 Surfaces
Surfaces are a special class of topological spaces that crop up in many places in the world of mathematics. In this free course, you will learn to classify surfaces and will be introduced to such concepts as homeomorphism, orientability, the Euler characteristic and the classification theorem of compact surfaces. First published on Thu, 18 Aug 2011 as
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Maths for science and technology
PLEASE NOTE: This course is being deleted from OpenLearn in April and will be replaced with a Badged Open version.You’re about to start a course in science and technology and you're wondering whether your level of maths is going to be enough to get you through. This free course, Maths for science and technology, will show you how to reflect on what you know, identify which skills you might need for your course, and help you to learn those skills using worked examples and activities.Author(s): Creator not set

Group theory
This free course consolidates and builds on group theory studied at OU level 2 or equivalent. Section 1 describes how to construct a group called the direct product of two given groups, and then describes certain conditions under which a group can be regarded as the direct product of its subgroups. Section 2 describes the key properties of the structure of cyclic groups. Section 3 introduces the notion of a set of generators of a group and a set of relations among the generators. It then looks a
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Number theory
This free course is an introduction to Number Theory. Section 1 provides a brief introduction to the kinds of problem that arise in Number Theory. Section 2 reviews and provides a more formal approach to a powerful method of proof, mathematical induction. Section 3 introduces and makes precise the key notion of divisibility. The Division Algorithm, concerning the division of one integer by another, is used. Its consequences, both practical and theoretical, make it a cornerstone of number theory.
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Univariate continuous distribution theory
This free course looks at a number of the basic properties of statistical models. Section 1 is concerned with the distributions of continuous random variables which are described by their probability density functions (pdfs) and cumulative distribution functions (cdfs). Section 2 is concerned with moments and covers, the concept of expectation or expected value, the familiar notion of the mean, also known as the first moment, two general definitions of moments, variance, random variables linked
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Metric spaces and continuity
This free course contains an introduction to metric spaces and continuity. The key idea is to use three particular properties of the Euclidean distance as the basis for defining what is meant by a general distance function, a metric. Section 1 introduces the idea of a metric space and shows how this concept allows us to generalise the notion of continuity. Section 2 develops the idea of sequences and convergence in metric spaces. Section 3 builds on the ideas from the first two sections to formu
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Rings and polynomials
This free course contains an introduction to rings and polynomials. We see that polynomial rings have many properties in common with the integers; for example, we can define a division algorithm, and this enables us to develop the analogue of the highest common factor for two polynomials. Section 1 explores the abstract definitions of a ring and a field. Sections 2 and 3 define polynomial rings where the coefficients of the polynomials are elements from a given field. Section 3 develops results
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Point estimation
This free course looks at point estimation, that is, the estimation of the value of the parameter of a statistical model by a single number, a point estimate for the parameter. Section 1 develops some aspects of maximum likelihood estimation. In particular, you will find out how to obtain the maximum likelihood estimator of an unknown parameter, using calculus. You will need to do lots of differentiation in this section. Section 2 introduces a number of important properties of point estimation.
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Introduction to the calculus of variations
This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler
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Linear programming – the basic ideas
This free course examines the formulation and solution of small linear programming problems. Section 1 deals with the formulation of linear programming models, describing how mathematical models of suitable real-world problems can be constructed. Section 2 looks at graphical representations of two-dimensional models, considers some theoretical implications and examines the graphical solution of such models. Section 3 introduces the simplex method for solving linear programming models and Section
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Author(s): The Open University

Lastly consider division. Dividing 8 by 2 means ‘How many times does 2 go into 8?’ or ‘What must you multiply 2 by to get 8?’. The answer is 4.

So to find 8 ÷ 2, you need to ask ‘What do I have to multiply 2 by to get 8?’. The answer is 4, since 2 × 4 = 8.

So 8 ÷ 2 = 4.

Similarly, to find 8 ÷ 2 you need to ask ‘what do I have to mul
Author(s): The Open University

Now that you have rules for addition and subtraction of negative numbers, think about multiplication and division.

Example 27

Describe each of the following in terms of the number line and the value of Thomas's piggy bank:

• (a) the mul
Author(s): The Open University

Activity 57

Evaluate each of the following and give an example from everyday life to illustrate the sum (e.g. Thomas's piggy bank).

• (a) 4 − 2

• (b
Author(s): The Open University

Next consider subtraction of a negative number. In terms of Thomas’s piggy bank, subtracting a negative number is the same as taking away one of his IOUs. If his mother says ‘you have been a good boy today so I’ll take away that IOU for £3’ this is equivalent to him being given £3.

So, − (3) = 3. Does this correspond with the number line interpretation of subtracting a negative number?

Consider the evaluation of 8 − 3. Continue to think o
Author(s): The Open University

Activity 56

Evaluate each of the following and give an example from everyday life to illustrate the sum (e.g. Thomas's piggy bank).

• (a) 4 − 6

• (b
Author(s): The Open University

Now think about adding a negative number, by looking at the number line again. Suppose you start at 0. Since 0 + 3 is the same as 3, you would expect that adding 3 to 0 on the number line should take you to the point marked 3 on the number line. So, 0 + 3 = 3.

Suppose Thomas's piggy bank were empty and he added an IOU for £3 to it. The value of the piggy bank would be 3.

Author(s): The Open University

Activity 52

Evaluate each of the following.

• [Image_Link]https://www.open.edu/openlearn/ocw/pluginfile.php/11742
Author(s):
The Open University

Activity 50

Evaluate 3 ÷ [Image_Link]https://www.open.edu/openlearn/ocw/pluginfile.php/117429/mod_oucontent/oucontent/2930/758510c5/b2a4fe7c/mu120_a_i021
Author(s):
The Open University

Activity 46

Evaluate each of the following.

• [Image_Link]https://www.open.edu/openlearn/ocw/pluginfile.php/11742
Author(s):
The Open University