Firstly, show that:
Secondly, are there any other operations you can do to and and the retains the same value? In other words:
- Find all functions such that for all
Students love to cancel wherever they can, so much so that the book Comic Sections (now out of print) had the following joke:
- The student law of universal cancellation: If the same symbol x occurs in any two different places on the one page it may be cancelled
So you get horrors like 😮
And yet strange things can happen. It is true that .
What is remarkable is that you can have as many sixes as you like and cancel them as many times as you like so, for example, . Can you prove this is true? Can you find 3 other similar fractions?
Many equations cannot be solved exactly without using special functions. For example, to solve requires the use of the function (or similar). This function is sometimes defined in terms of an integral from which their properties can be deduced. Thus is defined by and it is then clear that, for example,
There are many equations that can only be solved in terms of newly-defined functions. One such function that isn’t all that well known is the LambertW function where is defined as a solution (for ) of . This allows you to solve equations like which was asked about on the S.O.S. Mathematics CyberBoard
To solve let so that . Then
Thus and so which is our answer.
Using tables or software this gives 1.100.
But hang on, is that the only solution? No, because for small values of and grows much faster than so for large values of . Since both and are continuous on there is another value of for which . A quick fiddle with a calculator gives .
Research into the LambertW function to find out how this other solution can be given in terms of this function.
Joi Ito’s Page contains a wonderful joke which I really have to repeat here.
- After explaining to a student through various lessons and examples that
I tried to check if she really understood that, so I gave her a different example. This was the result:
A level syllabuses these days expect you to remember the exact values of sin cos and tan of certain angles. is easy enough as the calculator will give you the exact answer, but unless you know roughly what should be then the calculator will be no help.
But, help is at hand 😀 Memorising formulae is easier when there’s a pattern and the following table gives such a pattern.
Isn’t that amazing! I only came across this a few years ago but apparently it’s been around at least since the 1950’s.
What about tan? Since you just divide a value from the second row by the one below it (but please not for ; see Tuesday 16 March).
On tonight’s programme the following question was asked:
Which is not a prime number?
The choices are
The sad thing is that the contestant said he gave up as he would only be guessing (a wrong answer would lose him a lot of money). 😥