# Topic 1: Numbers and Algebra

## Concepts

Essential understandings

Number and algebra allow us to represent patterns, show equivalencies and make generalizations which enable us to model real-world situations. Algebra is an abstraction of numerical concepts and employs variables which allow us to solve mathematical problems.

Suggested concepts embedded in this topic

Generalization, representation, modelling, equivalence, patterns, quantity

AHL: Validity, systems.

Content-specific conceptual understandings:

Modelling real-life situations with the structure of arithmetic and geometric sequences and series allows for prediction, analysis and interpretation.

Different representations of numbers enable equivalent quantities to be compared and used in calculations with ease to an appropriate degree of accuracy.

Numbers and formulae can appear in different, but equivalent, forms, or representations, which can help us to establish identities.

Formulae are a generalization made on the basis of specific examples, which can then be extended to new examples.

Logarithm laws provide the means to find inverses of exponential functions which model real-life situations.

Patterns in numbers inform the development of algebraic tools that can be applied to find unknowns.

The binomial theorem is a generalization which provides an efficient method for expanding binomial expressions.

AHL: Proof serves to validate mathematical formulae and the equivalence of identities.

AHL: Representing partial fractions and complex numbers in different forms allows us to easily carry out seemingly difficult calculations.

AHL: The solution for systems of equations can be carried out by a variety of equivalent algebraic and graphical methods.

## SL Content

SL 1.1

Operations with numbers in the form $a \times 10^k$ where $1 \leq a < 10$ and $k$ is an integer.

SL 1.2

Arithmetic sequences and series.

Use of the formulae for the $n^{\mathrm{th}}$ term and the sum of the first $n$ terms of the sequence.

Use of sigma notation for sums of arithmetic sequences.

Applications.

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

SL 1.3

Geometric sequences and series.

Use of the formulae for the $n^{\mathrm{th}}$ term and the sum of the first $n$ terms of the sequence.

Use of sigma notation for the sums of geometric sequences.

Applications.

SL 1.4

Financial applications of geometric sequences and series:

compound interest

annual depreciation.

SL 1.5

Laws of exponents with integer exponents.

Introduction to logarithms with base 10 and $e$ .

Numerical evaluation of logarithms using technology.

SL 1.6

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof.

The symbols and notation for equality and identity.

SL 1.7

Laws of exponents with rational exponents.

Laws of logarithms.

$\log_a xy=\log_a x + \log_a y$

$\log_a \frac{x}{y} = \log_a x - \log_a y$

$\log_a x^m=m \log_a x$

for $a, x, y > 0$

Change of base algorithm.

$log_a x=\frac{log_b x}{log_b a}$ , for $a, b, x>0$

Solving exponential equations, including using logarithms.

SL 1.8

Sum of infinite convergent geometric sequences.

AHL 1.9

The binomial theorem: expansion of $(a+b)^n, n \in \mathbf{N}$

Use of Pascal's triangle and $C_r^n$

## AHL Content

AHL 1.10

Counting principles, including permutations and combinations.

Extension of the binomial theorem to fractional and negative indices, ie $(a+b)^n, n \in \mathbf{Q}$

AHL 1.11

Partial fractions.

AHL 1.12

Complex numbers: the number $i$ , where $i^2=-1$ .

Cartesian form: $z=a+bi$ ; the terms real part, imaginary part, conjugate, modulus and argument.

The complex plane.

AHL 1.13

Modulus–argument (polar) form: $z=r(\cos \theta + i \sin \theta)=rcis \theta$

Euler form: $z=re^{i \theta}$

Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

AHL 1.14

Complex conjugate roots of quadratic and polynomial equations with real coefficients.

De Moivre’s theorem and its extension to rational exponents.

Powers and roots of complex numbers.

AHL 1.15

Proof by mathematical induction.

Proof by contradiction.

Use of a counterexample to show that a statement is not always true.

AHL 1.16

Solutions of systems of linear equations (a maximum of three equations in three unknowns), including cases where there is a unique solution, an infinite number of solutions or no solution.

Last updated