tag:blogger.com,1999:blog-32975340200976553962024-02-19T22:49:23.305+00:00Completing The SquareThe Tin Drummer tries his hand at maths...Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.comBlogger73125tag:blogger.com,1999:blog-3297534020097655396.post-2864709119968743942011-05-02T11:07:00.001+01:002011-05-02T11:09:21.542+01:00HousekeepingI've been tidying and revising some of my older posts. Some errors have been corrected, some bad writing made less awful.<br /><br />However, although it's blog etiquette to notify your readers of when and how you have updated your posts, I'm not doing that because it will take too long and I cannot be bothered.<br /><br />Do feel free to point out errors that are still obscuring the light of truth...Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-42177452430855207252011-05-01T22:54:00.002+01:002011-05-01T22:58:32.014+01:00Completing the Square ReduxAn amazing site which explains completing the square much better than I ever could:<br /><br />http://www.mathsisfun.com/algebra/completing-square.htmlBill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-27311557382882217192011-05-01T16:07:00.002+01:002011-05-01T16:09:51.590+01:00The Modulus FunctionPut very basically, the modulus function finds the absolute value of a number.<br /><br />It is indicated by a pair of straight brackets around an x or a part of an equation.<br /><br />It just means that -x when operated on by the modulus function becomes x while x stays as x.<br /><br />It ignores negative values of x, or of the statement in x, such as (x-2).Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-78846968752094112011-05-01T16:02:00.002+01:002011-05-01T16:06:39.203+01:00Transformation of Graphs, contdWhen you are doing composite transformation of graphs, as you will at C3 and presumably C4 level, you do need to remember that sometimes the order of transformation counts. I have to admit to struggling with this (I don't have a tutor right now) so forgive me if I fail to offer a convincing explanation. But it's something like this:<br /><br />If you are stretching in the x-direction, that comes BEFORE any translation.<br /><br />If you are stretching in the y-direction, that comes AFTER any translation.<br /><br />That's what I've picked up from getting lots of answers wrong, anyhow. I think it's to do with the way that a y-stretch affects the whole equation, while an x-stretch affects anything directly attached to the x part of the equation.<br /><br />Some websites do suggest that you start at the innermost point of the equation and then work out - I don't know if this works for all composite transformations. I doubt it if what I've written above is right.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-84724024729476390932011-04-26T17:06:00.004+01:002011-04-26T17:20:03.752+01:00More LaTeX Experiments: Transformation of GraphsIf you take a graph and transform it in various ways, here is what you get:<br /><br />A reflection in the y-axis turns a graph of y=f(x) into the graph of y= f(-x).<br /><br />A reflection in the x-axis turns a graph of y=f(x) into the graph of y= -f(x)<br /><br />- <span style="font-style:italic;">note the difference</span> -<br /><br />A stretch of factor s in the y- direction turns the graph of y= f(x) into the graph of<br /><br />y = s f(x).<br /><br />A stretch of factor m in the x-direction turns the graph of y = f(x) into the graph of \[y = f(\frac{x}{m})\]<br /><br />A translation by \[ \left( \frac{k}{l} \right) \] turns the graph of y = f(x) into the graph of y = f(x-k) + lBill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-90781062722846088732011-04-26T14:07:00.003+01:002011-04-26T14:10:58.341+01:00Another TestI'm trying to see if I can use LaTeX to write formulae onto the blog, but I've always been a bit befuddled by it. Instead I've installed a javascript thing from www.watchmath.com and that's supposed to recognise code and turn it into proper maths.<br /><br />Well here goes.<br /><br />Let's have a look at the function f(x) = 1/x. By rights in LaTeX that should be \[ \frac{1}{x} \]Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-39148142150775111182011-04-26T14:04:00.001+01:002011-04-26T14:04:37.133+01:00Test\[ E(\mathbf{x}) = \sum_{i \in \mathcal{V}} \theta_i(x_i) + \sum_{ij \in \mathcal{E}} \theta_{ij}(x_i, x_j) \]Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-19983810516993662422011-04-26T12:09:00.002+01:002011-04-26T12:20:30.931+01:00C3 Functions Round-UpReminder: this is really about AQA C3, as OCR and Edexcel are slightly different. They don't change the truths of mathematics though.