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Center for languages and International Education, Institute of Education, University College London

chris.fenwick@ucl.ac.uk


Mathematics notes

I am in the process of writing complete notes on the topics listed below. The level of these notes are/will be suitable for both A-level students and 1st year university students. The topics I will inlude will be on Functions, Algebra, Differentiation, Integration, Series, and Complex Numbers.

In these notes I have tried / will try to do three things which are not often seen in textbook of maths at this level:

  1. to provide detailed conceptual explanations of things whose understanding are often taken for granted;
  2. to introduce variations on each topic in order to broaden and deepen one's learning of the topic being studies. Such variations are (virtually) never seen in standard textbooks, but can certainly be understood by A-Level and 1st year undergrad students. They taken from certain mathematics journals such as Mathematical Gazette, Two-Year College Mathematics Journal, Mathematical Spectrum, Mathematics Teacher, etc.;
  3. to include as many diagrams as possible throughout in order to visualise the mathematics or concept being discussed;
(it is for these reasons that my notes are significantly longer, and the file size significantly larger, than is usual for notes of these types). It is because of these three aspects that I believe that the notes below will be useful for 1st year university students studying maths, as well as A-level students.

Other maths stuff, such as software, other peoples' notes, and solution to some textbook exercises, can be found here

Note:


Quick access: Functions | Algebra | Differentiation | Integration | Series | Complex numbers


Notes:
On functions as transformations, and the transformation of functions

Table of content
  • 1.1 Functions as transformation;
  • 1.2 Describing the behaviour of f(x) under certain transformations,

    • 1.2.1 Transforming f(x) to f(x) ± k, where k is a constant,
    • 1.2.2 Transforming f(x) to f(x ± k), where k is a constant,
    • 1.2.3 Transforming f(x) to k.f(x), where k is a constant,
    • 1.2.4 Transforming f(x) to f(kx), where k is a constant,
    • 1.2.5 Transforming f(x) to |f(x)|,
    • 1.2.6 Transforming f(x) to f(|x|);

  • 1.3 One complete example;
  • 1.4 How to identify a function from a graph,

    • 1.4.1 Example 1: A linear function,
    • 1.4.2 Example 2: A square root function,
    • 1.4.3 Example 3: A quadratic function,
    • 1.4.4 Example 4: A sine function,
    • 1.4.5 Example 5: A piecewise function;

  • 1.5 Summary.

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Notes: On the factorisation of polynomials - part I: To come

Notes: On the factorisation of polynomials - part II

Table of content
  • 2.6 Aspects of factorisation III: The remainder theorem and long division,

    • 2.6.1 Division by linear divisors,
    • 2.6.2 The remainder theorem,
    • 2.6.3 Proof of the remainder theorem − version 1,
    • 2.6.4 Long division,
    • 2.6.5 Proof of the remainder theorem − version 2,
    • 2.6.6 Some comments about the form of the remainder theorem,
    • 2.6.7 Synthetic division for linear divisors,
    • 2.6.8 Linear divisors of the form ax − c,
    • 2.6.9 A study of the remainder theorem: An iterative application of the remainder theorem,
    • 2.6.10 The remainder theorem for quadratic divisors,
    • 2.6.11 Synthetic division for quadratic divisors,
    • 2.6.12 A modified synthetic division algorithm for quadratic divisors,
    • 2.6.13 Quadratic divisors of the form ax2 + bx + c,
    • 2.6.14 Extending the idea of the remainder theorem: Divisors of degree m < n.

  • 2.7 Selected studies on the remainder theorem;

    • 2.7.1 A study of the remainder theorem for linear divisors,
    • 2.7.2 A study of the remainder theorem for quadratic divisors,
    • 2.7.3 A generalisation of the remainder theorem,
    • 2.7.4 Divisibility rules.

  • 2.8 Aspects of factorisation IV: The factor theorem,

    • 2.8.1 The factor theorem,
    • 2.8.2 A study of the factor theorem,
    • 2.8.3 Repeated factors,
    • 2.8.4 Extending the idea of repeated factors,
    • 2.8.5 The rational root theorem,
    • 2.8.6 The rational root test in conjunction with the intermediate value theorem,
    • 2.8.7 Caveats to the rational root theorem,
    • 2.8.8 The rational root test in conjunction with synthetic division by quadratic divisors.

  • 2.9 A comment about irreducibility: When is it possible to factorise?

