Symbols are the basic building blocks of mathematics. After you have studied mathematics at advanced level for a while you will come to appreciate that certain symbols tend to mean certain things. For example x and y are used to represent variables, whereas a and b are used to stand for constants. Greek symbols are commonly used too.

A proof is a convincing demonstration that some mathematical statement is necessarily true, within the accepted standards of the field. A proof is a logical argument, not an empirical one. That is, the proof must demonstrate that a proposition is true in all cases to which it applies, without a single exception. An unproven proposition believed or strongly suspected to be true is known as a conjecture.

The concept of proof is central to mathematics at an advanced level.
NOTE: This section is included here to introduce the ideas of logical deductive proof but it may not be an explicit part of your syllabus or in your examination.

Laws of indices for all rational exponents. Use and manipulation of surds. Quadratic functions, equations and graphs. Completing the square. Simultaneous equations. Solution of linear and quadratic inequalities. Algebraic manipulation of polynomials, including expanding brackets and collecting like terms, factorisation. Graphs of functions; sketching curves defined by simple equations. Geometrical interpretation of algebraic solution of equations. Use of intersection points of graphs of functions to solve equations. Knowledge of the effect of simple transformations on the graph of y = f(x) as represented by y = af(x), y = f(x) + a, y = f(x + a), y = f(ax).

Much of the Basics section is aimed at helping bridge the gap between GCSE and advanced level. It could form the basis of summer work before the advanced level course begins.

Equation of a straight line, including the three common forms y = mx + c, y - y1 = m(x - x1) and ax + by + c = 0. The equation of a line through two given points and the equation of a line parallel (or perpendicular) to a given line through a given point. Conditions for two straight lines to be parallel or perpendicular to each other. Try the Geogebra page for an on-line coordinate geometry program where you can try out some ideas about linear equations.

How to generate sequences from the formula for the nth term;

how to find the nth term and sum of the first n terms of an arithmetic sequence;

how to use summation notation

Differentiation is used to find the gradient function (derivative) for a curve, the gradient at any point on a curve, and also to find the equation of the tangent or normal to a curve at a point on the curve. Differentiation, as part of calculus, is used in science and engineering, and was developed originally in the 17th century by Newton and Leibniz.

Integration may be seen as the reverse of differentiation. The principles of integration were formulated by Isaac Newton and Gottfried Leibniz in the late seventeenth century. Integration can be used to find areas and volumes of mathematically defined shapes and is used extensively in science.