There exist four men, A, B, C, and I. I observes an altercation between A and B, followed by A sitting down away from B. I conjectures that A was dismissed by B after A had accused B of deliberately jostling A on a crowded bus. Later I observes C and A. I conjectures that C is advising A on a matter of dress.
There exist a and b
b contacts a
a berates b
There also exists c
c advises a
This morning in a well known DIY store I heard two members of staff discussing the metre-cubed measure of volume. One said ‘what is a metre cubed?’ and the conversation then went:
‘Do you know what a square meter is?’
‘Well that’s, like, with 2 and a metre-cubed is, like, with 3.’
‘Oh I see’
If this is a hard concept then, in the words of a well known fictional army reservist, ‘we’re doomed, doomed I say.’
is not new but it doesn’t seem so easy to track down a simple derivation. A simple (formal) derivation of this runs as follows. First define,
where is a primitive such that . Then
But on substituting
That’s it really. As a corollary take and and then we see that
which is what we set out to demonstrate.
Following on from yesterday I found the following at:
Differentiate to get zero. Then put to show that the constant value of must be .
The usual way of proving Euler’s formula, , is to use the Maclaurin series expansions of , and . But suppose you want to introduce this formula to a class that has a good grasp of complex arithmetic and calculus, but hasn’t yet met Mclaurin’s series. The following is a non-rigorous demonstration of this result using only basic tools, and could form an extended homework.
Three basic results first need to be established. The first two were taken from the wikipedia page on Euler’s formula (1 August 2013).
First the student should show that
(the arbitrary constant isn’t important here). Second, that
noting that this can be shown by direct manipulation and does not require a knowledge of partial fractions. Thirdly, by differentiating both sides for example, that
Then, step-by-step, show that:
- (by putting ) .
Writing and taking antilogs then results in Euler’s formula:
If you can find a simpler `easy’ demonstration of this result I would appreciate hearing about it.