(Arguably) The Most Beautiful Equation inMathematics and (Unarguably) Proof that Math is

Dope

V. Hunter Adams

AbstractHey Kell. This is a derivation of Eulers (pro-nounced Oilers) equation using Taylor expansions. There areprobably some mathematicians that would get pissed at mefor doing this derivation as opposed to one based on differentmathematics, but I think this derivation is insanely cool, plusthis is the derivation that Euler actually used to discover hisequation.

Im sure you already know a lot of the stuff at the beginningof this derivation. Im not trying to be pedantic, I just thinkits really annoying when people dont start from first principleswhen explaining things, and its helpful to be clear about all ofthe terminology. Feel free to skip through the first sections.

I. FUNCTIONS

I know you know what a function is, but it might be oneof those things thats hard to put into words. Here are thewords:

Function: A relationship between a set of inputs and aset of outputs, with the property that each input is related toexactly one output.

For purposes of this derivation, only single-variablefunctions are needed. Examples include f(x) = x2, orf(x) = 3x3 +2x+1, or f(x) = sinx, or f(x) = ex. Its justan equation that you plug a single number into (x), and get asingle number out of (f(x), aka y).

Fig. 1. This is a function. For every value of x, there is exactly one valueof f(x) (or y).

II. DERIVATIVES

For all smooth functions (functions that dont have anypointy parts), you can take the derivative anywhere on the

Fig. 2. This is NOT a function since for every value of x there are twovalues f(x) (or y). One on the blue part, and one on the orange part.

function. The derivative is simply the rate of change of thefunction. If a function takes time as an input, for example,and gives position as an output; then the derivative is velocity.The derivative of the equation shown in Fig. 1 is:

d

dt(x2) = 2x (1)

It looks like this:

Fig. 3. Derivative of f(x) = x2 (to the left). Notice that it starts out negative,passes through the origin, then turns positive. Eyeballing the plot to the left,you can see that the slope does that.

If we take the derivative again of the velocity functionshown in Fig. 3, then we get acceleration. In this case, the slopeof the velocity is constant, so the acceleration is a constant.

d

dt2x = 2 (2)

And it looks like this: Since this line is flat, the slope is always

Fig. 4. Derivative of f(x) = x2 (to the left). Notice that it starts out negative,passes through the origin, then turns positive. Eyeballing the plot to the left,you can see that the slope does that.

zero. So if we take another derivative (or two more, or threemore, or infinity more), we will get zero. There are a classof infinitely differentiable functions for which the derivativenever dies. No matter how many derivatives you take of thattype of function, you will never get zero.

The thing to note here is just that the derivative is tellingyou about the curvature of the function. The first derivativetells you how fast the function is changing. The secondderivative tells you how fast the rate of change of the functionis changing. Etc.

III. TAYLOR SERIES

Lets say you have some function, well use f(x) = sinxas an example. This function shows up in equations for tons of

Fig. 5. Plot of f(x) = sinx

different things. That can be annoying, however, because sinis often really hard to deal with in equations. To get aroundthat problem, people use Taylor Series. The Taylor Series forsin can be used instead of sin itself in those equations. Its anapproximation of the real sin function.

Lets say, for example, that we want to approximate the sinfunction (shown in Fig. 5) around x = 0. The first thing wecould do is simply use the value of the sin function at x = 0(which is 0). See Fig. 6.

Fig. 6. Plot of sinx and sin (x = 0)

This is a pretty crappy approximation of the sin function.Its perfect exactly at x = 0, but then it diverges from the realsin function really quickly. If you only care about the solutionexactly at x = 0, then this works fine. Otherwise, you have tomake a better approximation.

