The Self-Inverse Function

This sounds really weird or complex but it's not.

Basically a self-inverse function is just a function that gives you the same answer when you do the function to the answer.

Wot?


Alright. Say you have a function f(x) = 4-x

Say x = 2

4-2 = 2.

Now apply the answer to the function

4-2 = 2.

Same answer.


Try again f(x) - 7-x

Say x - 5.

7-5=2.

7-2=5

We have x again.


And f(x) - 10-x

x=4

10-4 = 6

Put 6 into it:

10-6=4.

Brings us back to our starting point with x.


When you do the operation twice, finding the function of the answer, you get your starting value of x.
Reciprocals always give self-inverses too.

When Can an Inverse Function Exist?

You can't always have an inverse function (See previous post or previous but one or thereabouts).

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.

Something like y= x ² is a many-one function. 2² and -2² give the same value.

You can always draw a graph of a function.

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.

So.

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).

Only one-one type functions have inverse functions.

C1 Advice

Hmmm.

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.

1) Your mental arithmetic should be good
2) You should know square and cube numbers up to and including 5³
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.
4) You should remember to x stuff by -1 to get rid of unwanted negative numbers in your answers.
5) You need to be good at fractions, including cancelling down.
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.
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.

Functions II - The Inverse

You can reverse a function - not always, to give you what you started with.

This is known as the inverse function

Take y=x²

This function, or y=f(x), will give 4 for the input of 2.

So what will give us 2, from an input of 4?

It would be the opposite, the inverse - x = √y

This is sometimes written as f^-1.

It does get more complicated but that is the idiot's version.

A Recommendation

Normally 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.

An exception is the page on Trig. It gives some useful basics and some really handy animations which illustrate the key trigonmetric functions.

Functions

A 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=x² will give the answer 4 when both 2 and -2 are put in, but is still a function.

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.

Geometric Series Part II

Because 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.

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.

But what if you had a common ratio of 1? Well then your series stays exactly the same

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.

But if your common ratio is a fraction:

1/2, 1/4, 1/8, 1/16

Here the common ratio is 1/2. The terms of this sequence get smaller all the time, but will never reach 0.

When we think of the sequence properly as a series:

1/2 + 1/4 + 1/16 + 1/32 + 1/64

Then we can see that it is an infinite series, and will go on halving ad infinitum.
If we add the first few terms, we get 1/2, 3/4, 13/16, 27/32....

The sum is getting larger each time but by progressively smaller amounts.

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.

It converges on 1, and is therefore called a convergent series.

This happens with certain types of common ratio, which we will come back to.

Geometric Series

These are a bit like arithmetic series but instead of there being a common difference between terms there is a common power-type difference.

Consider 2,4,8,16,32,64...

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.

But the common difference is "doubling" - or rather, powers of 2.

The series goes 2¹, 2², 2³, 2⁴.....

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.

Sum of an Arithmetic Series

This 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).

The sum of an arithmetic series is done in a similar way. Think of a series as being:

a₁ + (a₁ + d) + (a₁ + 2d) +......

since d is a constant...

You want to find the sum of the first n terms.

S_n = a₁ + (a₁ + d) + (a₁ + 2d) +....

....(a₁ + (n-1)d)

This is probably quite a few terms (otherwise you wouldn't bother trying to find the sum, would you?).

So the wisest option would be to pair up the terms, like Gauss did, and find the sum of each pair.

The formula is derived from writing the above series out from beginning to n, and the opposite way, and adding the terms.

However, just as Gauss's addition always gave him 101, in our case we will always get 2a + (n-1)d

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)

BUT -

We have just added TWO series! Because we wrote it out twice to make adding the terms easier.

Therefore, the final formula will be: n(2a + (n-1)d)/2.


Quite fiddly to prove, but easy enough to use.

Series and Sequences 1 - nth terms

There are a few of these.

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.

A series is the same thing, but added together.

Here is an arithmetic sequence: 0,2,4,6,8,10....

Here is an arithmetic series: 0+2+4+6+8+10.....

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 n² - that's more like geometric series).

So finding the nth term of an arithmetic series or sequence is easy enough.

