Learning outcomes After studying this course, you should be able to: divide one number by another divide using decimals practise division skills learnt.
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
Introduction to analysis
This free course is an introduction to analysis which looks at real numbers and their properties, with a particular emphasis on inequalities. Section 1 starts by revising rational numbers and their decimal representations. Then, real numbers are introduced as infinite decimals. Section 2 looks at rules for manipulating inequalities and finding the solution set of an inequality. Section 3 looks at various techniques for proving inequalities. Section 4 introduces the concept of a least upper bound
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
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 ‘at random’ 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
Acknowledgements 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
Keep on learning   There are more than 800 courses on OpenLearn for you t 4 OpenMark quiz You might like to make some notes on the course for your own use later. Here is an example of a student's notes. 3.20 Multiplication rules 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 3.17 Subtraction of negative numbers 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 Try some yourself 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 3.15 Arithmetic with negative numbers In order to understand arithmetic with negative numbers, it is helpful to see how arithmetic can be represented on the number line. The strategy is to start with simple examples of whole positive numbers and then generalise to negative numbers. The same principles must apply! 3.13 More division with fractions How about other fractions? What is 6 ÷ Try some yourself Evaluate each of the following. 3.8 Order of decimal calculations The same rules about the order of calculations apply to decimals as apply to whole numbers. Brackets; Powers (e.g. squaring or cubing a number); Division and Multiplication (performed in the order written, left to Try some yourself 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 3.4 Order of calculations 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 3.2 Multiplying and dividing To multiply and divide by 10, 100, 1000, etc., write the digits in their place value columns. To multiply, move the digits to the left (replacing the numbers on the right with zeros) and to divide move them to the right (putting in a decimal point, and any zeros necessary for the place value). Multiplication and division by whole numbers in general can be carried out by combining this technique with a knowledge of the multiplication tables up to 10. Try some yourself Carry out the following calculations, without using a calculator. (a) A million pound lottery prize minus a three hundred pound administrative charge. 2.5 Measuring capacity The basic metric unit for capacity is the litre, usually denoted by the symbol l (though sometimes an uppercase L is used to avoid confusion with the number 1). In the SI system, units such as cubic metres (m3), cubic centimetres (cm3) and cubic millimetres (mm3) are used. These two systems are linked because: 1 ml = 1 cm3 The animation below i
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Activity 56
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Activity 46
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Calculations are performed in the following order:
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