An overview of Number and algebra: scientific notation, sequences and series, finance, exponents and logarithms, proof and the binomial theorem, plus the HL extensions to counting, complex numbers and proof.
Each one has its own page once something is published for it.
Number and algebra provides the language and tools used throughout the course. It begins with ways of writing and calculating with numbers of any size, moves on to sequences and series, which describe quantities that change in steps, and then develops exponents and logarithms, proof and the binomial theorem. Higher level adds counting methods, partial fractions, complex numbers, further methods of proof and systems of linear equations.
The topic is built in layers. Scientific notation (1.1) and the rounding conventions taught with it are used in almost every calculation in the course. Arithmetic and geometric sequences (1.2 and 1.3) introduce the two simplest patterns of change, adding a fixed amount and multiplying by a fixed factor, and their sums. Financial mathematics (1.4) applies geometric sequences to interest, depreciation and inflation.
Exponents and logarithms (1.5 and 1.7) supply the tools for solving equations in which the unknown is a power, such as finding how long an investment takes to double. Infinite geometric series (1.8) return to sequences and ask what happens when infinitely many terms are added. Simple deductive proof (1.6) and the binomial theorem (1.9) complete the SL content.
Subtopic | What it covers | Level |
|---|---|---|
1.1 Scientific notation | Numbers in the form ; rounding to decimal places and significant figures | SL |
1.2 Arithmetic sequences and series | Adding a fixed difference; , , sigma notation, applications | SL |
1.3 Geometric sequences and series | Multiplying by a fixed ratio; , , growth and decay models | SL |
1.4 Financial applications of geometric sequences | Compound interest, depreciation, inflation, the GDC finance solver | SL |
1.5 Exponents and introduction to logarithms | Laws of exponents, logarithms base 10 and , simple exponential equations | SL |
1.6 Simple deductive proof | Proving identities and results about integers; equations and identities | SL |
1.7 Laws of exponents and logarithms | Rational exponents, laws of logarithms, change of base, solving equations | SL |
1.8 Infinite geometric series | Convergence and the sum to infinity, | SL |
1.9 The binomial theorem | Expanding , Pascal's triangle, | SL |
1.10 Counting principles and the extended binomial theorem | Permutations, combinations, fractional and negative powers | HL |
1.11 Partial fractions | Splitting a fraction with two linear factors in the denominator | HL |
1.12 Complex numbers in Cartesian form | , arithmetic, the Argand diagram | HL |
1.13 Polar and Euler form | and ; geometric meaning of products | HL |
1.14 Complex roots and De Moivre's theorem | Powers and roots of complex numbers, roots of unity | HL |
1.15 Proof by induction, contradiction and counterexample | Further methods of proof | HL |
1.16 Systems of linear equations | Up to three equations in three unknowns; unique, infinite or no solutions | HL |
The HL subtopics extend the SL ones directly. Counting principles (1.10) generalise the binomial coefficients of 1.9. Complex numbers (1.12 to 1.14) extend the number system so that every quadratic equation has solutions, and De Moivre's theorem is proved by induction (1.15), which itself extends the deductive proof of 1.6.
Several of the course's key concepts run through the topic. Patterns and generalisation are central to sequences: a few terms suggest a rule, and the rule is written as a formula for that holds for every term. Change appears in the contrast between arithmetic sequences, which change by equal amounts, and geometric sequences, which change by equal percentages and model exponential growth and decay.
Modelling connects the topic to the world outside mathematics: salaries, populations, loans and the spread of disease can all be described by sequences, and a model's predictions have to be judged against reality. Equivalence underlies the laws of exponents and logarithms, and the different forms of a complex number. Validity is the heart of proof, which separates a result that has been checked in a few cases from one that is known to be true in every case. Quantity and approximation appear in scientific notation, in rounding, and in using a finite number of terms to estimate an infinite sum.
SL students study subtopics 1.1 to 1.9. HL students study the same subtopics and also 1.10 to 1.16. The HL material is more abstract: complex numbers, the extended binomial theorem and formal methods of proof are tools for later work in calculus and in the HL parts of other topics, such as series expansions and the geometry of lines and planes.
The entries of the mathematics: analysis and approaches formula booklet that 1 uses. Each opens in the explained booklet.
And 10 more in the formula booklet.
With a free account, Augment remembers where you stopped and what you have practiced.