You need the first term a₁ and the common difference d.

a_n = a₁ + (n-1)d

So say I wanted to find the 42nd term of the above sequence.

a₄₂ = 0 + 41d
= 0 + 41x2
= 82

The 42nd term of the sequence 0,2,4,6,8...is 82


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.

Slowness

Hmm, 50 posts in over a year isn't much is it? Well, I get bored easily.

You should see the other place...

I will try to post more, as I come back to maths after a few months' absence and try to tackle C3.

Radians

These 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.

The angle subtended is one radian.

There are 360 degrees in a circle and one degree really isn't very large.

A radian is much bigger (around 57 degrees). Every circle has 2π radians, like every circle has 360°. 180° is therefore π radians.

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.

It is also easy to convert one to the other.

360° = 2π radians

Therefore

1° = 2π radians/360.

and

say

12° = (2π/360) x 12

= 24π/360

= π/15

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.

Similarly

2π radians = 360°

1 radian = 360/2π

and

so

3 radians = (360/2π) x 3.

Easy.

Get In!

C2 = 92%

AS Level = A

TTD = Maths Genius

New Maths Specifications

As it's a mere five years since the last new A Level maths specifications came out the QCA are clearly in dire need of updating their syllabus. Accordingly they're now in the consultation stage for a new curriculum to begin in 2012. This means of course that uncompleted A Levels will come to the end of their shelf-life at this time. Hopefully that won't affect me. But I was interested to see some of their recommendations.

1) They want to abolish the non-calculator paper. I think C1 is great. You learn how to do a lot of manipulation of formulae and a lot of arithmetic in your head. Differentiation, integration, surds, quadratics (factorising) can all be easily done without recourse to a calculator. This, surely is good for fast, flexible thinking and for confidence. The down side is that it does mean C1 can't be that challenging - you can't really put radians into it for example, even though conceptually radians is a piece of the proverbial. The same goes for basic trig. Trig would be better introduced in C1 rather than stuffed into C2 (which is I think around 35-40% trig, all told).

2) They are trying to bring it down to four modules again. This is not a bad idea. The current A Level maths six unit system is complicated and contradictory, though it is flexible for people with different specialisms. It also throws up anomalies. For example as it currently stands maths is the only A Level where you can do 4 AS modules and two A2 ones - which I am doing, because in addition to C1,C2,C3 and C4 I took S1 for AS and am doing M1 for A2. M1 is an AS module. So I'm only doing C3 and C4 as actual A2 Levels. The situation can be reversed - Pure Maths AS is two AS modules and one Further! And you can also do AS modules in Further Maths (you could do FP1-4 and S1 if you hadn't done it). This leads to anomalies in the awarding of grades - the A* grade is only awarded on the basis of C3 and C4.

Four modules would simply take us back to where we started in the 90s with the equivalent of P1 and P2 - ie the four Core modules collapsed into two again. This is probably where this idea dovetails with the abolition of the calculator paper.

3) There are going to be no formulae to learn. This strikes me as disastrous. It makes an expecation of zero knowledge on the part of candidates - and this is an advanced qualification we are talking aobut here. The authorities do not view knowledge as important, but expect candidates to problem solve. Well you can only solve problems if you know how to approach them - which strategies you know of to use. I think the motivation for this is clear dumbing down, or as the QCA write, to "provide students with equality of opportunity and a common basis for progression.". Hmmm. An advanced qualification should be for people who are advanced in their knowledge and understanding of that subject - it should not pander to political ideas of equality, and believe me the QCA does just that. In the questionnaire I answered yesterday I was asked whether the new specifications promote gender and race diversity, or whether they were discriminatory against disabled people. For crying out loud, why is even pure knowledge infected with this leftist rot?

4) They want slightly to change the balance between pure and applied. This probably isn't a bad idea. Right now applied is 33% of AS and A2, but they would like it to be more flexible, up to 40%. With fewer papers I don't know how they will do this, except by weighting, although the QCA do suggest that they might allow certain exams to be longer - this would be an excellent idea. At the same time they think the pure content should be kept the same. So without changing weightings or lengths of exams or indeed mixing up applied content into pure modules, I'm not sure how this will work.

