Unit 4 sec 2.5 Single and paired data

Unit 4 sec 2.5 Single and paired data

22 November 2015

14:56

Looking just at column H, the values are all based on a single measure (weight) and can be described as single data, you could calculate an average: you will more about averages in section 3, were 2 different types of average RX warned alternatively you could measure how widely dispersed. The values are - in other words, whether the values are tightly clustered together. Why waste bright. Finally, you could plot the values to discern the overall pattern visually and you will be shown a number of useful statistical plots in unit 11. The purpose of doing these things would be to try and gain an insight into baby weights in general.

Suppose now that there was a 2^nd sample, birth weight grumpy different set of mothers, the babies in the 2^nd sample of being classed as premature. This is now a two sample, as opposed to one sample dataset. Other examples of making statistical comparisons might include making a comparison between 2 medical treatments or to commercial products. Again, a sample of measures would be taken from each of the results compared. (When making such a statistical comparison, there is no requirement that the 2 samples contain the same number of values, although they could do.)

A statistical question of interest might be held the baby’s weight (in column H) relate to the way of their mothers at the start of their pregnancy (column F). This question links to pieces of information for each of the people in this study - a case of paired data. In statistical investigation terms, this falls under the general heading of seeking a relationship. A number of important statistical ideas are linked to exploring relationships today with paired data.

In particular, how to classify and distinguish different types of data. Primary and Secondary data are terms that identify the data source -  primary data you collect yourself, whereas secondary data are taken from somewhere else.

Unit 4 sec 2.4 Spurious precision

Unit 4 sec 2.4 Spurious precision

21 November 2015

15:25

The values, examples of spurious precision. In the 1^st case, the data value seems to be the result of the person’s age being given as 29 years and 11 months, and 11 months being 11/12= 0.916666 years. It is likely that this value is entered as 29+11/12, which would automatically be displayed in its decimal form. In the 2^nd case, the spurious precision has arisen by conversion, 2 kg, of data measured in different units (pounds).

 Another way that spurious precision can arise is when figures are quoted to a greater number of significant figures than is warranted in the context.

The accused was a council employee who regularly stole a portion of the money she was coming from the fees paid into machines by motorists in car parks. Did you merely keep a careful record of all the money taken and add it all up accurately.

  Generally speaking, it is customary, when analysing data gathered by others, to assume that the claimed precision is justified unless there is definite evidence to the contrary. Displaying data it is frequently unnecessary to retain their full procession. A key principle here is that displayed data should be just precise enough to reveal the key features - offering the reader an answer containing too many significant figures can easily obscure these patterns in mass of numbers. If you are collecting primary data, you should bear in mind the kinds of difficulty with data discussed above in the case of secondary data, ideally you should be able to go back to the original data collector in check with them. Any suspicions you have about the data.

Unit 4 sec 2.3 Checking and cleaning data

Unit 4 sec 2.3 Checking and cleaning data

21 November 2015

15:12

 First, there some blank cells in the spreadsheet where data are missing. Second, you probably also noticed the value of 99, appearing in the columns that otherwise contain only small integer values.

An explanation for these, other than a typing mistake, is that numbers such as 99 are sometimes used as codes that signal value missing. Good documentation of the data file should make the presence and value of a missing data code clear, but with this secondary data this information can get lost.

 Column H contains the weights of the babies, in kilograms. The minimum weight is 2.05kg, which is not a great deal less than the next smallest weight, 2.22kg and 2.44kg. The largest baby’s weight, however, appears to be 34kg., most probably the decimal point was missed out on their weight of 3.4kg (but without confirmation from the original data collection source. There is no certainty that this is the explanation).

Outliers

one or more data values that are considerably smaller or larger than the other values in the same dataset are called Outliers. However, in the case of the tallest mother mentioned in the solution to activity 10, often there is no such obvious reason and the outlier may just be unusual, but not unreasonable of observation. Either way, a game, there are sophisticated techniques available to deal with outliers, but these are not explored.

Unit 4 sec 2.2 Discrete and continuous

Unit 4 sec 2.2 Discrete and continuous

21 November 2015

14:41

  The distinction between discrete and continuous measures is important as it provides useful information about the nature of the data collected. As an introduction to these terms. Here is a contact that should help you get a sense of how they are used.

