Eli5 Why do we need complex numbers in real life?
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Most of us don’t need them directly. Engineers and scientists and math folks do, because they explain real phenomena and are used to solve real problems. They are real mathematical ideas even if we call the actual numbers involved “imaginary” because in their “raw” form they don’t actually equate to anything on a conventional number line, or anything in the “real world.” Once we use them in mathematic operations thought they do explain physical events.
The only good use for them for most of us are like financial jokes and the like.
In your day to day life, you probably are only using integers. Banking and paying bills buying groceries etc are simple tasks.
So from that standpoint you as an individual don’t need to use complex numbers.
However, you use products and devices that were built using more complex mathematics. Your phone. Your car, your laptop, tablet and many other devices use complex numbers.
Complex numbers frequently use electronics and circuits. So when you use these devices you are relying on complex numbers.
There are a few different applications that come to mind. Admittedly, most of them require a really good understanding of physics and calculus. So I will try to write this as if none of that is prior knowledge. I hope that others can give more tangible examples.
Most of it boils down to Eulers identity. Basically, we can replace sin(x) and cos(x) with something that uses complex numbers, which is actually easier sometimes. For example, I studied sound waves in graduate school and we used complex numbers to model how a wave moves. As a very general statement, anything that rotates or oscillates can be modeled using complex numbers. I actually still use complex numbers daily!
Another example, complex numbers are used for lots of calculations in signal processing. Every time your phone receives a signal. It uses something called the Fourier transform, which uses complex numbers, to “translate” the signal to something it understands.
So if you are learning it in school now, or teaching it to someone, it may be difficult to motivate learn complex numbers without saying “trust me, in a decade this might matter”, but if you or they become an engineer or scientist, complex numbers have a lot of uses!
TLDR: engineers and scientists use them for a lot of complex (no pun intended) calculation because they actually make things easier sometimes. 🙂
Because complex numbers make solutions easier. For example, the first year physics students will understand, is a driven and damped oscillator. To mathematically describe it, we use complex exponentials to avoid a lot of trig identities, which then gives us a solution that can be physically interpreted. Of course the complex numbers have no physical meaning, since the first thing we do is we only consider the “real component” of the solution.
Complex numbers can be directly related to sinusoids (waves) because i * i = -1. Things like A^(i*x) repeat themselves!
This makes them very good for describing anything that involves regularly repeating information, which is actually an awful lot of stuff!
You don’t need them, but your graphics card does. The “Imaginary” number line is a very bad term, “perpendicular” would be a better one. Multiplying by *i* winds up being the exact same as rotating something 90 degrees counter-clockwise.
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.
Why do we need negative numbers? Will you ever have -3 cows?
I had 5 cows, 3 died, then I got 4 more, then 5 died again.
The caveman would: 5-3= 2 -> 2+4=6 -> 6-5=1
We however: 5 – 8 + 4 = -3+4=1
That -3 cows would make the caveman lose his shit. I had 5 lost 8 and got 4. We understand that this is useful. We get more freedom in doing calculations. Nobody cares that during the calculation -3 cows just show up.
Same with complex numbers. They give you more freedom during calculations. It may seem that problems are more complicated with complex, but its usually the opposite.
Or think like this: it would be hard to do algebra if we don’t allow negatives. In that case subtraction isn’t commutative. So moving terms left and right is a problem, until you say that subtraction is addition with negative numbers. Then you can shuffle things around easily.
Complex numbers (and theorems of complex analysis) makes many calculations easier.
Mathematical concepts correspond to human ideas.
Multiplication, as its taught in school, corresponds to areas of squares.
Negative numbers correspond to direction; you’re going in the backwards direction as opposed to forward.
In this sense, imaginary numbers represent rotation of shapes, among many other things.
If you multiply a number by the square root of -1, it corresponds to rotating that point by 90 degrees in a certain context.
Multiplying a number by just -1 rotates it by 180 degrees, you’re going backwards to where you came from.
This extends to any angle you want; if you multiply a number by the square root of the square root of -1, you get rotation of 45 degrees. Multiplying by the cube root of -1 gives rotation of 60 degrees. Multiplication by the cube root of the square root of -1 gives rotation of 30 degrees, and so on.
But imaginary numbers have many more uses than just this, just like multiplication is not just used to find the area of a square; this is just one of the first properties that crops up.
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