How Are The Most Important Numbers Related?

To get started, what are the most important numbers in math?

To be sure, there are quite a few. For the purposes of this problem, we'll focus on five numbers. The first three should be familiar to you, but the other two may not.

The first number is 0.

The second number is 1.

These should be familiar to pretty much everyone.

The third number is π (pi) (3.14159..., or about 3.14).

Pi is the ratio of a circle's circumference to its diameter. Pi should be familiar to most people who took math in elementary and junior high school.

The fourth number is e (2.71828...).

The number e is found most often in Algebra II-level and above math classes. It pops up in exponential and logarithmic problems, and is frequently used to calculate interest and growth on a bank account or loan.

The fifth and final number is i, the imaginary number (or the square root of negative one).

The number i, like e, often appears beginning in Algebra II curricula.

Other numbers are also important, such as 2, the square root of two and phi (the golden ratio), but we're going to focus on these five.

So how are they related?

To figure that out, we'll start with the two which are least familiar to most people, e and i.

Heads up, the next section can get REALLY mathy. I'll be going through a derivation. If you don't care to know it (though you should) or if you get lost, you can skip ahead and just accept the equation marked (1) as fact.

When e is used in exponential functions, it is usually seen as f(x) = e^x.

Using calculus, you can create a Taylor series for e^x. Don't worry if this is new to you. All you need to know is that a Taylor series basically takes a complicated function (like e^x) and makes it into a polynomial (like x x^2 + x^3...) that can go on forever.

The Taylor series for e^x is e^x = 1 + x + (x^2/2) + (x^3/6) + (x^4/24) + (x^5/120) + ...

If you're familiar with trigonometry, you probably know the words sine (sin) and cosine (cos). These are trigonometric functions that also have Taylor series.

The Taylor series for sin(x) is sin(x) = x - (x^3/6) + (x^5/120) - ... (notice the signs alternate).

The Taylor series for cos(x) is cos(x) = 1 - (x^2/2) + (x^4/24) - ... (notice the signs alternate).

If you're noticing a pattern, you're among famous company. Leonard Euler (and many other famous mathematicians) noticed it as well. Could there be a way to make e^x equal sin(x) and cos(x)?

This is where the imaginary number comes in. What if we had e^(ix) instead of e^x?

Our Taylor series would be e^(ix) = 1 + ix + ((ix)^2/2) + ((ix)^3/6) + ((ix)^4/24) + ((ix)^5/120) + ...

The powers of i repeat, so i to the first is i, i to the second is -1, i to the third is -i and i to the fourth is 1.

This also means i to fifth is still i, and i to the sixth is still -1.

So we can simplify to get e^(ix) = 1 + ix + -(x^2/2) + -i(x^3/6) + (x^4/24) + i(x^5/120) + ...

By rearranging the terms, we see that e^(ix) = [1 - (x^2/2) + (x^4/24) - ...] + [ix - i(x^3/6) + i(x^5/120) - ...].

The left brackets should look familiar. That's the Taylor series for cos(x)!

So now we have e^(ix) = cos(x) + [ix - i(x^3/6) + i(x^5/120) - ...].

Everything in the right side has an i, so we can factor that out.

Now we have e^(ix) = cos(x) + i[x - (x^3/6) + (x^5/120) - ...].

Now the right brackets are exactly sin(x)!

Our final equation is:

(1) e^(ix) = cos(x) + i sin(x) (1)

This is known as Euler's Formula, and it will be very useful in later discussions.

If that seems like a lot of math, it is. But that was the hard part. The rest is smooth sailing.

Now we introduce the third number, π. If we let x=π, we get the following equation:

e^(iπ) = cos(π) + i sin(π).

Pi is a radian measure (π radians = 180 degrees). If we know (or use a calculator to find) that cos(π) (or cos(180 degrees)) is -1 and that sin(π) (or sin(180 degrees)) is 0, our equation simplifies.

e(^iπ) = -1 + (i)(0)

Since we know anything times 0 is 0, we end up with the following equation.

e(^iπ) = -1

Now we want to introduce the second number, 1, by adding it to both sides of the equation.

e^(iπ) + 1 = -1 + 1

Our last step is to simplify the right hand side.

 
In our final step, the first number, 0, fell out of the equation!

So there it is, the relationship between the most important numbers. Some have called this equation the most beautiful and elegant in all of mathematics, and, so far, I agree. This one is my favorite, and I hope it is now one of yours.

Anyway, I hope you got something out of this. If not, well, there's always next time.

If you have any questions, or if anything isn't clear, just let me know in the comments.

I first came across this fact during my junior year of high school in my math team class. I later saw explanations in my senior year calculus class, and I saw the explanation given here in my freshman year of college in 18.01A Single Variable Calculus.

The image of the final step was found at http://www.informath.org.