Write this as e^(iπ) + 1 = 0 and you have my favourite equation. It relates all the fundamental mathematical numbers: e, i, π, 1 and 0.
... using the fundamental equivalence relation =, and the operations +, * & ^
Why is e^(pi i) = -1?
11–20 of 53 posts
Re: Why is e^(pi i) = -1?
#12Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf
It should be e^(pi i) = 1, but pi was unfortunately defined at half the appropriate value in the 17th century.
Re: Why is e^(pi i) = -1?
#13Imagine you're a complex number, which is just a type of 2-vector.
Exponentiation is to do with growth at a speed which is a multiple of how big you are already.
i is the multiplication which turns you through ninety degrees.
If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude.
Pi is how long it takes you to turn through a half circle.
So if you grow at right angles to yourself for time pi, you are pointing the opposite way.
Re: Why is e^(pi i) = -1?
#14I like Feynman's description, where he actually used 10^x first, just noting that 10^x for small real x was 1 + ln(10)*x (approximating from the derivative), assuming that this worked for small complex values too, and then extended to larger values by squaring.
Re: Why is e^(pi i) = -1?
#15So many texts on complex analysis simply define e^{i \theta} = \cos \theta + i \sin \theta without ever explaining how. This provides a good introduction to the reason behind with only minimal recourse to Calculus.
Another good description is discussed in "Visual complex Analysis" by Needham.
Re: Why is e^(pi i) = -1?
#16I have known it as the celebrities formula, because if you write it this way:
e^(pi i) + 1 = 0
It contains all the "celebrities" of the math world.
It's quite cool actually.
Re: Why is e^(pi i) = -1?
#17Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
That's a great explanation. Is this how they explain it in math classes? Because if not, then it should be. At least at a superficial level.
Re: Why is e^(pi i) = -1?
#18Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
Thats neat. Only thing is that (to me anyway) this takes the idea of the complex plane as being very fundamental as opposed to just something convenient. I'm not sure how to convince someone that 1 + i is the same as the coordinate (1, 1) without saying "thats just how we define it because things work out."
Re: Why is e^(pi i) = -1?
#19Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
Thats neat. Only thing is that (to me anyway) this takes the idea of the complex plane as being very fundamental as opposed to just something convenient. I'm not sure how to convince someone that 1 + i is the same as the coordinate (1, 1) without saying "thats just how we define it because things work out."
http://betterexplained.com/articles/a-visual-intuitive-guide...
http://betterexplained.com/articles/intuitive-arithmetic-wit...
The first link was posted here a while ago, I think.
Re: Why is e^(pi i) = -1?
#20Imagine you're a complex number, which is just a type of 2-vector. Exponentiation is to do with growth at a speed which is a multiple of how big you are already. i is the multiplication which turns you through ninety degrees. If you grow in a direction which is at right angles to yourself, you turn rather than increasing in magnitude. Pi is how long it takes you to turn through a half circle. So if you grow at right…
That's a great explanation. Is this how they explain it in math classes? Because if not, then it should be. At least at a superficial level.
Thanks! No, I worked it out for myself while I was thinking about what the exponential of a linear operator is. The explanation at my school was the one with Taylor Series.
By the time I got to university it was just taken as a given, so nobody explained it at all.
My explanation seems to have attracted a lot of positive votes in a few minutes. I've written a more detailed version here if anyone doesn't get it:
http://johnlawrenceaspden.blogspot.com/2009/10/ei-pi-1.html
If it gets much attention I might draw the diagrams that go with it.
I'm always bewildered how many scientific types (and even some mathematicians) find complex numbers mystical. They were originally discovered by mystic methods, but Argand showed us what was really going on, and they're no more weird than 2-dimensional vectors.
Actually a lot less weird than the Reals, which really are magical and mysterious, but which everyone seems to be quite happy with!