1 Number and algebra

Introduction

Number and algebra is the toolkit the rest of the course is written in. It covers how quantities are written and rounded, how they grow by steady additions or by steady percentages, how logarithms undo exponents, and how equations and systems of equations are set up and solved with technology. Almost every later model, from a loan repayment to a population forecast, rests on one of these ideas.

The applied side is strong from the start. Very large and very small measurements are compared in standard form, every rounded answer carries an error that can be measured, and the same geometric sequence that describes the heights of a bouncing ball also describes savings, debt and depreciation. At HL the topic adds three new kinds of object, complex numbers, matrices and eigenvectors, that let one calculation handle rotations, alternating currents and systems that change in steps.

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How the subtopics fit together

The subtopics form four strands. The map of the topic shows each strand as a row, with the SL subtopics shaded and the HL-only subtopics dashed; an arrow means the first subtopic is used in the second.

The sequences and finance strand is the backbone of SL. An arithmetic sequence adds the same amount each step; a geometric sequence multiplies by the same amount, which is exactly what compound interest and depreciation do. Loans and annuities then add a regular payment to that growth, and the GDC's TVM solver handles all of it. The exponents and logarithms strand feeds into this chain, because finding when an investment reaches a target means solving an equation with the unknown in the exponent.

The equations strand starts with solving systems and polynomial equations on the GDC. At HL the same systems are rewritten as a single matrix equation, and quadratics with no real roots lead to complex numbers. The number and accuracy strand runs underneath everything, because every answer in the course is rounded and its bounds and percentage error say how far it can be trusted.

The subtopics

Links to other topics

Geometric sequences are the step-by-step version of the exponential models in subtopic 2.5, Modelling with functions, and logarithms return in subtopic 2.10 to straighten curved data on log scales. The GDC methods of subtopic 1.8 find the parameters of the quadratic and cubic models of subtopic 2.5 from points on their graphs, and percentage error measures how good the trapezoidal rule's estimate of an area is in subtopic 5.8.

The HL objects travel furthest. Matrices describe transformations of the plane in subtopic 3.9 and networks in subtopic 3.15; eigenvectors give the long-term state of a Markov chain in subtopic 4.19 and the behaviour of coupled differential equations in subtopic 5.17. Complex numbers add and subtract in the same way as the two-dimensional vectors of subtopic 3.10, which is why both are drawn as arrows.