Don't we all? In that form it links the five most fundamental quantities in mathematics.
Personally though, I prefer it in the form e^(i pi) = -1. That doesn't have 1 and 0 (which are much more important than -1), but it is just that bit simpler.
Either way, it is a thing of beauty.
+1 heh!
Yes I too am not enamoured of the +1. Although fundamental to all arithmetic, it's just a workhorse and not as special as some of the other terms.
However I really do like to see the zero in my form of the equation. Zero is really, er, something. :-) I believe the Romans never really understood it.
In uk.d-i-y message , Mon, 23 Nov 2015
21:25:20, Fredxxx posted:
It's far too late to check the facts tonight, but ISTR that the more general statement that the complex roots of any polynomial with real coefficients occur in conjugate pairs, possibly multiple, may be true?
This is all just A-level maths. It's also one of the most elegant bits, which is why I like it.
I read on a comment on Peter Woit's 'not even wrong' blog that the Euler identity isn't even mentioned in some American degree courses.
Math degree courses (obviously). :)
I'd not expect to see it in History or French :-)
Sounds like you did rather more useful stuff in A-level maths than we did - too much useless geometry with us. 9-point circle anyone?
We did start some extra tew with more interesting stuff that would have been useful at Uni, but the teacher added too much extra homework with it, so I dropped it.
In article , Tim Streater writes
I thought it was were i came in.
This rings a bell. What is true is that a polynomial (with real coefficien ts) of odd degree always has at least one root that is real. It may have mo re. Those of even degree may not have any real roots.
In uk.d-i-y message , Tue, 24 Nov
2015 00:03:50, pamela posted:
Google for Multiplication of complex numbers expressed in polar coordinates and read
The N'th roots of -1 lie at the orders of a regular N-sided polygon of radius one and centred at the origin.
A N'th order polynomial has N roots, not necessarily all different for N>1.
Multiply out (x-1)(x-1)(x-2), and get a third order polynomial with roots 1, 1, & 2.
No, j!
Here we go again...
Thank you. It's been many years since I looked at complex numbers and this discussion plus that web page is starting to remind me of all those things I've forgotten.
By the way, that's a very nice site with clear non-trivial explanations. I like it!
ents) of odd degree always has at least one root that is real. It may have more.
A real polynomial of the Nth degree in X is dominated for large X by the X^ N term, so its values as X approaches +- infinity are themselves infinite. if N is odd, the value must therefore cross zero an odd number of times, g iving an odd number of real roots. And if N is even there must be an even number, possibly zero, of real roots.
In uk.d-i-y message , Wed, 25 Nov 2015 15:28:38, snipped-for-privacy@makewrite.demon.co.uk posted:
And x^2 + 1 = 0 is an example of that case.
HomeOwnersHub website is not affiliated with any of the manufacturers or service providers discussed here. All logos and trade names are the property of their respective owners.