Nobody can write about the history of science and mathematics without eventually bringing up Leonhard Euler (1707-1783). (Most Americans end up pronouncing his name like “oiler”.) So many important findings in math and physics that it’s hard to list them all, so I won’t try. I don’t really want to write a biography of him anyway: I just want to focus on one profound equation he discovered, and follow where it leads into some other interesting math…and of course physics.
Without further ado, here’s the most famous version of Euler’s equation:The numbers π and -1 you should know; the exponential number e may be less familiar. It’s another geometrical number like π, and it has a value approximately equal to 2.71828 (but since it goes on forever without repeating, I’ll spare you any more digits). There are several ways to get at e, but we don’t need to worry about them for now. The main thing is that it’s standard, built into scientific calculators, and well-understood. The imaginary unit i also appears in the equation: recall that So this is where things get interesting: e and π are real irrational numbers, but including i should give us a complex number, but it doesn’t: it gives the negative integer -1. (Irrational means these numbers can’t be expressed as the ratio of integers; integers are the whole numbers 0, 1, -1, 2, -2, 3, -3, etc.) Somehow the combination of two irrational numbers and the imaginary unit produces -1.
In case you’re thinking I’m making a big deal out of nothing, type “exp(pi)” into Wolfram Alpha: you should getwhich is significantly larger than 1, and a positive number to boot. In fact, you won’t ever get a negative number by raising e to a real number. Try these in Wolfram Alpha: exp(-10), exp(-1), exp(0), exp(1), exp(10). Something weird and cool happens when you include an imaginary number in the exponent, and Euler realized after some careful computation what was going on.
First of all, the number π is special: if you plug another number in its place (say 1), you won’t get a real number out:It’s a complex number: the sum of a real number with an imaginary number. (Refer to the figure on the right for a refresher on how to interpret complex number using coordinates.) However, watch what happens when we square the real part and the imaginary part and add them together:I’m rounding in all these cases; if you want to be more accurate, keep the full set of digits you get out of your calculator before squaring…and you’ll still get 1 as your answer. This will happen if you use any real number in place of π, and Euler determined that you can split the exponential into real and imaginary parts like this:where “cos” is the cosine function, “sin” is the sine function, and θ is a real number.
If you’re like me, you first learned about sine and cosine in the context of triangles, and that’s a pretty useful way to think of them here too. To start, let’s draw a circle with radius equal to 1. All points on the circle will be the same distance—1—from the center; after all, that’s really what a circle means. Take an arbitrary point on the circle and map its x- and y-coordinates; draw a line from the center of the circle to your point. Now complete the triangle by connecting your point with the x-axis and the center, as shown in the figure below:
If we associate the x- and y-axes with the real and imaginary parts of a complex number as before, the angle between the blue line and the x-axis, marked by θ, is the same as in Euler’s formula. The x-coordinate is the cosine of the angle, and the y-coordinate is the sine of the angle. Properly, we need to write the angle in radians, not degrees: one circle (360°) is 2π radians, so a half-circle (180°) is π radians. Looking at the circle above, you can see that an angle of π corresponds to x = -1 and y = 0, exactly what Euler’s formula predicts!
I won’t prove Euler’s formula (since it really needs calculus to do properly), but with a little more work we can see how useful it is. By drawing a larger or smaller circle, we can represent any complex number using a variation on Euler’s formula:where r is a real number representing the radius of the new circle. Instead of writing the real and imaginary parts of the complex number like x- and y-coordinates, we can use r and θ, which are known as polar coordinates. The angle θ is also known as the phase of the complex number, and r is its magnitude. That’s a very simple formula, and makes doing many calculations with complex numbers quite easy.
For example: if you want to mathematically represent rotations in two dimensions, you can do it using complex numbers. Take the coordinates of (say) the corners of a square, and write them as four complex numbers a, b, c, and d. To rotate the square by angle θ, multiply the number for each corner of the square by and you’ll get the coordinates of the rotated square:
Euler and Quantum Mechanics
Quantum physics doesn’t just use complex numbers out of convenience (like we did to perform rotations): complex numbers are absolutely necessary, and Euler’s formula is more than useful. The wavefunction in quantum mechanics consists of the real probability amplitude and a phase:The phase is generally undetectable in experiments, but the difference in phase between two particles is measurable. The difference leads to interference, which is key to understanding the famous double-slit experiment and the Aharonov-Bohm effect.
Euler’s Formula Using Quaternions
A few weeks ago, I wrote a post about a fancier version of complex numbers known as quaternions, discovered many years after Euler’s death. Instead of one imaginary number, quaternions have three (labeled i, j, and k), and we showed them to be useful for representing rotations in three dimensions. Therefore, as you might expect, there are three Euler-like formulas for quaternions, representing rotations around the x, y, and z axes:
In aerospace engineering, the axes are attached to the body of the airplane or spacecraft, and the angles θ, φ, and ξ are called pitch, yaw, and roll. The different controls aboard the plane are designed to control rotation in those directions: for example, during liftoff, you need to raise the pitch, but keep yaw to a minimum, while roll can be used to steer the plane in the right direction. (By the way, there’s another set of angles known as (guess what?) Euler angles, but these don’t correspond directly to the angles I’m using here.)
By now you may guess the way my twisted brain works, and know what our next logical step must be. If complex numbers generalize to quaternions, are there even higher-dimensional versions of complex numbers—with attendant rotations and Euler formulas to go with them? The answer of course is a resounding yes, though we have to introduce a new type of number to make everything work. However, by the time we’re done, we’ll have the math to handle relativity and particle physics, and you, Dear Reader, will arrive at the cutting edge of modern physics.