 The path itself is continuous, so precision on the path is possible, whereas the stepping stones placed on the path are discreet: they represent distinct, separate position with nothing in between any 2 consecutive steps. Using the path, you might mark your journey in terms of measured distance, whereas taking the same journey on the stepping stones involve counting out steps. In general, the distinction between measuring and counting is a useful way of identifying which majors are discreet and which are continuous.

  When it comes to statistical data, the same distinction can be made., foot length, on the other hand, has no such restriction - it is something that is measured on a continuous scale of measures and therefore produces continuous data.

One of the clearest distinctions between the numbers in the columns is that in some columns, the numbers seem to be discrete values well in other columns, the numbers seem to come from a continuous scale. Discrete data are data that can take one of a particular set of values: such data. Typically, though not necessarily, take integer values.

Here are some examples of discrete data

ª    the number of days in a week, on which one takes exercise.

ª    Number of × a particular website is visited in one day.

ª    The quality of a person’s recovery after a serious accident when called it 0 for full recovery, 1 for partial recovery and 2 for failure to recover.

Sometimes, as in the 3^rd example, discrete data arrays as a convenient way of recording data whose outcome is really some non-numerical category. A widely occurring example of this is when there are just 2 numerical values, and said to be binary data

why discrete data, continuous data can take all the in between values on the number scale. In theory, and depending on the context, they may take any numerical value from the set of real numbers, either negative or positive. Alternatively, they may be constrained to be positive or they may be limited it to a finite interval. Notice that all these columns contain data that take positive values.

Mass and weight

did you notice that the weights in table 2 given in kilograms, even though the kilogram is a unit of mass?

 The mass of an object is a major of the amount of matter that it contains, whereas its weight is a major of the gravitational force acting on it. Weight, being force is measured in newtons. However, you will often seem weights quoted in kilograms in everyday life, and this informal approach will sometimes be used in MU123 too.

You were deliberately not told in this activity. What level of precision to use? You might have written down 21cm and this would have carried the implication that the page width was nearer 21cm than 20cm or 22cm. again, they would be an implication that the actual measurements were nearer 211mm than 212mm. if you had access to more precise measuring device still, you might have been able to write down 211.0mm or 211.03mm, and so.

However, no matter how good you measuring device, you would never be able to say what the exact width of the particular sheet of paper was.

  The edge of a piece of paper is by no means straight and smooth - in that, the closer we work, the rougher the age be used to be. Clearly there is a limit to the precision with which is in meaningful to describe it width. However, the law particular purposes, there is no need for extreme precision, and recording the value of the width correct to, say, the nearest centimetre or the nearest millimetre may well suffice.

  In practice, for all manufactured items. There is a tolerance for the possible range of sizes that each item can be. According to the ISO standard, the width of manufactured A4 paper should be 210 ± 2mm. this is despite the mathematical exactness suggested in unit 3 of how ISO paper size relate to each other.

  There is a sense in which each of the columns C to H in table 2 can be considered to contain continuous data. It is just that the measurement and recording process has resulted in these columns of data being presented correct to the nearest month, year, centimetre, tenth of a kilogram, and hundredth of a kilogram, respectively.

 It follows that while the actual values of 2 items of continuous data can never strictly be identical, then started well use, into a contain degree of precision, may well be.

 You will not be right to think that all major dictator are actually discrete, but the idea of continuous data remains useful both contractually and when creating mathematical and statistical models of the world   

Unit 4 sec 2.1 primary and secondary data

Unit 4 sec 2.1 primary and secondary data

17 November 2015

23:22

You might consider collecting some data yourself. This might be particularly appropriate to question of interest is one very specific to you and your surroundings or on which you can collect relevant data quite easily. Data that you collect yourself a called primary data. For more substantial research questions, this tends to be a reasonable broke only in yourself refers to a research team in the University, company or other research unit.

Types of statistical question, before rushing into collecting data about any question, it is always a good idea to ask. Has anyone collected data on this before? The answer is often yes. Secondary data are data that already exists and can be used to adopt for your purpose.

Secondary data are plentiful and often readily available through the Internet, published literature and other sources. There are a number of consistently reliable sources such as UK government sister statistics which are generally professional way collected and presented and free from bias.