5)There is a lot of guff on the QCA site about being more stretching. They conceive it as using longer, less structured questions. This is clearly an excellent idea, where possible and where practicable. I don't know if they just mean in Further or in Core maths as well.


You can go and look all this up for yourself at the QCA website.

Long Time, No Blog

It's been a tricky old year so far, like a particularly knotty equation, so I haven't found the time for maths blogging, or even for maths.

So it was with some trepidation that I looked at my calendar about a month ago and saw the date for the AQA C2 exam. At that point I knew nothing of radians, logarithms I couldn't even spell, and geometric series I thought were just lots of pretty pictures (lots of geometry type stuff). So I spent a month fairly hectically doing the last four chapters of C2, missing out a few questions on the way and also, disappointingly, not really bothering to look at the proofs for the various formulae but just learning them.

C2 was yesterday. It was ok, though exams are always hard to assess until you get the marks. I answered all qs, but I think the differentiation with fractional indices one might have cost me a few %.

As before, the exam was thrilling. No I really mean this. I had bags of adrenalin, lots of excitement, masses of determination to show what I could do. I am like this with things that don't matter. Only important things have me quivering in the corner like a wobbly jelly who's just been made professor of wobbling at Oxford (ok ok that's a Blackadder joke sort of). Lining up beforehand with fifty sixth formers all pointing at me and whispering was a bit strange but it just reminded me of how utterly uninterested in it I was at their age. You do it for reasons, to get to uni, or because you happen to be good it, but you rarely do it just because you love it - you learn that later. Sometimes.

Integration, Differentiation

This is quite tricky to get your head round, I think.

Differentiation is what you do to a curve to obtain its gradient. It works by splitting the curve into infinitesimally tiny chunks and tending towards a limit (ie where you started from). I think I put the proof of it on an earlier post.

Basically it gives you the rate of change in the curve as a formula. Because curves are always changing, the gradient will be different at each point. You plug the x-coordinate into the differentiated gradient formula and there's your gradient at x=whatever.

But if you integrate that curve you get the area under the curve. It works in a similar way to differentiation: it takes the space under the curve and chops it up into tiny rectangles to approximate the area.

It seems to work out that in terms of a formula, integration is the opposite of differentiation (ie raise the power by one and divide by the value of the new power, instead of dropping the power by one and timesing by the new power).

The curve bounding an area with the x-axis is now the value of the area formula differentiated - ie it is the rate of change of the area itself.


Differentiation and integration are inverse processes: that's the theorem of calculus (which means stones used for counting, or something).

If you have a derivative, you can integrate it to get the indefinite integral - which you can then find the full equation for if you know a set of co-ordinates. So by knowing the derivative, you can find the equation of that curve.

If you have a derivative, often you can differentiate again to find a stationary point on the curve - the second derivative test.

I'm still not sure I get this, you know. I still find the connection between the two loose and hard to actually articulate. I can do it with the formulae no problem at all - but understanding the linkage and how rates of change relate to curves is much harder.

More Basic Trig

Every angle has a sin or cosine or tan (well, not quite every angle, in the case of tan). You can use this value to find the angle, doing it easily on a calculator.

You just press sin^-1 .456 or whatever it is, and that gives you the angle (if you have the calc. set to degrees that is).

Amen Trig Corner

Well I guess I don't get it, but then maths is like that. Someone tells you to do x, you do x, you get it right or you get it wrong: no one gives a toss. No one cares that you've just described the entire universe (less the supernovae or the black holes), no one cares that you've inscribed patterns across space - patterns which remain, whether you like them or not; whether you think the guy who made them matters or not: they remain.

They are not contingent like the semi colon, or culturally derived like the feminism or the theism: they are not subject to the whims of tenured professors: there they are and you can just something off if you don't like it, tbh.

But to just do x, if you have been told to: that defies and defeats the entire point of mathematics. They tell us this because they know we are thick and also because in a pyramidal structure like maths, you just cannot *understand* things that you need to know how to do, when you are little.