<br /><br />After quite a long break, I've started doing maths again. I've forgotten chunks of it, as you would expect. Here are some things I've forgotten (on the assumption that they're generally forgettable):<br /><br />1. Inverse functions are simply functions reversed. The domain becomes the range and vice versa. However, not all functions can have an inverse. If a function is many-one (as quadratic functions generally are) then they can't have an inverse - it would be one-many, which is not a function at all. <br /><br />2. To get an inverse function you can do two things. If x appears only once in the original function, then you can do a reverse flow chart, reversing the order and the operations. BUT BEWARE OF SELF INVERSE FUNCTIONS - if you meet a subtraction from or a divide into, those operations stay the same because they're self-inverse functions. IF x appears more than once you need to do some equation-fiddling. On a function, x is a/the domain value(s). y is the same as f(x) - ie it's the range of the function. So by re-arranging the equation to isolate x, you're turning the function INSIDE OUT and making x the f(x) instead of the variable at the beginning (the domain).<br /><br />3. composite functions are two functions put together. They're usually written as fg(x). What this means is you do both operations. You write out g(x) first, say it's x<sup>2</sup> + 4. Then you put it into f(x), which might be 2x - 3. The resulting composite function is 2(x<sup>2</sup>+4)-3. <br /><br />4. On some functions, there are values that cannot be defined and that affects the range as well as the domain. Take f(x) = 1/x for example. X cannot be 0. But this means there are going to be impossible values of f(x) as well. A good way of estimating this is to draw the graph of f(x) on your calculator!! Always use the calculator wherever possible, even if you need to sketch the graph by more formal methods.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-25680288649886643992011-04-06T15:38:00.000+01:002011-04-06T15:39:33.893+01:00PolynomialsThese are a very common sight in almost any study of mathematics. It’s worth taking a moment to define them properly and think about how they can be used.<br /><br />A polynomial is a mathematical expression involving a whole, positive power of x. The type of polynomial depends on the highest power of x in the expression. The different bits of the expression are known as terms, and the polynomial as a whole is the sum of the terms it contains.<br /><br />Take a quadratic expression: 4x² + 2x - 2. It has three powers of x: 2, 1 and 0. These are whole numbers and positive (except 0). This is therefore a polynomial. The same applies to cubic expressions: x³ + 4x² + 5x + 2 is a polynomial too. <br /><br />In fact, the same applies, whatever the power of x, as long as it is not a fractional or negative number. A polynomial does not need to have several terms, like most cubics and quadratics do. So, 1 is also a polynomial, being a term in x raised to the power of zero. The same applies to any constant (ie any number). 4x + 1 is a polynomial too, as x is x raised to the power of one.<br /><br />Polynomials are written in descending powers of x, though they do not have to be.<br /><br />The numbers in front of the x variables are called coefficients, and can take any value, including fractional or negative values, as can constants. <br /><br />The highest power of x in a polynomial defines the degree of that polynomial. So 1 is a polynomial of degree 0, because the highest power of x here is 0. 4x + 1 is a polynomial of degree 1, and quadratics are polynomials of degree 2.<br /><br />Polynomials can be created by expanding brackets. (x+2)² gives x² + 4x + 4 when expanded. In fact, both expressions here are polynomials. If the brackets were to give us an expression with a fractional power, then of course this would not be the case.<br /><br />When you make a polynomial like a quadratic equal to 0, then you have a polynomial equation. In the case of a quadratic, such as 4x² + 2x – 2, the resulting equation is usually written as: 4x² + 2x – 2 = 0. As an equation, it is now a description of a parabolic curve which intersects the x-axis at (-1,0) and (½, 0).<br /><br />Polynomials can be added, subtracted, multiplied and divided. The Factor and Remainder Theorems can be used to find out additional information about them. For example, if a number assigned to the variable x causes the polynomial to equal zero, that number is a factor of the polynomial (a root, a value of x where the curve of the equation crosses the x-axis). So we can demonstrate that -1 is a factor of 4x² + 2x – 2 by putting it in place of x. 4(-1)² + (-1x2) – 2 = 0. Therefore -1 is a factor of this polynomial. This is the essence of the Factor Theorem. The Remainder Theorem is a little more involved but states that if we divide the polynomial by x-a then the remainder is f(a) – in other words, the remainder is the sum of the polynomial when a is put in place of the variable x.