The way to improve the approximation of the sin functionat x = 0 is to use information about the derivative of the sinfunction at x = 0. In this way, we start to match the curvatureof the sin function at x = 0. Starting with the first derivative:

sinx sin 0 + x ddx

sinx|x=0 (3)= 0 + x (4)

Heres the plot: Its a better approximation. It sticks near the

Fig. 7. Approximation of the sin function at x = 0, using information aboutthe derivative of the sin function at x = 0.

sin curve for a while longer than the previous approximation.An even better approximation can be made by using informa-tion from the first, second, third, etc. derivatives of the sinfunction at x = 0. From the second derivative:

sinx sin 0 + x ddt

sinx|x=0 + 12x2

d2

dx2sinx|x=0 (5)

0 + x+ 0 (6)(No new information came from the second derivative, for thisparticular function). From the third derivative:

sinx 0 + x+ 0 16x3 (7)

The improved approximation is shown in Fig. 8:

We could keep doing this forever. Each time informationabout the next derivative is taken into account, the approxima-tion to the sin curve gets a bit better. Taking a few more intoaccount, the approximation looks like this:

Fig. 8. Approximation of the sin function at x = 0, using information aboutthe first, second, and third derivatives of sinx at x = 0.

Fig. 9. Adding more and more information about the derivative at x = 0improves the estimate of the sin curve.

As a matter of fact, if we used information from infinitynumber of derivatives, the approximation would convergeexactly to the originial sin curve. Think about that for a second.Using information about the function at one point, it is possibleto reconstruct the function over all of space. If this were a3D function that occupied the entire universe, it would bepossible to reconstruct the entire thing using information aboutthe function at the tip of your nose.

The procedure by which these derivatives are combined isgiven by:

f(x) n0

1

n!fn(x)|x=0xn (8)

In the case of the sin function, this summation convergesexactly if n = 0. For the sin function, the expansion can bewritten as:

sinx = x x3

3!+x5

5! (9)

=

n0

(1)n x2n+1

(2n+ 1)!(10)

If n = , then the summation and the sin function arecompletely, 100 percent identical.

IV. ONE STEP FARTHER

The sin function is not unique in that it can be expressedas a summation. By using equation (8), a lot of other functionscan also be expressed this way. Lets take two in particular:

the cosine and the exponential.

cosx = 1 x2

2!+x4

4!+ (11)

=

n0

(1)n x2n

(2n)!(12)

All that we did was use equation 8 to write out a few termsof the summation. Then we noticed a pattern and re-wrote thesummation as equation (12). Similarly for the exponential:

ex = 1 + x+x2

2!+x3

3!+ (13)

=

n0

xn

n!(14)

(Ill talk later about what e actually is). Again, by letting ngo to infinity in equations (12) or (14), we can exactly recoverthe cos and exponential functions. Im repeating myself, butthats a very important point.

Its also important to remember that equations (10), (12),and (14) are just sums of numbers. Because they are just abunch of numbers added up, all of the properties of additionstill hold. It doesnt matter, for example, the order in whichwe add the terms in the summation since, by associativity:

+ = + (15)

The order doesnt matter. The distributive property also stillholds. Any constant multiplied by the summation can berewritten as the summation of the constant multiplied by everyelement of the summation. That is:

C(+ ) = C+ C (16)

Equations (10), (12), and (14) are nothing more complicatedthan a bunch of numbers added together.

V. A BRIEF ASIDE

Sidestepping for a moment, Id like to give the complexnumber i a clear definition before going farther in the deriva-tion. The imaginary unit i is defined as

i =1 (17)

Clearly, there is no real number that satisfies that equation.Imaginary numbers are, in a mathematical sense, perpendicularto real numbers. It took people a long time to realize thatthese numbers were useful (in much the same way that it tookpeople a long time to come around to the idea of 0 or negativenumbers). In fact, the term imaginary was coined in the 17thcentury as a derogatory term.

Anyway, from the definition given in equation (17) wederive a few properties:

i0 = 1 (18)ii = i (19)

i2 = (i)(i) =11 == 1 (20)

i3 = (i)(i2) = (i)(1) = i (21)i4 = (i2)(i2) = (1)(1) = 1 (22)i5 = (i)(i4) = i (23)

... (24)

After i3, the cycle repeats with i4 = 1, i5 = i, etc.

