Introduction to differential equations
Differential equations are any equations that include derivatives and arise in many situations. This free course, Introduction to differential equations, considers three types of first-order differential equations. Section 1 introduces equations that can be solved by direct integration and section 2 the method of separation of variables. Section 3 looks at applications of differential equations for solving real world problems. Section 4 introduces the integrating factor method for solving linear
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Modelling and estimation
This free course is concerned with modelling and estimation and looks in particular at the binomial distribution. Section 1 starts by defining probability, introduces relevant notation and briefly discusses basic properties of probabilities. The section concludes by considering some of the general features of and ideas about modelling discrete random variables. Section 2 looks at one particular probability model for discrete data, the binomial distribution. Section 3 investigates how data can be
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Linear programming â€“ 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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Modelling events in time
This free course develops ideas about probability and random processes. Sections 1 and 2 introduce the fundamental ideas of random processes through a series of examples. Section 3 describes a model that is appropriate for events occurring â€˜at randomâ€™ in such a way that their rate of occurrence remains constant. Section 4 derives the main results from Section 3. Section 5 introduces the multivariate Poisson process in which each event may be just one of several different types of event. Sect
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Assessing risk in engineering, work and life
Risk is something that must be taken into account at all times when practising as an engineer. We accept risk as part of everyday life so there is a need to balance the risks of an activity against the benefits that it brings. This free course, Assessing risk in engineering, work and life, investigates how we can manage risks at work, in the home and in the wider community. Risk is tied in with accidents so it is vital to understand how accidents arise, how likely they are to occur and how t
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Babylonian mathematics
This free course looks at Babylonian mathematics. You will learn how a series of discoveries has enabled historians to decipher stone tablets and study the various techniques the Babylonians used for problem-solving and teaching. The Babylonian problem-solving skills have been described as remarkable and scribes of the time received a training far in advance of anything available in medieval Christian Europe 3000 years later. Author(s): Creator not set

This free course, Squares, roots and powers, reminds you about powers of numbers, such as squares and square roots. In particular, powers of 10 are used to express large and small numbers in a convenient form, known as scientific notation, which is used by scientific calculators.

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This OpenLearn course provides a sample of level 1 study in Author(s): The Open University

Except for third party materials and otherwise stated in the acknowledgements section, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.

Course image: russellstreet in Flickr made available under
Author(s): The Open University

The rules for multiplying positive and negative numbers can be illustrated by the table below.

Multiplying a positive number by a positive number gives a positive answer.

Multiplying a negative number by a positive number gives a negative answer.

Multi
Author(s): The Open University

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

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

Consider the evaluation of 8 âˆ’ âˆ’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) âˆ’4 âˆ’ 6

• (b
Author(s): The Open University

How about other fractions? What is 6 Ã· ? This means how many Author(s): The Open University

## Activity 46

Evaluate each of the following.

• Author(s): The Open University

The same rules about the order of calculations apply to decimals as apply to whole numbers.

## Calculations are performed in the following order:

Brackets;

Powers (e.g. squaring or cubing a number);

Division and Multiplication (performed in the order written, left to
Author(s): The Open University

## Activity 31

Insert brackets in the following calculations to emphasise the order in which a scientific calculator would perform them, then do the calculations by hand and on your calculator, with and without the bracke
Author(s): The Open University

You may have noticed that sometimes the order in which calculations are carried out seems to matter and sometimes it does not. When using a calculator, it is very important to know the order in which it will do calculations. It is not always the order in which you enter them.

Although written English is read from left to right, this is not the case for all written languages (Chinese is read top to bottom, right to left). With mathematics, the order of the written operations does not alw
Author(s): The Open University

Division is probably the most awkward of the four arithmetic operations. Since you may have a calculator, you do not need to be able to carry out complicated divisions by hand, but you do need to carry out simple divisions in order to check your calculator calculations. Division is the reverse process of multiplication. The quantity 12 Ã· 3 tells us how many times 3 goes into 12. Since 4 Ã— 3 = 12, 12 Ã· 3 = 4.

Author(s): The Open University

## Activity 22

Carry out the following calculations, without using a calculator.

• (a) A million pound lottery prize minus a three hundred pound administrative charge.

• Author(s): The Open University

When you are adding or subtracting whole numbers, an important thing to keep in mind is the place value of the figures. It is often a good idea to set out the numbers in columns before doing the arithmetic.

## Example 11

• (a) There are 4
Author(s): The Open University

In order to compare quantities, it is best to express them in the same units.

## Example 10

Three children have just measured their own heights in metric units. Isaac says â€˜My height is 1098â€™, Jasmine says â€˜My height is 112â€™ and Kim says â€˜Mine is 1.1â€™. What units were
Author(s): The Open University