7 ms·

Why is e^(pi i) = -1?

114 points by ulvund 17y ago

jrp 17y ago
I 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.
sophacles 17y ago
So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.
- rottencupcakes 17y ago
  We verify in a quick Google search: http://www.google.com/search?hl=en&safe=off&q=sin(pi)&aq=f&oq=&aqi=g10 http://www.google.com/search?hl=en&safe=off&q=sin(pi... that sin(pi) = 0.
- deleted 17y ago
  [deleted]
- amalcon 17y ago
  sin(pi)=0 I wouldn't feel ignorant or anything. The article left out that information seemingly on purpose, presumably so that people would try to figure out where the i went.
- travisjeffery 17y ago
  The formula is: e^(ix) = cos(x) - i sin(x) Since sin(π) == 0, i * 0 == 0 so you have cos(π) left which is -1. So in sum: e^(i π) = cos(π) - i sin(π) == -1 - 0 == -1
- catzaa 17y ago
  If you draw a Sine graph, you will see that sin(x) is zero at 0, pi, 2*pi, ... If you draw a cosine graph you will see that it is -1 at pi. Therefore in the above sum the sine part just falls away. It is easy to see that the above is true when you recall that cosine(theta) = adjacent side/(hypotenuse). Since pi=180 degrees, adjacent will be -1 hypotenuse 1 and opposite side 0.
travisjeffery 17y ago
The article didn't have any mention of Euler, but this is typically called Euler's Formula and is considered one of the most beautiful formulas to Mathematicians. Here's some more info if you want to find out some info, applications and such: http://en.wikipedia.org/wiki/Eulers_formula http://en.wikipedia.org/wiki/Eulers_formula
tom_b 17y ago
For those of you interested in a longer look at this, check out "E, the Story of a Number" by Eli Maor. Yes, I read when I should probably be hacking.
daredevildave 17y ago
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.
- JBiserkov 17y ago
  ... using the fundamental equivalence relation =, and the operations +, * & ^
GavinB 17y ago
Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf 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.
- amalcon 17y ago
  Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
  - sofal 17y ago
    I'm not convinced that redefining Pi as 2*Pi would cause an equal amount of complications (besides confusing everyone with the change). The radius is more fundamental to a circle. The diameter is just derived from the radius. The idea of a circle with a unit radius is so widely used that it is simply called a "unit circle" with the obvious implication that the radius is what the "unit" refers to. Every time I've done math with radians, the fact that Pi is only half way around has added a subtle but noticeable disconnect. I have to force my intuition to fit the definition.
    - amalcon 17y ago
      The radius is only more fundamental to the notion of a circle because of the modern definition of a circle (set of all points exactly r away from the given point). One could also use a definition like: "Continuous set of points such that each is distance d from exactly one other, and further from none." The modern definition is convenient because it's a special case of a convenient definition of ellipse, but there's nothing terribly fundamental about it. And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
  - abecedarius 17y ago
johnaspden 17y ago
Imagine 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.
- ilitirit 17y ago
  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.
  - johnaspden 17y ago
    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 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!
- roundsquare 17y ago
  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."
timwiseman 17y ago
So 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.
tomerico 17y ago
I 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.
tfincannon 17y ago
Pages like this will be so much better when we have MathML in the browser and can stop using fuzzy images for mathematical formulas.
- btilly 17y ago
  If you control the website just use http://www.math.union.edu/~dpvc/jsMath/ http://www.math.union.edu/~dpvc/jsMath/ and you're fine. Any users that want can install the fonts and will get a better experience. Given how much harder MathML is to author, the jsMath solution is always going to be more popular among random math folks. Particularly since they're already used to writing TeX.
bbg 17y ago
http://xkcd.com/179/ http://xkcd.com/179/ I'm sure almost everyone has seen this anyway, but why not post.
bayareaguy 17y ago
I enjoyed Lakoff and Núñez's section on this in their book Where Mathematics Comes From[1]. The relevent pages are available at Google Books[2]. 1- http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From 2- http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&lpg=432 http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&#...
mhartl 17y ago
The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc. In these terms, Euler's formula would be recast as e^(i τ) = 1 That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.
alan-crowe 17y ago
You remember those half-angle formulae sin 2A = 2 sin A cos A cos 2A = (cos A)^2 - (sin A)^2 If you are happy using sine of small x is x and cosine of small x is 1 as your base cases, you can write sine and cosine as mutually recursive functions: (defun sin (z) (if (small z) z (* 2 (sin (half z)) (cos (half z))))) (defun cos (z) (if (small z) 1 (- (square (cos (half z))) (square (sin (half z)))))) (sin pi) => 6.167817939221069d-7 quite close to zero (cos pi) => -1.0012055113842453d0 you can get this much closer to -1 by using 1-x^2/2 as your base case. It had never occurred to me to do exponential the same way, as (defun exp (z) (if (small z) (+ 1 z) (square (exp (half z))))) So it is a bit of a shock to try it and see it work just fine for complex numbers (exp (complex 0 pi)) => #C(-1.0012055113842453d0 6.167817939221069d-7)
fridgeposts 17y ago
A better question is how can you: (pi/2)^2e
rameezk 17y ago
umm i m no smart ass.. but i think e^(pi i) = -1 cuz LHS = cos PI + isin PI = -1 .... :O

