a solid, measured in square units. For a cone it includes the circular base plus the curved surface.

The distance measured along the sloping surface of a cone or pyramid from the apex to a point on the base edge. It is used in surface area formulas and differs from vertical height.

An average calculated from a frequency table by multiplying each class midpoint by its frequency, adding these products and dividing by the total frequency. It estimates the centre of grouped observations.

The middle value of an ordered data set, found for grouped data from the cumulative frequency table using the class containing the middle observation. It splits the data into two equal halves.

A number between zero and one expressing how likely an event is, found as favourable outcomes divided by all equally likely outcomes. A value of zero means impossible and one means certain.

The set of all possible outcomes of a random experiment, such as the six faces when a die is rolled. Each simple event is one element of this set.

The event that a given event does not happen, containing all outcomes of the sample space outside it. The probabilities of an event and its complement add up to one.

A well-defined collection of distinct objects, called its elements, considered as a whole. Sets are written with braces and may be described by listing elements or by a rule.

A set whose every element also belongs to another set. If A is a subset of B then each member of A is a member of B, and every set is a subset of itself.

The set formed by collecting all elements that appear in at least one of the given sets. Elements common to several sets are listed only once in the result.

The set of elements that belong to every one of the given sets at the same time. If the sets share no element, their intersection is the empty set.

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

The complete set of input values for which a function is defined. For a fraction, values making the denominator zero must be excluded from this set.

The set of all output values a function actually produces as its input runs through the domain. It is a subset of the codomain and need not include every possible value there.

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

The statement formed by swapping and negating both parts of a conditional statement. It is always logically equivalent to the original statement.

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