Number and Algebra

IB Higher Level Mathematics : Core

1. Number and Algebra

1.1

Sequences and series including arithmetic and geometric series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series. Applications of the above.

See Arithmetic and Geometric Sequences and Series and Sequences & Series for more information.

Definitions for sequences and series can also be found here.

1.2

Exponents and logarithms: laws of exponents; laws of logarithms.

Rules for Exponents is given here.

1.3

The binomial theorem for a positive integer exponent.

See Arithmetic Properties of Binomial Coeffiecients.

Also see Expanding Binomials, from Exercises in Math Readiness (EMR).

1.4

Proof by mathematical induction, standard results and other applications.
Forming conjectures to be proved by mathematical induction.

A proof of the principle of mathematical induction may be found at Countable Infinity.
Explanations are given at Mathematical Induction and Mathematical Induction, two different sites with the same name.

Some additional notes on Mathematical Induction can also be found here.

1.5

Complex numbers. The number

. The terms complex number, real part and imaginary part, conjugate, modulus and argument.

The form z = a + bi = (cos q + i sin q) and the use of z = r e^iq (Euler's formula).

The complex plane (Argand diagram).

See The Polar Form of a Complex Number : The unit circle, from S.O.S. MATHematics.
De Moivre's theorem is mentioned.

The same site has an explanation of e^iq = cos q + i sin q. See Euler's Formula.

See The Complex Plane, from S.O.S. MATHematics.

See also Complex and Complex Numbers for further details.

1.6

Sums, products and quotients of complex numbers.

2 + i is multiplied by i, shown in red, resulting in a rotation of p/2 in the complex plane.
The diagram was created using downloadable packages developed by METRIC :
Mathematics Education Technology Research at Imperial College.
Their learning modules, based on Mathematica, have resulted in a text :
Experiments in Undergraduate Mathematics.

1.7

De Moivre's Theorem (proof by mathematical induction). Powers and roots of a complex number.

For background material, see The Polar Form of a Complex Number : The unit circle.
Proof of the theorem (cosq + i sinq )ⁿ = cos n q + i sin n q is left as an exercise.

The 12 roots of 1.
Points representing the roots are symmetrically arrayed about the origin.
Graphed using Mathematica and METRIC packages.

1.8

Conjugate roots of polynomial equations with real coefficients.

Polynomial equations are discussed at Polynomials and Roots, from the University of Saskatchewan's Exercises in Math Readiness (EMR). The Advanced Exercises require the solution of equations having radicals as coefficients.