No-one tells an 8 year old about subjects, objects, predicates, the passive voice or the genitive case. Equally, you shouldn't need to tell an 8 year old about the mechanics of division in order to teach them how to do it. But so paranoid are we about maths, so in hoc to an educational vision based more in socialism than in intellectualism, that we cannot believe we can teach these skills without sending 8 year olds into A Level territory, and in the meantime, depriving them of the skills of actually dividing one number by another.

Division by repeated subtraction: for crying out loud are you REALLY going to stop bright 11 year olds dividing by decimals by using this infantile but ridiculously complex method? Are you REALLY going to prevent tough calculation by saying "well, just take away and take away and take away"?

More to the point: stop insisting on an "understanding" that defies most adults, let alone children. You do not need to know that a sentence consists of subject, object and predicate to write one. Why are we baffling kids with mathematical jargon? Why are we constructing entirely false notions of "understanding" which exclude the simple explanation "well, I am timesing x by y" or "well, I guess I am seeing how often z goes into a". - and hamstringing bright kids, and stopping them calculating?

It could'nt be because socialist academics who write the curriculum don't want kids to be able to do these things...could it?

A Proper Go At Explaining Trig

Alright, well, clearly it is the study of angles and triangles and by use of the unit circle, of circles too.

We talk about sin, cosine, and tan a lot in the basic stuff.

Although we learned SohCahToa at school, it really only applies to right angled triangles. But by using the unit circle (radius 1, centre at the origin) and drawing a right angled triangle inside it, we can calculate basic values of sine, cosine and tan, which hold for any angle and can be used in calculations involving any triangle.

The sine of an angle is a measurement of its distance from the horizontal axis, which can be negative or positive and which can be between 1 and -1 inclusive. If you think of an angle sweeping up around the circle, it reaches its furthest point from the axis (ie sin of 1) at 90 degrees.

The cosine of an angle is a meaurement of the distance from the vertical y axis of a point on the end of the angle; again, between 1 and -1.

The tan of an angle is a measurement of whether a line of angle x would meet a tangent to the circle drawn at right angles to the radius (the x axis) near, far, or not at all. The tan can have any value. Calculators really don't like being asked tan 90 or tan 270 because the lines of these angles will never meet a tangent.

There are some cool diagrams on the wikipedia page that show these things in action.

More Trig

I can't say I get it! But I can use the formulae.

I loathe doing stuff I don't get.

General AS Advice

Know your volumes and areas from GCSE.

I don't.

The Really Basic Basics of Trig

Trig is basically to do with triangles and circles. It works from the properties of angles relative to the sides of triangles and how these can be seen in circles (unit circles - with a radius of 1). It's used to tell you the size of angles and sides using other information you already know about the triangle.

The trig identities, I reckon, are probably best defined as ratios.

All fractions are ratios, as well as being processes.

For a right angled triangle (where we tend to begin with these)

SohCahToa

Sin =opposite/hypotenuse

Cos = adjacent/hypotenuse

Tan = opposite/adjacent


For each angle, there will be a different value for each one of these. So sin 42 will be the same whatever triangle you have. It regulates what the sizes of the opposite and hypotenuse will be only in terms of the ratio between them.

Sin and Cos (not sure about Tan) will always give a value between 1 and -1 inclusive.

On Fractions

I think these are great: more adaptable and more accurate than decimals.

But it's the division thing which always bugs me. You know: "turn the divisor upside down and multiply" - what the hell? Well, ok, I'll do it....

Well, doodling during a lesson the other day (yes I know I am supposed to be the teacher), I worked out a kind of rationalisation.

Take 3/6 divided by 1/2.

Flip over the 1/2 to make 2.

3/6 x 2 = 6/6 = 1.

So a half goes into 3/6 once. Makes sense, as 3/6 is a half.


But think of it without fractions.

6 / 3 = 2, right?

but 6 x 1/3 also = 2.

Because division is the inverse of multiplication, you are basically just doing the inverse. Also because multiplication is a scaling process, when you times by a number less than 1, the answer will be less than the number you were trying to scale. It scales downwards. 4x2 = 8 but 4x1/2 = 2.


And reciprocals are essential here. A reciprocal is any number you multiply any other number by to make 1. In practice this means that the reciprocal of a whole number is 1/that number.