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Notes: Differentiation 1: On the definition of the derivative, and the proof of some basic derivatives

Table of content
  • 1.1 Introduction;
  • 1.2 The derivative as the slope of a curve at a point,

    • 1.2.1 Approaching the derivative from one direction,
    • 1.2.2 A comment about tangents,
    • 1.2.3 Approaching the derivative from the other direction,
    • 1.2.4 A comment about limits,
    • 1.2.5 Becoming more rigorous;

  • 1.3 The derivative as a functions representing the slope of the curve as a whole,

    • 1.3.1 One way of understanding df/dx as a function,
    • 1.3.2 Another way of understanding df/dx as a function,

  • 1.4 On the formal definition of the first derivative;
  • 1.5 The derivative of xn from 1st principles;
  • 1.6 The derivative as a transformation from position to slope;
  • 1.7 The derivative as a measure of sensitivity;
  • 1.8 The derivative as a measure of distribution,

    • 1.8.1 Functions as distributions of values, and how these distributions relate to rates of change,
    • 1.8.2 The rate of change of distribution in more detail,

  • 1.9 The second and third derivatives (to come);
  • 1.10 Equations involving the derivative in nature: Selected examples;
  • 1.11 The derivative of other basic functions from 1st principles,

    • 1.11.1 The derivative of sin(x),
    • 1.11.2 A geometric proof that limθ→0(sin(θ)/θ)
    • 1.11.3 The derivative of cos(x),
    • 1.11.4 The derivative of tan(x),
    • 1.11.5 The derivative of sec(x), cosec(x), and cot(x),
    • 1.11.6 The derivative of ax,
    • 1.11.7 The derivative of logax,
    • 1.11.8 A summary about the concept of limits;

  • 1.12 Not all functions have a derivative,

    • 1.12.1 Functions continuous at a point but whose derivatives are not continuous at that point,
    • 1.12.2 Functions which are not continuous at a point,
    • 1.12.3 A continuous function that does not have a slope anywhere along its curve;

  • 1.13 A study on derivatives and tangents (to come);

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Notes: Differentiation 2: On the rules of differentiation

Table of content
  • 1.1 Introduction;
  • 1.2 The multiplication by a constant rule for differentiation,

    • 1.2.1 An informal proof of the multiplication by a constant rule for differentiation,
    • 1.2.2 A formal proof of the multiplication by a constant rule for differentiation,

  • 1.3 The addition and subtraction rule for differentiation,

    • 1.3.1 An informal derivation of the addition and subtraction rules for derivatives,
    • 1.3.2 A formal proof of the addition and subtraction rules for derivatives,

  • 1.4 On the product rule for differentiation: d(u.v)/dx,

    • 1.4.1 An informal derivation of the product rule,
    • 1.4.2 A note on the proof of the product rule of the previous section,
    • 1.4.3 The formal proof of the product rule,
    • 1.4.4 A proof of the power rule for positive integers using the product rule,
    • 1.4.5 The product rule for more that two functions,
    • 1.4.6 Repeated differentiation of f(x) = u.v − Leibniz's rule,
    • 1.4.7 Proof of Leibniz's product rule for differentiation: To come,
    • 1.4.8 When does d(u.v)/dx = (du/dx)*(dv/dx)?;

  • 1.5 On the quotient rule for differentiation: d(u/v)/dx,

    • 1.5.1 An informal derivation of the quotient rule,
    • 1.5.2 The formal proof of the quotient rule,
    • 1.5.3 An alternative proof of the quotient rule,
    • 1.5.4 An extension of the quotient rule,
    • 1.5.5 Proof of the power rule for negative exponents using the quotient rule,
    • 1.5.6 The quotient rule for more that two functions,
    • 1.5.7 When does d(u÷v)/dx = (du/dx)÷(dv/dx)?;

  • 1.6 On the chain rule for differentiation: d[f(g(x))]/dx,

    • 1.6.1 A conceptual description of the chain rule,
    • 1.6.2 More on the effect of g(x),
    • 1.6.3 A formal proof of the chain rule,
    • 1.6.4 The chain rule for higher derivatives;

  • 1.7 Implicit differentiation,

    • 1.7.1 The derivative of inverse trig functions,
    • 1.7.2 A proof of the power rule for rational exponents,
    • 1.7.3 Proof of the power rule for all real exponents,
    • 1.7.4 The derivative of the general exponential y = ax;

  • 1.8 A study in differentiation,

    • 1.8.1 A fictitious definition for dy/dx,
    • 1.8.2 Another fictitious definition for dy/dx,
    • 1.8.3 A problem,
    • 1.8.4 An alternative proof of the product rule and quotient rule;

  • 1.9 More difficult examples,

    • 1.9.1 Example 1
    • 1.9.2 Example 2
    • 1.9.3 Example 3
    • 1.9.4 Example 4
    • 1.9.5 Example 5

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Notes:
Complex numbers 1 − part 1: Introducing complex numbers, graphing complex numbers, doing arithmetic on complex numbers, and the polar form of a complex numbers.