VI. BACK TO THE DERIVATION

Look back at equations (13) and (14). In those equations,x is a real number. Let us use those summations to writethe summation for eix. By using ix instead of x, we are justmaking x imaginary. Precisely what that means is difficult tointuit, but the solution is easily obtained by substituting ix forx in the summations:

eix = 1 + (ix) +(ix)2

2!+

(ix)3

3!+

(ix)4

4!+ (25)

Now using the properties given in equations (18)-(23), we canrewrite this summation:

eix = 1 + (ix) x2

2 ix

3

3!+x4

4!+ i

x5

5!+ (26)

Because this is just a bunch of numbers added together,the order that we add them doesnt matter. Furthermore, thedistributive property (equation 16) holds. So, we can rearrangethe summation so that it looks like this:

eix = (1 x2

2!+x4

4!+ ) + i(x x

3

3!+x5

5!+ ) (27)

Look at the summations inside each set of parentheses,and look at equations (9) and (11). This is Eulers insight.The summation inside the left set of parentheses is exactly thesummation for cosx and the summation in the right set ofparentheses is exactly the summation for sinx. If we let n goto infinity (if infinity numbers of terms are included in eachof these summations), then the summations and the cos/sinfunctions are completely, 100 percent identical and can be usedinterchangeably. Rewriting one last time:

eix = cosx+ i sinx (28)

This is Eulers equation. From here, we can get some reallyinteresting relationships, including the most beautiful equationin mathematics.

VII. ANOTHER BRIEF ASIDE

Before finishing up, I want to sidestep one more time totalk about e and pi. pi has a really clean geometric definition.If you take any perfect circle and divide its circumference byits diameter, you will get pi. Thats because

C = 2pir = piD (29)

As you know, pi is real (not imaginary) and irrational (impos-sible to be expressed as a fraction, and with infinite decimals).Numerically, pi = 3.1415926 .

e doesnt have the same clean geometric definition as pi, butits similar in a lot of ways. Like pi, it is a real and irrationalnumber with infinite decimals. It can be defined in a numberof different ways. One definition is simply given by the TaylorSeries:

e =

n=0

1

n!(30)

If you do that summation, youll get the number e. It canalso be defined as the unique number a such that the graph of

y = ax has a slope equal to 1 at x = 0. Another definition fore is the solution to the equation

limn

(1 +

1

n

)n(31)

This equation shows up all over the place when youre dealingwith things that are compounding (interest, perhaps, or popu-lations). e shows up when youre dealing with things that areexponential.

The definition for e isnt as intuitive or geometricallysatisfying as that for pi. Like pi, its a real irrational number.e shows up all over the place as the solution to equationsinvolving compounding stuff, or exponentially growing stuff.Numerically, e = 2.718281828 .

VIII. THE MOST BEAUTIFUL EQUATION

For clarity, equation (28) is rewritten below:

eix = cosx+ i sinx (32)

For absolutely any number x that we plug into this equation,the left side will equal the right side. Let us plug in the valuex = pi. This yields

eipi = cospi + i sinpi (33)

The cosine of pi is -1, and the sine of pi is 0. So, rewriting:

eipi = 1 + i(0) (34)eipi = 1 + 0 (35)

Subtracting the -1 to the other side of the equation:

eipi + 1 = 0 (36)

The above equation relates e, pi, 1, i, and 0. These areconsidered the five most important numbers in mathematics.It relates the additive identity (0) which is real and rational,the multiplicative identity (1) which is real and rational,the most important constant in geometry (pi) which is realand irrational, the constant that forms the base of naturallogarithms (e) which is real and irrational, and the fundamentalunit of imaginary numbers (i) which is obviously imaginary.Furthermore, it includes the three core arithmetic operationsaddition, multiplication, and exponentiation.

This inspired Richard Feyman to call it our jewel andthe most remarkable equation in mathematics.

Stanford mathematics professor Keith Devlin said like aShakespearean sonnet that captures the very essence of love,or a painting that brings out the beauty of the human form thatis far more than just skin deep, Eulers equation reaches downinto the very depths of existence.