We were told about complex numbers, and did the algebra by solving problems with the general form of “a*n + b*i”. It does not really behave differently than standard algebra, it seems. What am I missing in the intention or meaning of the “i” part?
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Every number has a real part and an imaginary part. Like 5 would be 5 + 0i. It’s there because imaginary numbers are used as a way to make the negatives have square roots and stuff (I think I might’ve forgotten) and they usually loop back to a real counterpart.
Check out [this](https://youtu.be/Z49hXoN4KWg) video for more!
it’s basically two axis. Real numbers go horizontally and imaginary numbers go vertically. When you have those two axis you can calculate vectors and all kinds of stuff.
Imagine you start with just i.
Now, multiply it by i again, and you have -1, by definition i^2 = -1
Then, multiply by i again, and you have -1 * i = -i
Do it again, and you have -i * i = – ( i * i ) = – (-1) = 1
And finally, if you multiply by i one more time 1 * i = i
In this case, every 4th power of i takes you back to where you started, but more generally, multiplying by i tends to *rotate* your point through the complex plane. This is very different to how simple pairs of real numbers behave. With real numbers you can shift by adding, or stretch by multiplying, but they never rotate unless you do different things to different numbers.
This rotation property gives complex numbers quite a different structure than a simple pair of coordinates.
For example, you can look at the equation x^2 = 1, what are its solutions? 1 and -1 of course. But what about a different power, lets say x^4 = 1? In this case, you can start with our previous answers 1 and -1, and then look at what numbers can be squared to make them. For 1 its just 1 and -1, so no new solutions. But i * i = -1 but so does -i * -i = -1. So the equation x^4 = 1 has 4 solutions: 1, -1, i, -i, the exact values we were just discussing!
In general, for any x^N = 1, the solutions are N points evenly-spaced around a circle with radius 1 in the complex plane, and they always include 1. So, if you want to know the solutions of x^3 = 1, you start at 1, rotate that point 1/3 of a turn to get your second solution, rotate by another 1/3 of a turn to get the third solution, and a final rotation would bring you back to 1.
>What am I missing in the intention or meaning of the “i” part?
**You are missing the elegancy, and that’s it.**
Hamilton’s approach was exactly like you thought – 2D reals equipped with special algebra. This is in principle the same as the complex number, in the sense that the special algebraic structure R^2 is homomorphic to C.
Historically speaking, the imaginary unit was introduced to complete closed polynomial roots. In other words, given any polynomial equation, you wanted to find a number system that is closed for all the roots.
For example, the roots of x^2 +1=0 is non-existence in ‘conventional’ number system (ie, the reals.) To workaround this issue, you need to add some auxiliary part. An elegant way to do that is by defining the *i* (this was Euler’s idea).
i isn’t a variable, it’s a constant.
a+bi is three constants. a+by is two constants and a variable.
y^2 is just that, y^2, it doesn’t get simpler and it can never turn into an x. i^2 is just -1.
The algebra (symbol manipulation) is the same but the math meaning is totally different.
This is a great starting point! It’s not that different from having two different spatial dimentions: (almost) all of the basic algebra still works. I believe this is often the starting point for teaching complex numbers to avoid confusion, ease students in: you start with what you are familiar with, go over the rules that still work as before.
But the biggest difference is when defining products. The product for (ax + by) *(cx+dy) is not really defined, if we consider x and y separate spatial dimensions. But (an + bi) *(cn+di) is defined. If you consider that n=1, the real unit, this simplifies to (a + bi) *(c+di) = ac +(ad+bc)i + bd*i^2, and since we have defined i as the square root of -1, then also i^2 = -1, so
= ac + (ad+bc)i -bd = ac-bd + (ad +bc)i
This is pretty much the only major difference from just having a separate spatial dimension. But this very basic rule of i^2 = -1 and products being defined adds a ton of useful tools to algebra. This is quite typical in mathemathics: you add a simple rule, which might seem almost non-consequential, but it interacts with the pre-existing system in various, interesting and useful ways.
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