Diskret Matematik A

Introduction

This page gives references to relevant pre-university mathematics for students who are going to be studying a discrete mathematics course at Swedish university level A. There is a lot of material here, and doing it all is at least 4 weeks of study. It is important to note that you are not formally required to study any of the material here, but everything here is relevant to you, so I recommend that you use this page to brush up on topics on which you are rusty or insecure.

The links on this page is to the Mathtutor website which was created by a group of teachers, mathematicians and new media producers from the Universities of Leeds, Loughborough and Coventry in England in an attempt to bridge the gap between school and university studies in mathematics.

Mathtutor pages were designed for distribution on DVD-Rom, and so it has particular system requirements, even when viewed online. You will need a PC running Windows, Internet Explorer and Windows Media Player in order to see the video tutorials, and Adobe Reader to view the summary texts. The interactive diagnostic tests and exercises also use some software called an Active-X control, which is downloaded the first time you view them.

Most important

The areas where first-year students traditionally have the biggest problems are the following:

In the 'Working with numbers'-section: rules of artihmetic and working with expressions involving square roots.

In the 'Basic algebra'-section: working with powers, removing brackets, solving quadratic equations and simplyfying fractions where the numerator and denominator involve unknowns.

I advise that everybody has a look at these important topics.

Working with numbers

Click here for the Mathtutor website revising arithmetic

The front page of the site contains a list of topics and a short description of each topic. When you click on 'View Online' you get a pop-up where the list of topics is 'live' and you can click on a topic allowing you to access either a revision sheet by clicking on 'Summary text' or even a video tutorial. There is also a self-marking online diagnostic test for each topic to help you assess whether you have understood the topic. Unfortunately the online diagnostic tests only work properly in Internet Explorer on Windows with Active-X controls enabled, so you will need this particular set-up if you want to test yourself. All the topics in arithmetic covered by the Mathtutor website are important for you. They are:

• Fractions: Basic ideas
  Keywords: What is a fraction? Equivalent fractions, the numerator (sv: täljar), the denominator (sv: nämnar).


• Fractions: adding and subtracting
  Keywords: How to add and subtract mixed fractions by finding a common denominator and turning them into improper fractions.


• Fractions: multiplying and dividing
  Keywords: How to multiply fractions, and how to divide fractions by turning the second fraction upside down.


• Decimals
  Keywords: The decimals corresponding to a fraction. Rounding to given numbers of decimal places or significant figures. Irrational numbers.


• Percentages
  Keywords: How to carry out calculations involving percentages. The use of the percentage button on your calculator.


• Ratios


• Rules of arithmetic
  Keywords: Evaluating expressions with mixed arithmetic operations. Order of precedence of arithmetic operations. Brackets. Addition, subtraction, multiplication and division with negative numbers.


• Surds and other roots
  Keywords: Powers (sv: potenser). Roots (sv: rötter). Rational/irrational numbers. How to simplify expressions involving roots.

Using a calculator effectively

It is important to be able to use a computer or pocket calculator to do correct arithmetic. Many basic calculators need help to give the right answer to a numerical calculation. Try to evaluate the expression

3 + 2 × 4
on your calculator. The answer should be 11. Did the calculator tell you the result was 20? If so, it is because your calculator has not been programmed to know that multiplication takes precedence over addition. You have to help it!

Similarly, try to work out
-2⁴
on your calculator. The answer should be -16. Did the calculator tell you the result was 16? If so, it is because it wrongly has interpreted your input as (-2)⁴.

You need to be aware of which arithmetic operations have precedence (The order of precedence is BODMAS: Brackets, pOwers, Division/Multiplication, Addition/Subtraction), and you need to know how to get the right result out of your calculator.

Click here for a quiz testing correct basic use of a calculator.
To revise how to use a calculator: read the 'Revision Bite'-factsheet on the quiz page, then do the quiz.

Basic Algebra

Click here for the Mathtutor website revising algebraic skills

The front page of the site contains a list of topics and a short description of each topic. When you click on 'View Online' you get a pop-up where the list of topics is 'live' and you can click on a topic allowing you to access either a revision sheet by clicking on 'Summary text' or even a video tutorial. There is also a self-marking online diagnostic test for each topic to help you assess whether you have understood the topic. Unfortunately the online diagnostic tests only work properly in Internet Explorer on Windows with Active-X controls enabled, so you will need this particular set-up if you want to test yourself. The topics important for you in basic algebra covered by the Mathtutor website are:

• Mathematical language
  Keywords: Mathematical symbols, variables, the Greek alphabet.


• Powers or indices
  Keywords: Powers or indices (sv: potenser) and the rules of arithmetic for expressions involving them.


• Substitution and formulae(sections 1, 2 and 3 of the summary text only)
  Keywords: The formula for the area of a circle and volume of a sphere. How to substitute values into a formula. Using the formulas to find the area/volume of a given circle/sphere.


• Expanding and removing brackets
  Keywords: How to remove brackets from expressions, multiplying brackets together, dealing with nested brackets.


• Factorising quadratics
  Keywords: Multiplying two brackets, factorising quadratics, dealing with quadratics where the coefficient of x² is not 1. Quadratics where the constant term is zero. The formula for the the difference of two squares (sv: konjugatregeln).


• Transposition of formulae
  Keywords: How to rearrange expressions/equations into a different, but equivalent form.


• Linear equations in one variable
  Keywords: Examples of linear equations and how to solve them. In any equation there is an unknown quantity, x say, that we are trying to find. In a linear equation this unknown quantity will appear only as a multiple of x, and not as a more complicated function of x such as e.g. x², x³ or sin(x).


• Quadratic equations (sections 1, 2 and 4 of the summary text only)
  Keywords: Solving quadratic equations of the form ax² + bx + c = 0 by factorisation and by using a formula.


• Simplifying fractions
  Keywords: How to simplify fractions where the numerator and denominator involve variables.

Extra topics in basic algebra for students who are going study Calculus or Analysis at a later stage:

• Solving inequalities
• Cubic equations
• Polynomial division
• Partial fractions
  
  
  

Functions

Click here for the Mathtutor website revising the basics about functions

The front page of the site contains a list of topics and a short description of each topic. When you click on 'View Online' you get a pop-up where the list of topics is 'live' and you can click on a topic allowing you to access either a revision sheet by clicking on 'Summary text' or even a video tutorial. There is also a self-marking online diagnostic test for each topic to help you assess whether you have understood the topic. Unfortunately the online diagnostic tests only work properly in Internet Explorer on Windows with Active-X controls enabled, so you will need this particular set-up if you want to test yourself. Your discrete mathematics A course contains a block about functions, so you only need to revise the very basic Mathtutor website topics here, namely:

• Introduction to functions
  Keywords: What is a function? Plotting the graph of a function. When is a function valid? Domain (sv: definitionsmängd) of a function. Range (sv: värdemängd) of a function.


• Linear Functions
  Keywords: What is a linear function? How to recognise a linear function, how to find the slope (sv: lutning) and the y-intercept of its graph.


• Polynomial Functions (sections 1, 2 and 3 of the summary text only)
  Keywords: What is a polynomial function? Graphs of polynomial functions.

Extra topics about functions for students who are going to study Calculus or Analysis at a later stage:

• Exponential and logarithm functions
• Trigonometric functions
• Composition of functions
• Inverse functions