<br /><br />These are the absolute basics. Of course it does get a lot more complicated! But it is worth familiarising yourself with these principles before moving on.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com1tag:blogger.com,1999:blog-3297534020097655396.post-63616214572402300022010-01-28T11:47:00.003+00:002010-01-28T12:01:41.978+00:00What is Multiplication?Can be a tricky question, this. <br /><br />Primary schoolchildren are taught that it is repeated addition, which makes a lot of sense.<br /><br />4 x 4 = <br /><br />4 + 4 + 4 + 4 =<br /><br />16<br /><br />Four lots of four/groups of/sets of.<br /><br />But multiplication is also a scaling quantity. It ratchets things up massively quickly. If you type 2x2 into your calculator, then keep multiplying every answer by 2, the calculator will quickly run out of digits. To compare this with addition misses the point that multiplication is the centre of all geometric operations.<br /><br />It does through the squaring and cubing and so on effect. If you take four centimetres, and for each of these four, you add another four, you get 4 x 4. It is known as squaring because the Greeks used to see it geometrically as the way to find the area of a square. For each centimetre across, there are four up (and vice versa).<br /><br />x<sup>2</sup> is a powerful, recurring idea, which has its role in pretty much the entire universe. (E = mc<sup>2</sup>)<br /><br />It's also worth noting the effect of multiplication by a fractional quantity. <br /><br />4 x 1/4 will obviously increase the fraction, but it will have a decreasing effect on the whole number. Unlike multiplication by two whole numbers, where the answer is greater than either, in this operation, the answer will always be smaller than one of the inputted values.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-86030775099272848092010-01-19T21:30:00.003+00:002010-01-19T21:53:17.716+00:00Tips For Revising for a ModuleIt's exam season: in fact, it's just gone. So many of us will be turning our thoughts to retakes - or in my case, to reapplying for exams we had to drop out of.<br /><br />So how do you study for an A Level maths exam?<br /><br />1) Don't necessarily do all the questions. For some, like me, it's essential to do every question in the book and all the past papers. For many, that's just repetition. You need to do the first few and the last few. Past papers are good to study as well as to do, to work out which questions often come up.<br /><br />2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years! <br /><br />3) MAke sure you know the formulae which are NOT in the formula book for the module you are doing. There are usually several really common ones - like trig identities for example. Check each chapter in the textbook to see which formulae you need. It is easy to forget these, especially if you have been *ahem* working on the same module a while.<br /><br />4) DON'T do anything the night before. The formulae will start to jump around in your head.<br /><br />5) Revise the previous modules a little. You will find some identities and formulae you studied before that are not explicitly repeated come in handy. For example, the cosine and sine rules from C2 can be very handy when doing the questions on reciprocal trig functions in C3. Also the rules and techniques of radians are very useful indeed, though quite a bit of practice on this is done through C3.<br /><br />6)If you do past papers and you can't get a tutor to mark them for you, you can do it yourself via the online answer papers which are published. It's not an exact science because you are not amazing at maths, unlike the markers, which is why you're studying it now. So BE HARSH on yourself and be conservative. Markers will often be generous with follow-through marks so you can at least be prepared. That's the pessimistic side of me coming through.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-72235103673331446282010-01-18T19:45:00.004+00:002010-01-18T19:55:59.508+00:00e (2.71828)No, this is not a post for luvved-up early 90s teenagers.<br /><br />e is a bizarrely cool number, sometimes known as the natural logarithm.<br /><br />Like π it is an irrational number - ie it cannot be expressed exactly as the ratio of two whole numbers (and thus as a fraction).<br /><br />How do we get it then?<br /><br />Imagine an exponential curve. A curve that is n<sup>x</sup>. Say, y= 3<sup>x</sup>.<br /><br />As usual, the gradient will generally differ as x differs.