# Topic 1: Number 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 to solve mathematical problems.

Suggested concepts embedded in this topic

Generalization, representation, modelling, equivalence, approximation, quantity

AHL: Systems, relationships.

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 quantities to be compared and used for computational purposes with ease and 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

Mathematical financial models such as compounded growth allow computation, evaluation and interpretation of debt and investment both approximately and accurately.

Approximation of numbers adds uncertainty or inaccuracy to calculations, leading to potential errors but can be useful when handling extremely large or small quantities.

Quantities and values can be used to describe key features and behaviours of functions and models, including quadratic functions.

AHL: Utilizing complex numbers provides a system to efficiently simplify and solve problems.

AHL: Matrices allow us to organize data so that they can be manipulated and relationships can be determined.

AHL: Representing abstract quantities using complex numbers in different forms enables the solution of real-life problems.

## SL Content

SL 1.1

SL 1.2

Arithmetic sequences and series.

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 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.

Numerical evaluation of logarithms using technology.

SL 1.6

Approximation: decimal places, significant figures.

Upper and lower bounds of rounded numbers.

Percentage errors.

Estimation.

SL 1.7

Amortization and annuities using technology.

SL 1.8

Use technology to solve:

Systems of linear equations in up to 3 variables

Polynomial equations

## AHL Content

AHL 1.9

Laws of logarithms:

AHL 1.10

Simplifying expressions, both numerically and algebraically, involving rational exponents.

AHL 1.11

The sum of infinite geometric sequences.

AHL 1.12

Calculate sums, differences, products, quotients, by hand and with technology. Calculating powers of complex numbers, in Cartesian form, with technology.

The complex plane.

AHL 1.13

Conversion between Cartesian, polar and exponential forms, by hand and with technology.

Calculate products, quotients and integer powers in polar or exponential forms.

Adding sinusoidal functions with the same frequencies but different phase shift angles.

Geometric interpretation of complex numbers.

AHL 1.14

Multiplication of matrices. Properties of matrix multiplication:

associativity, distributivity and non-commutativity.

Identity and zero matrices.

Solution of the systems of equations using inverse matrix.

AHL 1.15

Eigenvalues and eigenvectors.

case where there are distinct real eigenvalues).

Last updated