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.