1/6 x 6 = 1

But the reciprocal of any fraction is that fraction turned upside down.


3/4 - 4/3.

This is because you can imagine any whole number as n/1.

I guess you could say the reciprocal of a number is the inverse of that number...

So, 10/2 = 5. Ie 2 goes into 10 five times.

And 10 x 1/2 = 5. ie 10 lots of 1/2 make 5.

And 10 / 1/2 = 10 x2 = 20. ie a half goes into 10 20 times, which is the same as multiplying ten by two.

The fact that this holds makes it a lot easier to divide fractions, so it is a trick, but a useful one. There are proper proofs of it, but this isn't supposed to be a proof, only a sort of meditation on the subject.

On The Importance Of Factorising

Without it, even linear equations become extremely difficult to re-arrange.

Take 6xy + 3x - 2y = 7

Re-arrange this for x.

MOVE the non x bit.

6xy + 3x = 7- 2y.

SPOT the common factor

x(6y + 3) = 7 - 2y.

DIVIDE by (6y+3)

x = 7-2y
       ----
       6y+3

PIECE of the proverbial.

Indices And Differentiation

Yesterday, courtesy of my beautiful and awesomely intelligent maths tutor, I suddenly worked out a way of making differentiation with negative indices slightly easier.

When inputting the x value to find the gradient at that point, to make the final stage of adding it all up much easier, turn your x^-2 into 1/x². It makes calculations a lot, lot easier.

Laws of Indices

Are pretty simple.

a^m x aⁿ = a^m+n

a^m / aⁿ = a ^m-n

(a^m)ⁿ = a ^mxn


But a^m + aⁿ DOES NOT equal anything other than what it says.
There isn't much you can do to collect the terms if the indices are different, or if the constant is different (ie x^m + y^m)

a^m + a^m = 2(a^m), of course.

Fractional Indices (2)

Well yes it gets a bit harder.

You see...a ^n/m


means ^m√aⁿ


...but I will try to explain this later.

Fractional Indices (1)

Think about it.


Think about it.


x is not just x. x is x¹. x is x raised to the power of one: x multiplied by itself no times at all; x just being x. So then x ¹ is x on its own. x is, therefore, x.

So then, x^1/2 is going to be a number that, multiplied by itself, makes x.

In other words, the square root of x.


√x = x ^1/2


It does get a bit more complicated than this....

Negative Indices

Negative Indices

Now these seem simple enough: you raise a number to a power and it usually means multiplying a number by itself n times.

a³ = a x a x a

All numbers to the power of 1 are themselves, and all numbers to the power 0 are one.

a⁰ = 1



Easy enough.


But then you can also have negative powers.

a^-3

What? You can’t multiply a number by itself a negative number of times!

Well, no, clearly. But you can see how negative powers come about.


It’s to do with the powers rule.

aⁿ x a^m = a^n+m

You can see this if you write out aⁿ and a^m in full.

aⁿ x a^-n = a^{n+ -n}

= a^n-n
=a¹
=a

Therefore, the negative powers are used to denote reciprocals (the number you multiply n by to get 1 – so 1/6 is the reciprocal of 6 – and it is always 1/n.

So.


3³ =27

3^-3 = 1/27 (or 1/33 – the reciprocal).


Negative powers are easy to manipulate, they just seem a bit weird until you think that positive powers are going up by multiplying the number by itself:

2⁴ is 2 x 2 x 2 x 2

But if you go down, towards 0, the same process is division by two.

64...32...16..8...4..2..

This continues as you go down below zero:

......2, 1, ½, ¼, 1/8, 1/16......

(2¹, 2⁰, 2^-1, 2^-2, 2^-3, 2^-4)

And you can see that these are reciprocals of the powers of 2.

The same applies for the powers of each number.

C1 and S1 taken!

Apologies for the lack of posting. I sort of became involved in my actual revision!

But I took C1 and S1 on Friday. C1 was ok but S1 was really tricky. Not sure how I did there.

Anyhow. I'm doing C2 now but I will try to post some more pure-related posts. I'm not sure I want to think about stats again!