 However, other sites are set up to by organisation that may want to sell you some product or promote the particular set of ideas. In some cases, the data that they present may be subject to bias or distortion, and search site are best avoided as sources of reliable secondary data.

 Data are usually percent as “datasets”. A dataset is collecting data usually presented in the tubular form, or as a single row, or something as a single column.

An importing convention when presenting any dataset, where the primary or secondary, is to provide an accurate reference to the data source (so that the reader can get the details if they wish).

Backache in pregnancy

secondary dataset collected at the London Hospital (new Royal London Hospital). It was designed to help answer questions concerning backache in pregnant women, including: how common it is and how severe? What factors affected? What factors alleviate it?

In order to make the datasets manageable for your work in this unit, the number of respondents has been reduced from 180 women to 33, and the number of items of information reduced from 33 to 13.

 This dataset will be used to illustrate most of the issues concerned with handling data in this section, and that the end, the queue of the data values from the original source have been changed. Several of the corns in this table have been entered into the module software resource dataplotter but for reasons that will be explained shortly, pure the data values from this table have been changed.

Take a quick look at these data. This 1^st thing to notice is that each row corresponds to results from one patient, and the column -expects the first to a specific item measured.

Unit 4 sec 1.2

Unit 4 sec 1.2 The statistical investigation cycle

17 November 2015

22:57

As it was mentioned in the introduction to this unit, there are 4 clearly identifiably for stages in most statistical

investigations, which can be summarised as follows

the 4 stages of a statistical investigation are.

Stage 1 Pose a question.

Stage 2 collect relevant data

Stage 3 Analyse the data

Stage 4 Interpret the results

It may be helpful to think of these stages set out as a cycle, the PCAI cycle. Here the problem starts in the real world and is resolved by making the journey into the statistical world and back again.

Note that it is important to be as specific as possible posing a statistical question. The better the investigation can be attuned to the question, and the ‘the chance of obtaining a useful answer. The less focused the question, the wider the investigation and the greater the chance that nothing very informative will come out of it.

Not surprisingly, different statistical techniques applied to different stages of the statistical modelling cycle.

Ø P: pausing a clear question should normally be the 1^st stage of any statistical work. The decision as to which techniques are to be used subsequently will depend on the sort of question that has been asked at the start of the investigation.

Ø C: collecting relevant data will involve issues such as choosing samples and designing questionnaires.

Ø  A: this stage, analyzing the data is where techniques like calculating averages and plotting graphs and charts will take place.

Ø  I: the final stage, interpreting the results, take the action back to the original context from which the initial question was posed. Do the data analyses how to answer the original question? If “yes” then you can stop, there. If “no”, then you may need to travel around the cycle once again, perhaps this time with a slightly modified question using different analytical techniques.

The 1^st was the categorization of statistical investigations into 3 types: summarizing, comparing or seeking a relationship.

 2^nd, the PCAI statistical

investigation cycle was introduced. This cycle, which may be gone around more than once, consists of 4 stages, pausing a question (P), collecting relevant data (C), analyzing data (A) and interpreting the result (I).

Unit 4 sec 1.1

Unit 4 sec 1.1 question questions, types of statistical question

17 November 2015

22:05

You only need to glance at a newspaper, magazine, television or the Internet to see that statistical information is all around you. The key aim of this unit is to present statistical ideas is more than simply facts and techniques - statistical thinking is presented as a helpful way of seeing the world quantitatively.

Mathematical thinking. It can also be viewed in this way and, indeed, many of the remarks and fight statistical in this unit can be equally applied to mathematics in general.

Here are some of the ways in which statistical is unavoidable in our lives

ª    Numbers: Each person operates within a variety of key life roles, such as at work, at home, as a consumer and in the wider community. In each of these environments, you are presented with information, often in the form of the numbers, that must be processed and interpreted you are to be successful. A functioning worker, family member, consumer and citizen.

ª    Graphs and charts: statistical information often takes a visual form. You need to know how to interpret these ‘data pictures’, both in terms of the overall trends and patterns, they suggest, and also by knowing how to pull out and examined some of the relevant details.

Increasingly, almost every subject that you might wish to study has become more quantitative making it ever more important to have a sound grasp of basic statistics.

   Most of this statistical information arises as an attempt to answer questions of various kinds. But they are often end up raising just as many questions as they answer.

