Secondary 4 Mathematics Extended Part (M1/M2)

The layered classification of numbers: natural numbers, integers, rationals, irrationals and reals, expanding layer by layer.

A number that can be written as a ratio of two integers; integers, terminating and recurring decimals are all rational.

A number that cannot be written as a ratio of integers, with an infinite non-recurring decimal expansion, such as the square root of 2 or π.

The collective name for rational and irrational numbers, matching the points on the number line one to one.

An expression containing a root sign; a non-exact square root is the common way of writing an irrational number.

A surd whose radicand has no square factor and whose denominator contains no surd — the standard form for answers.

Surds with the same radicand after simplification; they combine by adding or subtracting their coefficients.

Multiplying numerator and denominator by a suitable expression so that the denominator contains no surd.

The expression differing from the original only by the middle sign, e.g. √5 + √2 and √5 − √2; their product leaves only rationals via the difference of squares.

A decimal whose digits repeat in a fixed pattern; it can always be written as a fraction and is therefore rational.

A polynomial with two terms, such as x + 2; the expansion of its powers is the subject of this lesson.

The theorem expanding a binomial to a positive integer power into a sum of n + 1 terms whose coefficients are combination numbers.

The result of writing a power as a sum of terms; the expansion of a binomial to the n-th power has n + 1 terms.

The triangle of expansion coefficients: each row starts and ends with 1, each interior number being the sum of the two above.

The coefficient of each term of an expansion, equal to the corresponding combination number C(n,r).

The number of ways to choose r from n distinct objects without regard to order, written C(n,r).

The general form of the (r + 1)-th term: C(n,r) times a to the (n − r)-th power times b to the r-th power.

The numerical factor in front of the letters of a term; in expansions it is decided by the combination number and the numerical coefficients together.

The term containing no letter (index zero); to find it, set the letter's index in the general term to zero.

The notation using the Greek letter Σ for summing a list of numbers; the binomial theorem can be written compactly with it.

The angle subtending an arc equal in length to the radius; 180° = π radians, the standard angle unit in the extended course.

The reciprocal of sine, written cosec x = 1/sin x.

The reciprocal of cosine, written sec x = 1/cos x.

The reciprocal of tangent, written cot x = 1/tan x = cos x/sin x.

Formulas expanding trigonometric functions of a sum or difference of two angles, e.g. sin(A + B) = sin A cos B + cos A sin B.

The special case of the compound formulas with equal angles: sin 2A = 2 sin A cos A; cos 2A has three forms.

An equality holding for all admissible angles, marked with the triple bar ≡; a proof must transform one side into the other.

An equation containing trigonometric functions; within a given interval it usually has more than one solution, all to be listed by quadrant.

One of the four regions cut out by the axes; the quadrant decides the signs of the trigonometric values.

The interval length over which a function repeats; the periods of sine and cosine are 360° and of tangent 180°.

The degree to which data spread away from the centre, paired with central tendency as the two key features of a data set.

Maximum minus minimum; the quickest measure but the most sensitive to extreme values.

The third quartile minus the first quartile — the spread of the middle half of the data, robust against extremes.

A diagram showing the minimum, Q1, median, Q3 and maximum on one line, handy for comparing groups; also called a box plot.

The average of the squared deviations, measuring spread; equal to the square of the standard deviation.

The positive square root of the variance, sharing the unit of the data; the larger it is, the more spread out the data.

The relative position obtained by subtracting the mean and dividing by the standard deviation; zero means exactly average.

The sum of all values divided by their count — the common measure of central tendency.

The tendency of data to cluster around a centre, described by the mean, median and similar measures.

A collection of values obtained from one observation or survey.

The number of ways to choose and order r objects from n distinct ones, written nPr.

The number of ways to choose r objects from n distinct ones without regard to order, written nCr.

The notation n! = n × (n − 1) × … × 1 with 0! = 1; factorials underpin the permutation and combination formulas.

If a task can be done in several non-overlapping categories, the total number of ways is the sum of the category counts.

If a task consists of successive stages, the total number of ways is the product of the stage counts.

Events that cannot occur together; the probability of their union is the sum of their probabilities.

Events whose occurrence does not affect each other's probability; their joint probability is the product.

The probability of A given that B has occurred, written P(A|B) = P(A∩B)/P(B).

A branching diagram of a multi-stage experiment with probabilities on the branches; multiply along paths and add matching outcomes.

A diagram of overlapping circles showing how events intersect and union, used with set notation A∩B, A∪B, A′ in probability.

A variable that maps the outcomes of a random experiment to numbers, usually denoted by a capital letter.

Taking only finitely many or countably many values; the probabilities of a discrete random variable can be listed one by one.

A table or formula listing each value of a random variable with its probability, all probabilities summing to one.

A single trial with only two outcomes, success and failure, the success probability being p.

The distribution of the number of successes in n independent Bernoulli trials, written X ~ B(n, p).

The distribution of the number of random events per unit of time (or space), written X ~ Po(λ).

The probability-weighted average of a random variable's values — the long-run average result; E(X) = Σ x P(X = x).

A measure of how widely a random variable fluctuates around its mean; both distributions in this lesson have ready-made formulas.

The constant that fixes a distribution's shape: n and p for the binomial, just λ for the Poisson.

The average number of occurrences per unit of time (or space) — the single parameter of the Poisson distribution.