<br /><br />Is there a curve which has a gradient of 1, where x=0? There is, and, coolly, this curve has a gradient of 1 where x=0 and ALSO passes through y=1, ie the value of the function is 1.<br /><br />This curve is y = e <sup>x</sup>. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve. <br /><br />As you can imagine, that makes differentiating with e easy...<br /><br /><a href="http://www.wmueller.com/precalculus/e/e1.html">There is an excellent page here which shows you some interesting examples of e in action.</a>Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-2223025837158608322010-01-18T19:32:00.003+00:002010-01-18T19:45:06.305+00:00Reciprocal Trig FunctionsSine, cosine and tangent are functions. You input a value - an angle in radians, or degrees, and for each angle there is a value. They are repeating functions.<br /><br />They also have reciprocal functions.<br /><br />The reciprocal of sine x is cosec x and is (1/sin x)<br /><br />The reciprocal of cosine x is secant x and is (1/cos x)<br /><br />The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-1003868210252755882009-11-30T20:48:00.003+00:002009-11-30T20:54:41.719+00:00The Self-Inverse FunctionThis sounds really weird or complex but it's not.<br /><br />Basically a self-inverse function is just a function that gives you the same answer when you do the function to the answer.<br /><br />Wot?<br /><br /><br />Alright. Say you have a function f(x) = 4-x<br /><br />Say x = 2<br /><br />4-2 = 2.<br /><br />Now apply the answer to the function<br /><br />4-2 = 2.<br /><br />Same answer.<br /><br /><br />Try again f(x) - 7-x<br /><br />Say x - 5.<br /><br />7-5=2.<br /><br />7-2=5<br /><br />We have x again.<br /><br /><br />And f(x) - 10-x<br /><br />x=4<br /><br />10-4 = 6<br /><br />Put 6 into it:<br /><br />10-6=4.<br /><br />Brings us back to our starting point with x.<br /><br /><br />When you do the operation twice, finding the function of the answer, you get your starting value of x.<br />Reciprocals always give self-inverses too.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com2tag:blogger.com,1999:blog-3297534020097655396.post-90613279035179644062009-11-30T20:41:00.002+00:002009-11-30T20:48:12.932+00:00When Can an Inverse Function Exist?You can't always have an inverse function (See previous post or previous but one or thereabouts).<br /><br />A function is defined as any mapping which is one-one or many-one. This means that for any input value a unique output value is generated.<br /><br />Something like y= x <sup>2</sup> is a many-one function. 2<sup>2</sup> and -2<sup>2</sup> give the same value.<br /><br />You can always draw a graph of a function. <br /><br />A function CAN'T be a one-many mapping. It doesn't make a lot of sense to most of us to have some kind of operation which could generate loads of different answers for exactly the same input.<br /><br />So.<br /><br />If a function would generate a one-many correspondence, it follows that it's not a function. So a many-one function CANNOT have an inverse, because its inverse would be one-many (ie not a function at all).<br /><br />Only one-one type functions have inverse functions.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-4207933155376606052009-11-19T14:35:00.002+00:002009-11-19T15:28:30.557+00:00C1 AdviceHmmm.<br /><br />I was bored so I thought I'd offer a little advice for anyone revising for C1. As it is a non-calculator paper, you need to consider a few things.<br /><br />1) Your mental arithmetic should be good<br />2) You should know square and cube numbers up to and including 5<sup>3</sup><br />3) You should be totally au fait handling surds. This is an early part of C1 and you might have forgotten it by the time of the exam. There are some fiddly rules surrounding eliminating surds from equations so learn 'em.<br />4) You should remember to x stuff by -1 to get rid of unwanted negative numbers in your answers.<br />5) You need to be good at fractions, including cancelling down.<br />6) You also should be confident expanding brackets (and other GCSE stuff). This might be annoying if you did GCSE some years ago, like myself. It's worth buying a GCSE revision textbook and looking the Higher level stuff up.<br />7) Because of the non-calculator thing, the answers to C1 questions are generally nice, like 3 and 5 and 2. If you work something out not in surd form and it is a bizarre fraction a good rule of thumb is to go back and check your working.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-70264824262106701582009-11-02T15:40:00.005+00:002009-11-02T16:09:31.458+00:00Functions II - The InverseYou can reverse a function - not always, to give you what you started with.