    Before rushing into answering any question, it is always a good idea to ask.

Summarising: how can the information be reduced.

Working at a lot of facts and figures does not always provide you with a clear picture of what is going on. To avoid all the world, it is often a good idea to find a way of summarising the information -perhaps by reducing the many figures to just one representative number.

It makes sense to monitor the water quality by taking regular measurements of the quality of the river water. Quite quickly, search and mass of data is generated that it can become difficult to see any underlying patterns. What is needed in some way of reducing many figures into just a few representative ones.

The 2^nd, and equally powerful, way of summarising data is to represent the numbers pictorially using statistical charts or plots -a central theme of unit 11.

Here are some more examples of investigations of the form. How many? Or how much?

¨    How many people die from road accidents each day in the UK?

¨    What is the typical cost of a tube of toothpaste?

¨    How old are the students studying mu123?

These are the sorts of questions were a summary in the form of a simple average can really clarify things.

Comparing: is there a difference

one possible explanation might be that the traffic calming measures have worked. However, there are several problems with this conclusion. 1^st, sample sizes of only 20 too small to be reliable; one speeding car in the 1^st sample may have made all the difference. 2^nd, it is likely that the speeds of different vehicles vary quite a lot, so differences are to be expected. Anyway. 3^rd, the difference between the 2 averages. It was small. And finally, the lower speeds might have been brought about by some other factor, such as a greater density of graphite in the phrase of the experiment. Perhaps because it was school term time.

In general, investigations involving comparing 2 averages will depend on several factors, such as the sizes of the samples on which the averages are based, the degree of variation that one might reasonably expect to see in such values, and whether the size of the observed difference a significantly large to act upon.

·      2 more people, on average, down from road accidents on weekdays or at weekends?

·      How does the cost of brand x toothpaste compare to that of brand y?

·      I students studying mu123. Although are younger than students on an introductory art model?

Seeking a relationship: what sort of relationship is there?

Sometimes a statistical question is not about differences between 2 or more sets of results, but about investigating a possible relationship between quite separate things.

There appears to be a relationship between 2 factors, it is often useful to determine what the relationship is. That is how much does one factor changed relative to the other?

§  Are the numbers of the role deaths in different countries linked to their respective maximum speed limits?

§  How does the cost of tubes of toothpaste depend on their size?

§  What is the connection between the numbers of hours that students work on level III module in mathematics and their final grade?

Classifying statistical investigations

3 types of investigation have been described above

§  Summarising

§  Comparing

§  seeking relationship.

Well summarising investigations are fairly easy to pick out, it can be less easy to distinguish the other 2.

Depending on how the investigation was approached, this could be based either on comparing or unseating a relationship. This will be an investigation based on comparing. However, an alternative experimental design could be to choose a sample of people randomly, measured the running speed and leg length of each person, and see if there is a relationship between these 2 measures.

Exploring questions like those above gives a purpose and direction to statistical learning.

  

unit 4 introduction

Unit 4 introduction

13 November 2015

21:58

This is the 1^st of 2 units in the module that deal with statistical ideas. What 1^st out some of the kinds of questions that can be addressed by statistical methods and then at some issues that arise when appropriate data have been collected. An introduction to some important statistical techniques for finding numerical summarises.

 This unit presents statistical techniques in the contact of practical investigations. Any purposeful statistical investigation, there are ball useful stages that can help you to organise your planning of the tasks that need to be carried out. In fact, then make up a statistical version of the general mathematical modelling cycle.

In this unit you will see how to use these 4 stages in the context of using calculations to reveal patterns in data.

Unit 3 sec 4.2

Unit 3 sec 4.2 Aspect ratios

12 November 2015

23:52

The cape of a rectangle can be conveniently described using the idea of aspect ratio.

The aspect ratio of the rectangle is the ratio of its longest side to the shorter side. For example, the aspect ratio of the left-hand rectangle is 25 : 15 , when simplifies to 5 : 3. The aspect ratio of the right-hand rectangle is 10 : 6, when simplifies to 5 :3, so these 2 rectangles have the same aspect ratio. So the 2 rectangles have the same gate though the 2^nd is smaller.

A rectangular image can be in line all reduced to any rectangle that has the same aspect ratio as the original image. Every different aspect ratio is required, then the image has to be cropped.

