Euler's Formula

Around 1740, Euler discovered the following connection between the exponential function and trigonometry (Needham, 1998; Maor, 1994):

eiθ = cos(θ) + isin(θ)

In his honor, this formula is often referred to as Euler's formula. This formula is quite famous and widely regarded as one of the most beautiful formulas in mathematics. Of particular note is this formula when θ=Π, which yields the following identity:

eiΠ + 1 = 0

According to Maor (1994), this identity is remarkable because it concisely relates the five most important constants in mathematics: 0, 1, i, Π, and e. These five constants represent four major branches in classical mathematics: arithmetic (0 and 1), algebra (i), geometry (Π), and analysis (e). Additionally, the formula also utilizes the three most important operations in mathematics: addition, multiplication, and exponentiation. In regards to this formula, Edward Kasner and James Newman stated the following:

"There is a famous formula-perhaps the most compact and famous of all formulas-developed by Euler from a discovery of De Moivre: eiΠ +1=0....It appeals equally to the mystic, the scientist, the philosopher, the mathematician" (Maor, 1994).

Below are two arguments for why this formula is true.

Moving Particle Argument

The applet below provides one argument for Euler's formula. The left window of the applet considers the left hand side of the formula, eiθ. Recall that in the explanation of mathematics section we discussed how a defining property of the exponential function is that it is its own derivative. Thus, d/dx e kx = kekx where k is some constant. When we extend the exponential function to include imaginary numbers, we assume this property must still hold (Needham, 1998).

With this in mind, consider a particle moving in the complex plane whose position is defined parametrically by the function z(t)=eit. Then, the particle's velocity, v, is given by v(t) = dz/dt = ieit. Note that the velocity is the position multiplied by i and multiplying by i is equivalent to rotating through a right angle*. Since the initial position of the particle is z(0)=1 and its initial velocity is v(0)=i, the particle starts at 1+0i and begins moving upward (Needham, 1998). Using your mouse drag the blue point in the left window in the direction of v in order to draw the motion of the particle. How would you describe the path the particle travels?

* To illustrate why multiplying by i is equivalent to rotating through a right angle, consider the following example:

i(2+i) = (2i+i2) = (-1+2i)

Expressing the complex numbers as vectors and taking the dot product of <-1,2> and the original complex number <2,1> yields -1(2)+2(1)=0. Since the dot product is 0, the vectors are orthogonal.


The right window of the applet considers the right hand side of the formula, cosθ+isinθ. Using your mouse drag the black point in the right window around the unit circle in the complex plane. What relationships exist between the real and imaginary parts of the complex number associated with the point and cosθ and sinθ?



After exploring both windows, consider how they relate to each other. How does this deepen your understanding of Euler's formula?

To gain more insights about this argument and Euler's formula, click here .

Series Argument

Another argument for Euler's formula uses series. Recall that in the explanation of mathematics section we derived the general formula for a Taylor series as well as explored the series expansions of the functions ex, cos(x), and sin(x) using an applet. From this, we know:

ex   =  1 + x + x2/2! + x3/3! + x4/4! + x5/5! +...
cos(x) = 1 - x2/2! + x4/4! - x6/6! +...
sin(x) = x - x3/3! + x5/5! - x7/7! +...

Euler took the series expansion of ex and replaced x with ix yielding:

eix   =  1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! +...

Since i=√-1, this can be rewritten as:

eix =  1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! +...

Euler then rearranged the terms, so that all the real terms were grouped together and all the imaginary terms were grouped together. This can be dangerous as changing the order of terms in a series may change the series or affect its sum. However, this was not fully understood during Euler's time (Maor, 1994). Rearranging the terms yields the following:

eix =  (1 - x2/2! + x4/4! +...)  + i(x - x3/3! +  x5/5! +...)

Note that in the parentheses on the left is the series expansion of cos(x) and in the parentheses on the right is the series expansion of sin(x). Therefore,

eix = cos(x) + isin(x)


Significance

Euler's formula unifies and simplifies many results in diverse areas of mathematics and the sciences ("Euler's Formula For Complex Exponentials", n.d.). This is because many problems, even those that don't appear to involve complex numbers, are most conveniently solved when viewed through a complex lens (Needham, 1998). Some of the applications of Euler's formula include, but are not limited to, the following. Euler's formula:

However, perhaps more important than its applications, is the formula's remarkable nature and beauty.


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