BSEH Secondary Mathematics - Basic key terms

A real number that cannot be written as a fraction of two integers; its decimal expansion is neither terminating nor repeating, such as the square root of 2 or pi.

A number expressible as p divided by q where p and q are integers and q is not zero; its decimal form either terminates or repeats a fixed block of digits.

For positive integers a and b, there exist unique whole numbers q and r with a equal to bq plus r, where r is at least zero and less than b.

Every integer greater than one can be expressed as a product of primes, and that prime factorisation is unique apart from the order of the factors.

The highest common factor: the largest positive integer that divides each number in a given set without leaving a remainder, found by prime factorisation or Euclid's algorithm.

The lowest common multiple: the smallest positive integer that is a multiple of every number in a given set; for two numbers it equals their product divided by their HCF.

An irrational root of a rational number written with a radical sign, such as root 3; it can often be simplified by removing perfect-square factors from the radicand.

Rewriting a fraction so its denominator contains no surd, usually by multiplying numerator and denominator by the conjugate of the denominator to create a rational result.

Two expressions of the form a plus root b and a minus root b; their product is rational, which makes them useful for removing surds from denominators.

A decimal expansion that stops after finitely many digits, occurring exactly when the fraction in lowest terms has a denominator whose only prime factors are 2 and 5.

A decimal in which a digit or block of digits recurs endlessly; it represents a rational number and can be converted back into a fraction.

The highest power of the variable appearing with a nonzero coefficient in the polynomial; it determines the polynomial's name and the maximum number of its real zeros.

A value of the variable that makes the polynomial equal zero; graphically it is the x-coordinate where the curve crosses or touches the x-axis.

For a quadratic ax squared plus bx plus c, the sum of zeros equals minus b over a and the product equals c over a; these identities let you build or check equations.

An equation of the form ax plus by plus c equals zero with a and b not both zero; its graph is a straight line and each point on it is a solution.

A pair of simultaneous linear equations that has at least one solution; the lines either intersect at one point or coincide completely.

A pair of simultaneous linear equations with no common solution, occurring when the two lines are parallel and have equal slopes but different intercepts.

Solving simultaneous equations by drawing both lines on one set of axes and reading off the coordinates of their point of intersection.

The rule giving the roots of ax squared plus bx plus c equals zero as negative b plus or minus the square root of b squared minus 4ac, all divided by 2a.

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

The classification of a quadratic's solutions according to the discriminant: positive gives two distinct real roots, zero gives equal real roots, negative gives no real roots.

A sequence in which each term after the first is obtained by adding a fixed constant to the previous term; examples include 3, 7, 11, 15.

The constant added to each term of an arithmetic progression to obtain the next term; it equals any term minus the term immediately before it.

The general term of an arithmetic progression, given by a plus n minus 1 times d, where a is the first term and d is the common difference.

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 rule giving the distance between two points as the square root of the sum of the squared differences of their x-coordinates and y-coordinates.

The rule giving the coordinates of the point that divides the segment joining two points in a given ratio, using weighted averages of the endpoint coordinates.

The special case of the section formula for a ratio of one to one, giving each coordinate of the midpoint as the average of the corresponding endpoint coordinates.

Three or more points that all lie on a single straight line; they can be tested by checking whether the sum of two pairwise distances equals the third.

Triangles with equal corresponding angles and proportional corresponding sides; their areas are in the ratio of the squares of corresponding sides.

Also called Thales' theorem: a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

If a line divides two sides of a triangle in the same ratio, then that line is parallel to the third side.

Triangles identical in shape and size, with all corresponding sides and angles equal; standard tests include side-side-side, side-angle-side, angle-side-angle and right-hypotenuse-side.

In a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides; its converse identifies right angles.

A line that touches a circle at exactly one point; it is perpendicular to the radius drawn to that point of contact.

A line that cuts a circle at two distinct points, forming a chord between them; as it moves outward it approaches the tangent position.

The two tangent segments drawn from a single external point to a circle are equal in length and make equal angles with the line joining the point to the centre.

A straight segment joining two points on a circle; the perpendicular from the centre to a chord bisects that chord.

The angle measured upward from the horizontal line of sight of an observer to an object situated above the observer's eye level.

The angle measured downward from the horizontal line of sight of an observer to an object situated below the observer's eye level.

The six ratios of pairs of sides in a right-angled triangle: sine, cosine, tangent, cosecant, secant and cotangent, defined relative to a chosen acute angle.

An equation involving trigonometric ratios that holds for every angle in its domain, such as sine squared plus cosine squared equals one.

Pairs of acute angles adding to 90 degrees; the sine of one equals the cosine of the other, and tangent pairs with cotangent, secant with cosecant.

The portion of a cone or pyramid remaining after its top is cut off by a plane parallel to the base; its volume uses the two base radii and the height.

One third of pi times the square of the base radius times the height; a cone is one third of the cylinder with the same base and height.

The total area covering a sphere, equal to four pi times the square of its radius; its curved surface alone is the same value.

The average computed from class intervals using midpoints, either directly, by assumed mean, or by step deviation, weighting each midpoint by its frequency.

The middle value found from the cumulative frequency table using the class containing the middle observation and a formula involving its lower limit and frequency.

The value occurring most often, estimated for grouped data from the modal class using its frequency and the frequencies of the classes on either side.

The running total of frequencies up to and including a given class; plotting these against class boundaries produces the ogive used to locate the median.

For moderately skewed distributions, mode approximately equals three times the median minus twice the mean; it estimates one average when the other two are known.

The ratio of the number of outcomes favourable to an event to the total number of equally likely outcomes in the sample space.

The set of all possible outcomes of a random experiment; its size is the denominator when outcomes are equally likely.

The event that a given event does not occur; the probabilities of an event and its complement sum to one.

An impossible event has probability zero because it contains no outcomes, while a sure event has probability one because it contains every outcome in the sample space.