Scale factors

if an image  that measured 3 cm × 2 cm is enlarged to 9 cm × 6 cm, then they went and height both triple. We say that the scale factor is 3. Similarly, the same image is instead reduced to 1.5 cm × 1 cm, then the web and height both half, and the scale factor is half.

Whether line is that whether or height of the image, although I felt anything that appears in the image. The scale factors displayed on photocopiers are usually expressed as percentages. For example, if you want a photocopier to produce an image that is double the height of the original image, then you need a scale factor of 2, so you will certainly copier to enlarge by 200%.

 Videos

aspect ratio is also an important issue. Many older video programs that are made with the aspect ratio of 4 : 3, but in recent years 16 : 9, has become the most common video standard throughout the world. When a 4 : 3 image is displayed on a 16 :9 screen, the image has to be pillar boxed (displayed with black bars on the side), stretched or cropped.

Paper sizes

we consider the aspect ratio of sheets of paper. You are probably familiar with the paper size A4, A3, and so on. The largest paper size in this series is A0, and the next largest is A1, and so on. This series of paper sizes. It is known as the ISO 216 standard.

The paper sizes in the series by design so that they all have the same aspect ratio. This means that an A4 image, can be scaled up to an A3 one with no need for cropping. They were also designed to have the additional property that each side of paper is exactly the same size and shape as the 2 of the next small sizes placed side-by-side. For example, if you fold an A3 sheet of paper in half, then it becomes the same size as it is sheet of A4. There are various advantages of this property. For example, an envelope sized to fit an A5 sheet of paper will fit an A4 sheet folded in half, or an A3 sheet folded in quarters, and so on.

The aspect ratio that is needed at the paper sizes are to have the properties described above can be worked out as follows. Suppose that the aspect ratio needed is a:1 . Where A represents some number. Consider a sheet of paper with this aspect ratio. If it is shorter side has length w cm, say, then its longest side has length aw cm, since aw: w = a:1. Since paper sizes have the same aspect ratio, these ratios must be equal. So that we can compare them, let us make the 1^st ratio have 2^nd number a, the same as the 2^nd ratio. To do this, we multiply both numbers in the 1^st ratio by a.

Therefore, a must be √ 2. So the aspect ratio that is needed is √2:1 . It involves an irrational number. Each size of paper in the ISO216 standard has an aspect ratio of approximately √2: 1.

unit 3 sec 4.1

Unit 3 sec 4.1 what is a ratio

12 November 2015

23:40

   Why say the ratio of oil to vinegar is 3:1.

This ratio is equivalent to

30: 10, and  120:40, and 1.5:0.5

ratios can contain more than 2 numbers. For example, to make a particular type of concrete you need one part cement into parts sand for parts gravel.

1:2:4

you could make one shovel full of cement to shovel full of sand and for shuffleboard gravel and by several foes of cement with tens shovel films of sand and 20 shovel films of gravel and someone depending on how much concrete you need.

   This is the ratio between the distance on number and the corresponding distance on the ground. The ratio is changed to an equivalence ratio in the same way that fraction is changed to an equivalent fraction.

To find a ratio equivalent to a given ratio

multiply or divide each number in the ratio by the same non-0 number.

A ratio is simplest form, when each number in the ratio is a whole number, and these numbers are collected down as much as possible, that is, they have no common factors.

When you are working with ratios that contained 2 numbers, it is sometimes helpful to convert them to the form number 1. For example, this can help you compare different ratios. You can convert a ratio to this form. By dividing both numbers by the 2^nd number.

Rating ratios in the form number :1 also help you to find approximate ratios, which can be useful when you want to compare quantities. Sometimes you need to divide a quantity in a particular ratio

Other forms of ratio

ratios icon changes to numbers are sometimes written as fractions. For example, the ratio 3:2 can be written as a fraction. The 1^st and 2^nd number in the ratio become the numerator and denominator of the fraction, respectively. This is why you sometimes see ratios given as single numbers. The single number that represents a ratio is just a number that is obtained when the ratio is written in the form ‘number :1’.

The fact that ratios can be written as single numbers. Also explains why you often see phrases such as the larger ratio.

unit 3 sec 3.4

Unit 3 sec 3.4 fractional indices

12 November 2015

23:29

   Meanings are given for these indices, and you so that with these meanings. The index was at work on negative in 0 indices.

