Can be a tricky question, this.
Primary schoolchildren are taught that it is repeated addition, which makes a lot of sense.
4 x 4 =
4 + 4 + 4 + 4 =
16
Four lots of four/groups of/sets of.
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.
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).
x2 is a powerful, recurring idea, which has its role in pretty much the entire universe. (E = mc2)
It's also worth noting the effect of multiplication by a fractional quantity.
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.
Thursday, 28 January 2010
Tuesday, 19 January 2010
Tips For Revising for a Module
It'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.
So how do you study for an A Level maths exam?
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.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
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.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
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.
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.
So how do you study for an A Level maths exam?
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.
2) Make sure your calculator has new batteries. Yes - graphical calculator batteries do NOT last for years!
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.
4) DON'T do anything the night before. The formulae will start to jump around in your head.
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.
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.
Monday, 18 January 2010
e (2.71828)
No, this is not a post for luvved-up early 90s teenagers.
e is a bizarrely cool number, sometimes known as the natural logarithm.
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).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
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.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
e is a bizarrely cool number, sometimes known as the natural logarithm.
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).
How do we get it then?
Imagine an exponential curve. A curve that is nx. Say, y= 3x.
As usual, the gradient will generally differ as x differs.
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.
This curve is y = e x. It's not just at y=1 where the value of the function equals the gradient, it's every point on the curve.
As you can imagine, that makes differentiating with e easy...
There is an excellent page here which shows you some interesting examples of e in action.
Reciprocal Trig Functions
Sine, 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.
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
They also have reciprocal functions.
The reciprocal of sine x is cosec x and is (1/sin x)
The reciprocal of cosine x is secant x and is (1/cos x)
The reciprocal of tan x is cotangent x (cot x) and is (1/tan x).
Monday, 30 November 2009
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.
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.
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.
Thursday, 19 November 2009
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.
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.
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