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

  • 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

  • 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:

    • $\log_a xy=\log_a x + \log_a y$

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

    • $\log_a x^m=m \log_a x$

    • for $a, x, y > 0$

• AHL 1.10

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

• AHL 1.11

  • The sum of infinite geometric sequences.

• AHL 1.12

  • Complex numbers: the number $i$ such that $i^2=-1$ . Cartesian form: $z=a+bi$ ; the terms real part, imaginary part, conjugate, modulus and argument.

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

  • The complex plane.

  • Complex numbers as solutions to quadratic equations of the form $ax^2 + bx + c, a \neq 0$ , with real coefficients where $b^2 - 4ac < 0$ .

• AHL 1.13

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

  • Exponential form: $z=re^{\theta}$

  • 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

  • Definition of a matrix: the terms element, row, column and order for $m \times n$ matrices.

  • Algebra of matrices: equality; addition; subtraction; multiplication by a scalar for $m \times n$ matrices.

  • Multiplication of matrices. Properties of matrix multiplication:

  • associativity, distributivity and non-commutativity.

  • Identity and zero matrices.

  • Determinants and inverses of $n \times n$ matrices with technology, and by hand for $2 \times 2$ matrices.

  • Awareness that a system of linear equations can be written in the form $Ax=b$ .

  • Solution of the systems of equations using inverse matrix.

• AHL 1.15

  • Eigenvalues and eigenvectors.

  • Characteristic polynomial of $2 \times 2$ matrices.

  • Diagonalization of $2 \times 2$ matrices (restricted to the

  • case where there are distinct real eigenvalues).

  • Applications to powers of $2 \times 2$ matrices.