  Meanings can also be given to fractional indices in such a way that the index laws work for these indices.

Raising a number to the power half is the same as taking it square root, raising the number to the power 3^rd is the same as taking its cue route and so on.

This rule, together with the index was that you have already met, can be used to give a meaning of any fractional index.

Raising a number to the power is the same as raising the nth root of the number of the power m.

The tool rules in the boxes above hold all the appropriate numbers.

  Powers like this do have precise meanings, which you can learn about in detail in more advanced mathematics modules. You can work out the value of to square root to as accurately as you like by using as many decimal places of the decimal form of square root of 2 as you like. The indices here, 1.414 and 1.414213, and so on are rational, as they are determining decimal in power can be any real number. All the index laws that have you have seen in this unit called for indices and base numbers that are any real numbers, (except that the numbers must be appropriate for the operations. For example, you could not divide by 0 or take the square root of a negative number).

Powers and surds on your calculator

the final activity of this section, you can practice using your calculator from calculations involving powers, scientific notations and surds.

unit 3 sec 3.3

Unit 3 sec 3.3 Surds

12 November 2015

23:15

  The roots of numbers that you are asked to find in our section were rational, but most have irrational groups. In particular, the square root of any natural number that is not perfect square is irrational.

  Is numbers like these cannot be written down exactly posted terminating decimals of fractions, we often lead then just as they are in calculations and in the answer to calculations.

    The advantage of this approach is that it allows us to work with exact numbers, rather than approximations. This is a surd is a numerical expression containing one or more irrational groups of numbers. × are usually omitted, though sometimes it is necessary to helpful to include them. Also, wary number and roots are multiplied together, it is conventional to write the number first. It is also helpful to write surds in the simplest form possible . You can simplify describing in this way whenever the number under the square root sign had a factor that is a perfect square greater than 1.

  The square root of a day was simplified by 1^st using the fact that the perfect square ball is the factor of 80. The working can be shortened by instead using the fact that the larger perfect square. 16 is a factor of 80.

  So the most efficient to begin with the largest growth factor that you can spot, but it turns out, is that there is a larger one, then you can simplify the route in stages. Another way in which you can sometimes simplify surds is to simplify products to or more square roots. Where different square roots are multiplied together. You can use the rule.

  You can not usually simplify some of the different such as √3 + √5.

To simplify surds

v simplify roots of integers with square factors.

v Simplify products and quotients of roots

v add or subtract roots that are the same.

unit 3 sec 3.2

Unit 3 sec 3.2 Roots of numbers

12 November 2015

23:08

  As you have seen, the number you square is to is called the square root of 2.

In general square root of a number is a number that when multiplied by itself gives you original number.

  Every positive number has to square roots, positive one and a negative one. This symbol ±, which means plus or minus, can be useful when you are working with square rates.

 The positive square root of the positive number is donated by the symbol √.

 There are other types of routes apart from square roots. A cube root of a number is a number search that if you multiplied 3 copies of it together you get the original number. A 4^th route is a number is a number search that if you rise to power of 4 you get the original number.

  You can use this information to find the square root of the product of a quotation of number of the numbers with square roots 3 and 5 respectively. So the positive square root of 9×25. It 3×5.

  The square root of a product the same as a product of square roots; square rate of rotation is the same as a quotation of square roots.

unit 3 sec 3.1

Unit 3 sec 3.1 what is irrationals number?

12 November 2015

22:57

  This can be expressed in the form of an integer divided by an integral.

  You saw that all rational numbers have decimal forms that are either terminating or recurring, and so the following number is not a rational number.

The measure the length, you 1^st need to decide on unit of measure. The unit can be a centimetre, a meter, and or any other conveniently does not matter what it is, as long as it is used consistently.

Suppose that redesign to measure licensing centimetres. Here is the ransom some lines measured using this unit. Consider the diagonal winds in the tiling pattern. The pattern is made up all square tiles, each with 1 cm long and each tail is half green and half yellow.

 The diagonal line from the sides the green square. So the length of the sides of the green square, measured in centimetres is not a rational number.

