Eli5 Why do we need complex numbers in real life?
In: 11
Tl;Dr I feel people purposely try to make it sound mystical. They’re basically good because they let us write about rotations in a really nice, succinct way.
Picture a yo-yo, or anything that springs up and down. You may not be familiar with the idea, but when it’s higher, it has more ‘potential energy’, and when it’s lower, it has less (it’s converted into things like kinetic energy, and vis-versa when it springs back up).
We obviously can link this potential energy with its height as you spring it up and down over time. It might look like a sine wave (those curves that go up and down).
Maybe we want a graph that shows where our yo-yo is in Y (height) over time, T (seconds), but where would our energy sine-wave plot go? Well, we can make a third axis for energy, but it can look really complicated – it’s going up and down in time on the Y axis, and up and down in time on the energy axis. (Note: picture this as a single line on one 3D plot, rather than 2 separate 2D plots – it may be difficult, that’s the point!) Of course, in bigger complicated systems, this gets a lot harder to explain with words.
This sort of 3-axis behaviour is real hard to describe with 1 equation. Sure, we could have Y = something * time and maybe E = something else * time, but it’s not quite the same as something that bundles together Y and E.
This is the part that’s hard to describe without a diagram:
If we say that energy is instead in its own little world (since it had different units to height anyway), maybe we say it’s imaginary – it’s separate to our normal world of units we use for height. Imaginary numbers can be linked to real numbers through a rotation very elegantly. So, now we have time going forward, T, linked to Y (height) and in our original 2 axes, and it goes up and down. Using a third imaginary axis, which is accessed by rotating the Y value at some rate in time, picture that as well as going up and down in Y, it’s also rotating around the time axis too (using that third dimension). It may appear to add a left or right movement to the plot, perhaps forming a slinky-like shape. This new dimension, the imaginary component, can describe energy.
(Note: this particular case wouldn’t make a slinky shape, but that’s a very common shape to see for engineering systems – this case is a bit too linear to describe with a nice shape, I think… Which is weird).
It’s not always that simple, but the gist is, rather than describing everything with x y and z (3 variables), we use x and y, as well as some rotation (represented by combinations of i terms, which aren’t variable), which can be condensed to a lovely complex equation that uses one less variable.
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