114 points by ulvund 17y ago

jrp 17y ago

I 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.

sophacles 17y ago

So to broadcast my math ignorance to all: Why does cos(pi) + i sin(pi) == -1 ?? Why does i disappear in this? Edit: Thanks to the replies below.

rottencupcakes 17y ago
We verify in a quick Google search: http://www.google.com/search?hl=en&safe=off&q=sin(pi)&aq=f&oq=&aqi=g10 http://www.google.com/search?hl=en&safe=off&q=sin(pi... that sin(pi) = 0.
deleted 17y ago
[deleted]
amalcon 17y ago
sin(pi)=0 I wouldn't feel ignorant or anything. The article left out that information seemingly on purpose, presumably so that people would try to figure out where the i went.
travisjeffery 17y ago
The formula is: e^(ix) = cos(x) - i sin(x) Since sin(π) == 0, i * 0 == 0 so you have cos(π) left which is -1. So in sum: e^(i π) = cos(π) - i sin(π) == -1 - 0 == -1
catzaa 17y ago
If you draw a Sine graph, you will see that sin(x) is zero at 0, pi, 2*pi, ... If you draw a cosine graph you will see that it is -1 at pi. Therefore in the above sum the sine part just falls away. It is easy to see that the above is true when you recall that cosine(theta) = adjacent side/(hypotenuse). Since pi=180 degrees, adjacent will be -1 hypotenuse 1 and opposite side 0.

travisjeffery 17y ago

The article didn't have any mention of Euler, but this is typically called Euler's Formula and is considered one of the most beautiful formulas to Mathematicians. Here's some more info if you want to find out some info, applications and such: http://en.wikipedia.org/wiki/Eulers_formula http://en.wikipedia.org/wiki/Eulers_formula

tom_b 17y ago

For those of you interested in a longer look at this, check out "E, the Story of a Number" by Eli Maor. Yes, I read when I should probably be hacking.

daredevildave 17y ago

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.

JBiserkov 17y ago
... using the fundamental equivalence relation =, and the operations +, * & ^

GavinB 17y ago

Pi is Wrong: http://www.math.utah.edu/~palais/pi.pdf 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.

