and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
An “imaginary” number is one that, if you square it, the result is negative. The other kind of number, “real” numbers, cannot work like this; if you square them, you get a positive value, because a negative number times a negative number is a positive number.
“Imaginary” numbers are called that because, as a matter of something you can actually directly *measure* like a length or a temperature or a current, no such number can be measured. But they are not imaginary in the usual sense of being “just made up, completely fantastical” (or, at least, not any more than any other number.) Instead, the difference between “real” numbers and “imaginary” numbers is that they tell us different things. “Real” numbers tell us raw data, the direct observable stuff. “Imaginary” numbers, on the other hand, are a way to talk about the *phase* of something. This is extremely useful because a lot of our universe can be described using waves, and waves can affect each other depending on their phase.
Two waves are perfectly in phase when their peaks exactly line up with each other, same for their troughs. Two waves that are exactly in phase will have “constructive interference” and thus add all of their amplitude together, so (for example) two sound waves that have the same frequency and amplitude, and are perfectly in phase, will be twice as loud as they were individually. On the other hand, two waves could be exactly reverse: where one has a peak, the other has a trough, every time, making them perfectly out of phase (aka 180 degrees out of phase). When that happens, it’s called (complete) “destructive interference” and it causes the smaller wave to cancel out part of the bigger wave. If the two are perfectly identical other than their phase, then they will entirely cancel out, leaving it seeming like there’s no waves at all. Most of the time, waves are only partly out of phase, somewhere between 0 and 180 degrees out of phase, meaning they partly add and partly subtract.
A “complex” number, which has both a real part and an imaginary part (usually written “a+bi,” where a is the real part and b is the imaginary part), can encode this phase information alongside the actual amplitude of the wave. This allows us to do very quick calculations in a simple way (using exponents), without needing to faff about with angles and cosines and sines and such. As a natural consequence of this approach, we can easily determine the actual physical situation (e.g. places where waves will cancel out or amplify each other).
This has a lot of uses. Lasers, for example, are coherent light beams. Or for a very practical example, the design of a concert arena’s speakers needs to account for places where the waves from two speakers would cancel out. You don’t want your audience to be left with big silent spots because their seats happen to be in a dead zone! Quantum physics uses this all the time, and various imaging, electronics, and sound applications exist that make use of waves in one way or another.
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