<br /><br />This is known as the inverse function<br /><br />Take y=x<sup>2</sup><br /><br />This function, or y=f(x), will give 4 for the input of 2.<br /><br />So what will give us 2, from an input of 4?<br /><br />It would be the opposite, the inverse - x = √y<br /><br />This is sometimes written as f<sup>-1</sup>.<br /><br />It does get more complicated but that is the idiot's version.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-87895604063210928882009-11-02T13:37:00.003+00:002009-11-02T13:42:18.914+00:00A RecommendationNormally I would not advise students of maths under degree level to go anywhere near Wikipedia. It's not the reliability issue as such, it's that it gets very complicated very quickly.<br /><br /><a href="http://en.wikipedia.org/wiki/Trigonometry">An exception is the page on Trig.</a> It gives some useful basics and some really handy animations which illustrate the key trigonmetric functions.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-34153212148947807632009-11-01T08:09:00.002+00:002009-11-01T09:17:06.391+00:00FunctionsA simple definition of a function is an operation which, for every value put in, will give a single value. It can give the same answer for different inputs. y=x<sup>2</sup> will give the answer 4 when both 2 and -2 are put in, but is still a function.<br /><br />What a function can't do is give multiple answers to the same inputted value. A graph that appears to show this is not the graph of a function.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-37429791761115621842009-10-31T13:10:00.003+00:002009-10-31T13:33:59.535+00:00Geometric Series Part IIBecause timesing is a funny operation, a scaling up or down operation - repeated addition or however you like to look at it, it does strange things to series.<br /><br />A common ratio of 2, like in the previous post, gives you a series which just keeps getting more and more massive, into infinity. It therefore has no sum - because it just keeps going on. You can find the sum of the first n terms of course.<br /><br />But what if you had a common ratio of 1? Well then your series stays exactly the same<br /><br />2,2,2,2,2,2,2,2,2,.....into infinity. It's the same sort of thing as above. Finding a complete sum of the series isnot going to be possible, because there will simply be an infinite load of 2s.<br /><br />But if your common ratio is a fraction:<br /><br /> 1/2, 1/4, 1/8, 1/16<br /><br />Here the common ratio is 1/2. The terms of this sequence get smaller all the time, but will never reach 0. <br /><br />When we think of the sequence properly as a series:<br /><br />1/2 + 1/4 + 1/16 + 1/32 + 1/64<br /><br />Then we can see that it is an infinite series, and will go on halving ad infinitum.<br />If we add the first few terms, we get 1/2, 3/4, 13/16, 27/32....<br /><br />The sum is getting larger each time but by progressively smaller amounts.<br /><br />The sum is getting closer and closer to 1, but without ever quite reaching it. If it had an infinite number of terms, then its sum would indeed reach 1. Its sum to infinity is 1.<br /><br />It converges on 1, and is therefore called a convergent series.<br /><br />This happens with certain types of common ratio, which we will come back to.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-19046807608778700162009-10-31T13:00:00.002+00:002009-10-31T13:09:32.103+00:00Geometric SeriesThese are a bit like arithmetic series but instead of there being a common difference between terms there is a common power-type difference.<br /><br />Consider 2,4,8,16,32,64...<br /><br />This series appears to double each time, which means that there is no common number which gives you the nth term when added to the n-1th term.<br /><br />But the common difference is "doubling" - or rather, powers of 2. <br /><br />The series goes 2<sup>1</sup>, 2<sup>2</sup>, 2<sup>3</sup>, 2<sup>4</sup>.....<br /><br />With geometric series, therefore, we don't talk about a common difference, but a common ratio - the thing you times each term by to get the next one. So this series has common ratio 2.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-57072518381820565422009-10-30T10:01:00.005+00:002009-10-30T11:07:21.123+00:00Sum of an Arithmetic SeriesThis was supposedly demonstrated by Gauss, aged eight or something, when he was set a problem by a teacher desperate to get on with something more interesting to add together all the numbers from 1-100. He's supposed to have realised that if you paired up the numbers, they all had the same sum ie 101. It was then a matter of spotting the number of pairs and multiplying the two (5050).<br /><br />The sum of an arithmetic series is done in a similar way. Think of a series as being:<br /><br />a<sub>1</sub> + (a<sub>1</sub> + d) + (a<sub>1</sub> + 2d) +......