amalcon 17y ago
Pi is defined as the circumference of a circle of unit diameter. This isn't somehow less correct than the alternative of making that circle one of unit radius. It causes some annoying extra factors (as the PDF points out), but the alternative would cause different complications. The area of a circle, for example, would be r^2*pi/2.
- sofal 17y ago
  I'm not convinced that redefining Pi as 2*Pi would cause an equal amount of complications (besides confusing everyone with the change). The radius is more fundamental to a circle. The diameter is just derived from the radius. The idea of a circle with a unit radius is so widely used that it is simply called a "unit circle" with the obvious implication that the radius is what the "unit" refers to. Every time I've done math with radians, the fact that Pi is only half way around has added a subtle but noticeable disconnect. I have to force my intuition to fit the definition.
  - amalcon 17y ago
    The radius is only more fundamental to the notion of a circle because of the modern definition of a circle (set of all points exactly r away from the given point). One could also use a definition like: "Continuous set of points such that each is distance d from exactly one other, and further from none." The modern definition is convenient because it's a special case of a convenient definition of ellipse, but there's nothing terribly fundamental about it. And yes, every time you've done math with radians, that problem has haunted you. That's because we decided to use radians instead of "diameter-ans", to keep with the above definition.
- abecedarius 17y ago

johnaspden 17y ago

Imagine 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.

ilitirit 17y ago
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.
- johnaspden 17y ago
  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 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!
roundsquare 17y ago
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."

timwiseman 17y ago

So 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.

tomerico 17y ago

I 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.

tfincannon 17y ago

Pages like this will be so much better when we have MathML in the browser and can stop using fuzzy images for mathematical formulas.

btilly 17y ago
If you control the website just use http://www.math.union.edu/~dpvc/jsMath/ http://www.math.union.edu/~dpvc/jsMath/ and you're fine. Any users that want can install the fonts and will get a better experience. Given how much harder MathML is to author, the jsMath solution is always going to be more popular among random math folks. Particularly since they're already used to writing TeX.

bbg 17y ago

http://xkcd.com/179/ http://xkcd.com/179/ I'm sure almost everyone has seen this anyway, but why not post.

bayareaguy 17y ago

I enjoyed Lakoff and Núñez's section on this in their book Where Mathematics Comes From[1]. The relevent pages are available at Google Books[2]. 1- http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From http://en.wikipedia.org/wiki/Where_Mathematics_Comes_From 2- http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&lpg=432 http://books.google.com/books?id=YXv6SEjTNKsC&pg=PA432&#...

mhartl 17y ago

The explanations mapping complex exponentiation to rotations are basically right. In this context, it's worth noting that the conventional choice of circle constant is off by two. We should be using tau = τ = C/r as the circle constant, rather than pi = π = C/D. Read τ as "turn" and all those radian angle measures suddenly make sense. Ninety degrees? Instead of the confusing π/2 we have 90° = τ/4 = one quarter turn. And so on: 60° = τ/6 = one sixth of a turn, 180° = τ/2 = one half turn, etc. In these terms, Euler's formula would be recast as e^(i τ) = 1 That is, the exponential of the imaginary unit i times the circle constant τ is unity: one full rotation.

alan-crowe 17y ago

You remember those half-angle formulae sin 2A = 2 sin A cos A cos 2A = (cos A)^2 - (sin A)^2 If you are happy using sine of small x is x and cosine of small x is 1 as your base cases, you can write sine and cosine as mutually recursive functions: (defun sin (z) (if (small z) z (* 2 (sin (half z)) (cos (half z))))) (defun cos (z) (if (small z) 1 (- (square (cos (half z))) (square (sin (half z)))))) (sin pi) => 6.167817939221069d-7 quite close to zero (cos pi) => -1.0012055113842453d0 you can get this much closer to -1 by using 1-x^2/2 as your base case. It had never occurred to me to do exponential the same way, as (defun exp (z) (if (small z) (+ 1 z) (square (exp (half z))))) So it is a bit of a shock to try it and see it work just fine for complex numbers (exp (complex 0 pi)) => #C(-1.0012055113842453d0 6.167817939221069d-7)

fridgeposts 17y ago

A better question is how can you: (pi/2)^2e

rameezk 17y ago

umm i m no smart ass.. but i think e^(pi i) = -1 cuz LHS = cos PI + isin PI = -1 .... :O