I have long been fascinated by the mysterious mathematical formula called Euler's Identity. I enjoy it so much that I even have it framed on my wall:
Peculiar taste in art, I know. The print is entitled "Beauty," and somewhere in my apartment is a piece of paper with the artist's name and his description. It's amazing that I can lose something in my tiny residence.
In a recent conversation at Khan Academy, where I have been brushing up on my math and programming skills, someone asked what the relationship was between complex numbers and the trigonometric functions of sine and cosine. I had a vague understanding that they were both related to rotation, and I knew the Identity, but I had never really explored the Identity in depth to truly understand it. I have been thinking about it for two weeks now, and I'm still only about 80% of the way there. Another commenter asserted that it cannot be derived without recourse to Calculus, and maybe that's true if you need a formal proof, but I think an intuitive understanding should be within reach of anyone who knows a little trigonometry. Such as I have puzzled out, I share with you.
To begin with, i is defined as the square root of -1. Exponents of i follow a pattern:
i^0 = 1 (as all numbers raised to the 0th power)
i^1 = i (raising any number to the 1st power yields that number)
i^2 = -1 (√-1 * √-1)
i^3 = i * i^2 = -i
i^4 = i^2 * i^2 = 1 (back where we started)
The exponents of i will continue this four-part cycle of 1, i, -1, -i for as long as you care to keep multiplying. When something creates a cycle like that, it is tempting to believe that it is describing a circle. Let's take a look:
A complex number (a real number and an imaginary number, expressed like so: 2 + 4i) can be graphed on an Argand diagram. Apparently, if you relabel one of the axes on a Cartesian plane, you get to name it after yourself. An Argand diagram is just a normal graph with the x-axis representing the real numbers line and the y-axis representing an imaginary numbers line. If you graph those exponents of i on the Argand diagram, you wind up with four points, one unit away from the origin at 1+0i, 0+1i, -1+0i, and 0-1i:
If you were to write the complex numbers as ordered pairs on the Argand Diagram, you would get this:
i^0 = (1,0)
i^1 = (0, 1i)
i^2 = (-1, 0)
i^3 = (0, -1i)
Cosine and Sine each have a cycle of their own (angles expressed in radians, this doesn't work as nicely with degrees):
cos(0) = 1, sin(0) = 0,
cos(π/2) = 0, sin(π/2) = 1
cos(π) = -1, sin(π) = 0
cos(3π/2) = 0, sin(3π/2) = -1
Curious: Cosine and Sine show the exact same pattern. Suppose we make an assumption that:
i^x = cos(x*π/2) + sin(x*π/2)i
It is evident from the work we have already done that it will hold true for all integer values of x. What if x is continuous? Does it remain true when x = 0.5? Let's go back and look at the equation that started us down this road. Euler's Identity is actually a special case of Euler's Formula:
e^(ix) = cos(x) + sin(x)i
Substituting π for x makes the right-hand side of the equation -1, which does indeed give us the Identity. So it looks like we're almost there. The formula I derived is very similar to Euler's, but we still have that e to deal with. Where did it come from, and what is it doing?
Although natural logarithms and the constant e are usually introduced in an Algebra course (Algebra 2 or College Algebra usually), the origin of that "magic" number is something you're most likely to get in a Calculus course, as it involves the concept of limits. The limit, at least as it applies to a simple matter like e, isn't very difficult to understand though, so for the benefit of anyone who hasn't studied Calculus,
The least complex formula for calculating interest on a loan is this: A = P * (1 + r/n)^n, where P is the principal—the amount of money you started with, r is the interest rate, and n is the number of times the interest is calculated in whatever window we're considering (this is called "compounding"), usually a year. So $1000 at an interest rate of 7% compounded quarterly gives this equation:
A = $1000 * (1 + .07/4)^4 = $1071.86
The more times you calculate the interest, the higher your result goes. So it is in the interest of the investor for that calculation to be performed as often as possible. Monthly is good; weekly is better. Daily is superb. What if we were calculating every hour? Every minute? Every second? There are 31,536,000 seconds in a non-leap year.
A = $1000 * (1 + .07/31536000)^31536000 = $1072.51
65 cents over the course of a year isn't much to get excited about, but if your principal were $1,000,000 instead of a meager thousand, you'd add three zeros to the interest as well: $650 is a nice bit of cash in exchange for a little more computer time. But we're still looking at a finite number of calculations. We could keep subdividing down to milliseconds, nanoseconds, femtoseconds, and the interest ekes ever so slightly higher the faster we go.
What if we compound continuously? What does that look like mathematically? The time interval is infinitesimal; n approaches infinity. That's what a limit is, and that's how we calculate the value of e:
As n approaches infinity, (1 + 1/n)^n ≈ e ≈ 2.71828…
Where did the r go, you ask? It's still there. In the definition of e, the interest rate is 100%. If we reduce the interest rate, say back down to .07, we will ultimately wind up with e^.07. Try running the formula a few times with different r values and very large n values and see how the results compare to e^r. For our original problem:
A = $1000 * e^.07 = $1,072.51
That's the same answer we got when we were compounding every second, which supports the assertion that e^r is accurate. There is another hidden number in the expression. The time window across which we are compounding is in there, too. When we're talking about money, we usually want to know the annual increase. The window is 1 year. If we wanted to know how much the quantity would grow over ten years, we could multiply the interest rate: .07 * 10 = .7, yielding this formula:
A = $1000 * e^(.07 * 10) = $2013.75
The natural logarithm isn't just for money, though. It describes any process of continuous growth: bacteria colonies, radioactive decay, and population growth can all be modeled using e.
e describes growth, with its exponent indicating the rate of growth times the duration of the growth. So when we see e^i*π, we have i causing a rotation for a duration of π. Since 2π is the circumference of a circle, it follows that π is half of that. If our circle begins at 1 and rotates around the origin, then a half rotation places us at -1, which is what Euler's Identity tells us. The intuition verifies the assertion, but how do we get from i² to e^iπ?
And that's where I'm stuck. I can certainly show, using limits and geometry, that subdividing the circle into smaller and smaller triangles by using smaller and smaller values for x in Euler's formula will approach the shape of the unit circle. That doesn't help me to make the leap of whether i^x really has a valid relationship to e^ix. Intuitively, I feel that it does, but the logic just hasn't quite resolved yet.
I hope I will get there eventually, but for now I'll just have to continue to sit back and enjoy the Beauty of it.
If you'd like to read more about these topics, I recommend the writings of Kalid Azad at BetterExplained. His blog is dedicated to developing an intuitive understanding of difficult math concepts.