 This is true no matter what unit of measurement. You choose. Of course, in practice, you can approximate these lines by rational numbers, but a sensible system of numbers should include the numbers that are the exact length of these lines stop so the rational numbers by themselves, do not form a workable system of numbers. We must include the decimal with an infinite number of digits after the decimal point, but no repeating block of digits. These numbers are called the irrational numbers. They are the numbers that are not rational.

 The irrational numbers also include the positive number is square is 3, which is denoted by square root of 3, and many other age irrational numbers. Number irrational number is pie.

This is an important number in mathematics, and you will see you use frequently in some of the lighter units in the module, stop. There is nothing special about this one, except that its digits have a pattern, one that is different from the type of past and found in the decimal form of rational numbers.

  Your irrational numbers, together with the rational number from the real numbers. These numbers are significant to represent the length of any line or curve. Each point on the number line represents a real number, so the number line is often called the real-line.

There are infinity men irrational numbers and infinitives make many irrational numbers stop

 complex numbers include all the numbers above and also many imaginary numbers, searches the square root of -1. The idea of imaginary numbers might seem strange, but the complex numbers have a huge number of useful practical applications stop

unit 3 sec 2.5

Unit 3 sec 2.5 scientific notation.

06 November 2015

23:46

  Some of the numbers used in mathematics, science, medicine, and economics are very big and very small.

Nos like these and more usefully expressed in scientific notations (which is also called standard form).

Strategy to express a number of scientific notation

1.    Place the decimal point between the cheese of one and second significant digits together a number between 1 and 10.

2.    Count to find the power of the 10 by which this number should be multiplied (or divided) to restore it to the original number.

  The reason is simple that this notation has been agreed as the one that everyone will use. Using consistent notation makes it easier to compare numbers carry out calculations.

   To convert a number from scientific notation back to ordinary notation, you just need to carry out the multiplication or division. Since the multiplication or division is a power of 10. This involves moving the decimal point.

Calculations using numbers in scientific notation

you will use your calculator to carry out most calculations involving numbers in scientific notation, and you will be asked to practice this in an activity at the end of this subsection. There may be occasions perhaps when you are making a quick estimate, when it is more convenient to work out the answers by hand, using the index was that you met earlier.

Unit 3 sec 2.4

Unit 3 sec 2.4 Negative numbers

06 November 2015

23:39

That does it also makes sense to extend it to the? This is, do all the powers in the following less mean something? The powers with positive indices have the pattern shown below. If you assume that this pattern continues leftwards, then it must be the number that you get by dividing

The pattern continues in this way, then the 0 and negative indices must have the meanings. But do these meanings make sense? Do they work with the index was that you met in subsection 1.5. There are summarised below, and you can think of them into further index was. 

Unit 3 sec 2.3

Unit 3 sec 2.3 Multiplying and dividing fractions

06 November 2015

23:09

Strategy due to multiply fractions

multiplying the numerators together and multiply the denominators together.

Example 5 multiplying fractions

carry out the following fracture multiple occasions stop

(A)       ×       (B) 2 ×     (C)  ×     (D)  ×

Solution

(A)       ×  =

(B)       Here, you can either use this strategy, as is done below, or use the fact that 2 lots 3 seventh is 6 sevenths.

2 ×  =  ×  =

(C)        ©Here, the number 2 is a factor of the numerator of the 1^st fraction and also a factor of the denominator of the 2^nd fraction, so it is a factor of both the numerator and denominator of the product stop it is easier to cancel factors like this before multiplying

 ×  =  ×   =

(D)       To multiply by a mixed number, 1^st, converted to a top-heavy fraction.

 ×  =  ×  =  =

The answer can be left as  if the wish

Your role by doing this can be conveniently described using the idea of reciprocal of a number. The number and its reciprocal multiplied together to give 1. As you can see from example 1, defining the reciprocal of a fraction, you just turn it upside down

Strategy to divide by a fraction

multiply its reciprocal.

Example 6 dividing fractions

carry out the following fraction divisions

(A)       ÷       (B)  ÷     (C)  ÷ 2

Solution

(A)       ÷  =  ×  =

(B)       Here, once you have turned the 2^nd fraction upside down, there are factors that you can cancel before multiplying.

 ÷   =  ×  =  ×  =  =

(C)       ÷ 2 =  ÷  =  ×  =