Table of content
  • 1.1 A Brief history of complex numbers;
  • 1.2 On quadratic equations having Δ < 0: Defining imaginary and complex numbers,

    • 1.2.1 Solving quadratics with Δ < 0 − Part 1,
    • 1.2.2 Defining the imaginary number,
    • 1.2.3 Solving quadratics with Δ < 0 − Part 2,
    • 1.2.4 Defining a complex number,
    • 1.2.5 Deriving the quadratic formula using complex numbers,
    • 1.2.6 The arithmetic of i,
    • 1.2.7 The conjugate of a complex number,
    • 1.2.8 Certain properties of the conjugate;

  • 1.3 Complex roots can be located on a Cartesian graph − Part 1,

    • 1.3.1 Locating the complex roots of a quadratic equation,
    • 1.3.2 Locating the complex roots of a cubic equation;
  • 1.4 The Argand diagram, |z| and arg(z);
  • 1.5 Complex roots can be located on a Cartesian graph − Part 2,

    • 1.5.1 Locating the complex roots of a quadratic equation,
    • 1.5.2 Generalising the analysis of the complex roots of a real-valued quadratic,
    • 1.5.3 Locating the complex roots of a cubic equation;
  • 1.6 On addition and subtraction of complex numbers,

    • 1.6.1 Addition and subtraction of complex numbers,
    • 1.6.2 The geometric effect of addition/subtraction on complex numbers;
  • 1.7 On Multiplication of complex numbers,

    • 1.7.1 Multiplying two complex numbers,
    • 1.7.2 The geometric effect of multiplication on complex numbers,
    • 1.7.3 The effect of continual multiplication by a complex number;
    • 1.7.4 Finding the square root of a complex number;
  • 1.8 On division of complex numbers,

    • 1.8.1 The division of two complex numbers,
    • 1.8.2 The geometric effect of division on complex numbers;
  • 1.9 On Multiplication of complex numbers,

    • 1.9.1 Some attempts to order complex numbers, and why they fail,
    • 1.9.2 Trying to preserve the ordering of complex numbers under addition and multiplication,
    • 1.9.3 Another attempt to order complex numbers, and why it fails,
    • 1.9.4 Yet another attempt to order complex numbers, and why it fails,
    • 1.9.5 Conclusion;
  • 1.10 The quadratic formula for a quadratic equation whose coefficients are complex numbers;
  • 1.11 The Polar form of a complex number,

    • 1.11.1 The polar form of a complex number,
    • 1.11.2 Proof of the quadratic formula using polar form of a complex number,
    • 1.11.3 Choosing the correct argument for z,
    • 1.11.4 The geometric effect of powers of i;
  • 1.12 On exponentiation of complex numbers: DeMoivre's theorem,

    • 1.12.1 DeMoivre's theorem,
    • 1.12.2 Proof of DeMoivre's theorem (up to rational powers),
    • 1.12.3 The periodicity of a complex number,
    • 1.12.4 Multiplication and division of complex numbers in polar form,
    • 1.12.5 The geometric effect of multiplication and division of complex numbers in polar form;

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Notes:
Complex numbers 1 − part 2: Roots of complex numbers, trig identities via complex numbers, the exponential form of a complex number, and properties of |z| and Arg(z)

Table of content
  • 1.13 On finding roots of a complex number,

    • 1.13.1 The general roots of a complex number;
    • 1.13.2 Deriving the equation for the roots of a complex number;
    • 1.13.3 Extending exponentiation to rational numbers − The case of irreducible p/q;
    • 1.13.4 Deriving DeMoivre's theorem for an irreducible exponent p/q;
  • 1.14 Issues when finding roots of a complex number,

    • 1.14.1 Issue 1: An inconsistency in DeMoivre's theorem − The rooting operations gives different results;
    • 1.14.2 Issue 2: An inconsistency in DeMoivre's theorem − The case of reducible p/q;
    • 1.14.3 Issue 3: The non-distributive nature of taking roots of a complex number;
  • 1.15 On the roots of unity of a complex number,

    • 1.15.1 The structure of the roots of unity;
    • 1.15.2 Finding π from a study of roots of unity ;
    • 1.15.3 Sums of roots of unity;
    • 1.15.4 The cyclic nature of roots of unity;
    • 1.15.5 More complicated examples;
    • 1.15.6 Products of roots of unity;
    • 1.15.7 Primitive roots of unity;
    • 1.15.8 Certain properties of roots of unity;
  • 1.16 On deriving trigonometric identities via complex numbers,

    • 1.16.1 Trig identities involving powers of trig functions;
    • 1.16.2 Trig identities involving multiples of θ;
    • 1.16.3 Examples on deriving trig identities;
    • 1.16.4 An alternative approach to deriving the identity for tan(nθ);
    • 1.16.5 A Proof without words;
    • 1.16.6 Finding exact values to trig and inverse trig equations;
  • 1.17 On the connection between general roots of a complex number and roots of unity − To come,

  • 1.18 On deriving trigonometric identities via complex numbers,

    • 1.18.1 Euler's formula;
    • 1.18.2 Proof of DeMoivre's theorem for all real powers;
    • 1.18.3 The geometric intepretation of eiθ;
    • 1.18.4 The relationship between cos and sin and eiθ;
    • 1.18.5 More examples;
    • 1.18.6 Roots of complex numbers in exponential form;
    • 1.18.7 More complicated examples;
  • 1.19 On deriving trigonometric identities via complex numbers,

    • 1.19.1 Properties of |z|;
    • 1.19.2 Properties of Arg(z);

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