Introduction to the Euler Identity
In mathematics there are some numbers that deserve special attention. The Euler Identity connects five of the most important numbers in mathematics, conventionally named e, pi, i, 0, and 1. The Euler Identity evaluated at pi gives e^(i*pi) - 1 = 0. For those who are already vaguely familiar with these numbers, this is quite disorienting at first glance, but a quick proof might remedy the vertigo. First we need to review the numbers e, pi, and i one by one.
The number e is the most challenging to describe. Roughly, it is the choice of base of an exponential function that makes the function equal to its derivative. Suppose f is a smooth, continuous function of x on the interval (a,b). There are 2 related functions called the derivative of f and the integral of f.
|Figure 1. The derivative and integral|
Recall from calculus that at any point, x, in the domain of f we can derive the slope of the line tangent to the graph f at that point. We call this quantity the “derivative” of f and denote it by
(d/dx)(f(x)), or f'(x)
This function can be obtained through the evaluation of the limit of a sequence of differences. Similarly the area under the graph of f between the two values of x, a and b, can be found by evaluating a limit of a sequence of sums. We call this quantity the “integral” of f and denote it by
Fortunately there are easily remembered formulas for the integral and the derivative. The integration formulas are more or less antidifferentiation formulas; the only change is the addition of a constant k. Recall that if the function is a polynomial p_n*x^n
(d/dx)[p_n*x^n] = n*p_n*x^(n-1)
[n*p_n*x^(n-1)] dx = p_n*x^n + k .
For the trigonometric functions we have
(d/dx)[sin(x)] = cos(x)
(d/dx)[cos(x)] = -sin(x)
[sin(x)] dx = -cos(x) + k
[cos(x)] dx = sin(x) + k .
The exponential function f(x) = p^x has base p. Its derivative has the form
(d/dx)[p^x] = k*p^x
for some constant k. The constant k will be larger if p is larger. For example,
if p = 2 then k_2 = 0.693147...
if p = 3 then k_3 = 1.098612...
This invites speculation, does there exist a base, which we will call e, for which k_e = 1, giving (d/dx)[e^x] = e^x ? There does, e = 2.7182818459... . Figure 1. shows a plot of y = e^x.
|Figure 2. Three exponential functions|
A base p exponential function, p^x, has an inverse function called the base p logarithm, log_p(y). Inverse means that
log_p(y) = x if and only if y = p^x .
Notice that if you use either one of these equations for x or y inside the other, then you arrive at the identities
p^(log_p(y)) = y , and log_p(p^x) = x .
The logarithm and the exponential do and undo each other like plus and minus. There is of course a base e logarithm. It is denoted ln(*). Suppose we take
y = ln(x) ,
exponentiate both sides and rewrite it as
x = e^y .
Using the chain rule to differentiate with respect to x gives
(e^y)*(dy/dx) = 1
(dy/dx) = 1/e^y = 1/x ,
or since y = ln(x)
(dln(x)/dx) = 1/x .
Integrating both sides reveals that
ln(x) = [1/x] dx + k .
Remember this, it is an important stepping stone to Eulers Identity.
Perhaps the most famous number ever is pi. For any circle the ratio of the diameter to the circumference is ~= 3.1415... denoted by the Greek letter pi. This means that it takes pi diameters, or 2*pi radii, called 'radians', to wrap around the outside of the circle. (see Figure 3.) The angle, phi, subtended by the center, and two points on the circles diameter is conveniently characterized by the ratio of the circumference enclosed by the two points to the total circumference. We express this ratio as "radians of circumference enclosed by the angle", or simply "radians". For a given angle, this ratio is the same on any circle regardless of how big or small. We call this property similarity. Similarity is found in other geometric figures, in particular in triangles.
|Figure 3. Radian angle measure & similarity|
Recall from trigonometry that if we take one of the two circumferential points to lie on the x axis and label the other as the coordinate (x,y), then we can construct the right triangle with (0,0), (x,y), and the point (x,0) lying on the x-axis immediately below (x,y). (see Figure 4. ) The lengths of the sides of the triangle are given by x, y, and r = sqrt(x^2 + y^2). We can scale the size of the triangle by repeating the construction with the same angle on a circle with a greater or smaller radius, but the two triangles are 'similar'. That means that the ratio of any two given sides will be the same in both. These ratios uniquely characterize the triangle and thus the angle phi.
|Figure 4. Polar Coordinates & similarity(Note:I couldn't find a more accurate image on the internet. here 'x' represents phi, the angle)|
We define the ratios
sin(phi) = y/r , cos(phi) = x/r, and tan(phi) = y/x.These ratios are the same for any right triangle having angle phi. A transformation from rectangular coordinates to a system of polar coordinates easily follows from these definitions. Our transformation is given by
(x,y) = (r*cos(phi), r*sin(phi))
(r,phi) = (sqrt(x*x + y*y), arctan(y/x))
for some angle theta and radius r. It is often useful to choose r = 1; this set of points is called the "unit circle".
The imaginary number, i, is also important, and deeply connected with a discussion of the unit circle. We denote the square root of negative one by the letter i. A complex number is of the form z = a + i*b, where a and b are real numbers. It is possible that a or b equal zero. That is, purely real or purely imaginary numbers are still complex numbers, just as the x-axis and y-axis are still contained in the plane. Taking this notion a brief step further we could interpret the real and imaginary components of z as rectangular coordinates. Conventionally x is chosen as the real axis and y as the imaginary axis. From the earlier discussion we know that z also has a polar representation for some (r,phi). Pick a z on the unit circle, making r = 1. Then
z = cos(phi)+i*sin(phi)
Differentiate with respect to the angle phi
dz/dphi = -sin(phi)+i*cos(phi) = i(cos(phi)+i*sin(phi)) or
dz/dphi = i*z .
Interchanging z and dphi gives
dz/z = i*dphi .
Integrating both sides gives
ln(z) = i*phi + k ,
and exponentiating both sides gives the Euler Identity
e^ln(z) = e^i*phi or z = e^i*phi = cos(phi) + i*sin(phi) .
It is counterintuitive to accept, however it is easy to evaluate the Euler Identity. Try evaluating it with phi = pi.
e^i*pi = cos(pi) + i*sin(pi) = -1 + 0 or e^i*pi + 1 = 0 .
And there in a single relation we have e, i, pi, 0, and 1, the cornerstones of our number systems! Fortunately this is as close to mysticism that mathematics ever gets.