Associated Topics || Dr. Math Home || Search Dr. Math

### Euler Equation

```
Date: 09/13/97 at 17:19:49
From: Oliver Dale

I have seen the famous Euler equation e^i*pi-1 = 0 and know something
abut its derivation. Specifically, it follows from the infinite series
that define sin, cos and e^x. My question is: does the equation still
work if we decide to work in degrees (i.e. is e^i*90-1=0 true)?

I ask because the mysterious appearance of e, i, and pi in one
equation isn't so strange if it is partly definitional. I believe the
question boils down to this: do the infinite series definitions of sin
and cos

(cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ..., and
sin(x) = x - x^3/3!  + x^5/5!  - x^7/7! + ...)

work if x is expressed in degrees, or are they only valid when x is a
```

```
Date: 09/13/97 at 19:04:56
From: Doctor Anthony
Subject: Re: Euler function and radians

The Euler equation is  e^(i.pi)+1 = 0

The series you quote for cos(x) and sin(x) require x to be in radians.
Radian measure is non-dimensional, being the ratio of two lengths,
arc length divided by radius, just as sin and cos are non-dimensional,
again being ratios of lengths.  It is necessary, of course, that the
two sides of the equation have the same dimensions - in this case both
are non-dimensional.

Degrees are not mathematical in any real sense.  They are arbitrary
conventions and, historically, mathematicians could have agreed to
have 500 'degrees' in a circle or any number you care to think of.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 09/18/97 at 21:39:35
From: Oliver Dale
Subject: Re: Euler function and radians

a little confused.  I have heard in a number of places that the Euler
function is in some way arbitrary. From the information in your email,
it seems to me that one could fairly easily prove the truth of the
equation. To me that indicates that the Euler function tells us
something "real" and not constructed. Am I missing something?

Oliver Dale
```

```
Date: 09/19/97 at 07:41:35
From: Doctor Anthony
Subject: Re: Euler function and radians

The Euler equation is indeed something real. The derivation is not too
difficult if you are familiar with the basics of complex numbers and
exponential functions.

dz/dx = -sin(x) + i.cos(x)

= i(cos(x) + i.sin(x))    (since i^2 = -1)

= i.z

So   dz/z = i.dx          now integrate both sides

ln(z) = i.x + const.  from (1) when x= 0, z= 1
so const. = 0
ln(z) = i.x

z = e^(i.x)    but z = cos(x) + i.sin(x) so

cos(x) + i.sin(x) = e^(i.x) .........(2)

Put x = pi in (2) and we get:

-1 + 0 = e^(i.pi)

and so e^(i.pi) + 1 = 0

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Imaginary/Complex Numbers
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search