<br /><br />since d is a constant...<br /><br />You want to find the sum of the first n terms.<br /><br />S<sub>n</sub> = a<sub>1</sub> + (a<sub>1</sub> + d) + (a<sub>1</sub> + 2d) +....<br /><br />....(a<sub>1</sub> + (n-1)d)<br /><br />This is probably quite a few terms (otherwise you wouldn't bother trying to find the sum, would you?).<br /><br />So the wisest option would be to pair up the terms, like Gauss did, and find the sum of each pair.<br /><br />The formula is derived from writing the above series out from beginning to n, and the opposite way, and adding the terms.<br /><br />However, just as Gauss's addition always gave him 101, in our case we will always get 2a + (n-1)d<br /><br />Since we want the sum to n of the series, there will be n of these pairs. So the sum is n(2a + (n-1)d)<br /><br />BUT - <br /><br />We have just added TWO series! Because we wrote it out twice to make adding the terms easier.<br /><br />Therefore, the final formula will be: n(2a + (n-1)d)/2.<br /><br /><br />Quite fiddly to prove, but easy enough to use.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-41606862300428839392009-10-29T10:17:00.005+00:002009-10-29T10:34:48.047+00:00Series and Sequences 1 - nth termsThere are a few of these.<br /><br />Let's just start with a definition. A sequence is a load of numbers in a list, with there being a common difference between the numbers.<br /><br />A series is the same thing, but added together.<br /><br />Here is an arithmetic sequence: 0,2,4,6,8,10....<br /><br />Here is an arithmetic series: 0+2+4+6+8+10.....<br /><br />In an arithmetic series or sequence, there is always a common difference between the numbers. In the ones above, the common difference is 2. It is always a constant (and not anything weird like n<sup>2</sup> - that's more like geometric series).<br /><br />So finding the nth term of an arithmetic series or sequence is easy enough.<br /><br />You need the first term a<sub>1</sub> and the common difference d.<br /><br />a<sub>n</sub> = a<sub>1</sub> + (n-1)d<br /><br />So say I wanted to find the 42nd term of the above sequence.<br /><br />a<sub>42</sub> = 0 + 41d<br /> = 0 + 41x2<br /> = 82<br /><br />The 42nd term of the sequence 0,2,4,6,8...is 82<br /><br /><br />Because the difference is a constant with arithmetic sequences, this formula is simple enough to grasp. To find the nth term you need the first term, and then the number of terms before the nth one timesed by the common difference because there are that many lots of the common difference.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-71093104365201403842009-10-28T10:55:00.002+00:002009-10-28T10:56:16.305+00:00SlownessHmm, 50 posts in over a year isn't much is it? Well, I get bored easily.<br /><br />You should see the other place...<br /><br />I will try to post more, as I come back to maths after a few months' absence and try to tackle C3.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0tag:blogger.com,1999:blog-3297534020097655396.post-37864195273613819792009-10-28T10:34:00.002+00:002009-10-28T10:55:04.159+00:00RadiansThese are little blighters used to describe the angles of circles. Like degrees, they can be used to express arc widths, or can be used in equations (especially trig equations). Basically they're just like degrees, but are based on π and the radius of the circle (hence radian). Imagine a line from the centre to any point on the circle. That's a radius. Then, in any direction, draw an arc around the surface of the circle with the same length as the radius. Stop. Draw a radius from this point back to the centre.<br /><br />The angle subtended is one radian.<br /><br />There are 360 degrees in a circle and one degree really isn't very large.<br /><br />A radian is much bigger (around 57 degrees). Every circle has 2π radians, like every circle has 360°. 180° is therefore π radians.<br /><br />Most calculations involving radians will also involve fractions. Using π enables you to be exact without lots of fiddly decimals. The same is true of fractions. In some ways then, using radians is satisfyingly precise.<br /><br />It is also easy to convert one to the other.<br /><br />360° = 2π radians<br /><br />Therefore<br /><br />1° = 2π radians/360.<br /><br />and<br /><br />say<br /><br />12° = (2π/360) x 12<br /><br /> = 24π/360<br /><br /> = π/15<br /><br />and it's usually sufficient to leave it like this for C2. At other levels you might want to do the calculation to n dps or whatever.<br /><br />Similarly<br /><br />2π radians = 360°<br /> <br />1 radian = 360/2π<br /><br />and <br /><br />so<br /><br />3 radians = (360/2π) x 3.<br /><br />Easy.Bill Haydonhttp://www.blogger.com/profile/08357811679771159469noreply@blogger.com0