[Potironette]

Admittedly, a lot of that went over my head, but just to clarify...

>An imaginary number is anything with "i"
>A real number is anything without "i"
>A complex number is not an imaginary number nor a real number but a mix of the two?

The norm of a complex number (a + bi) is sqrt(a² + b²). This is an extension of the notion of absolute value: you'll notice it's equal to the absolute value for numbers without an imaginary part.

Err..what do you mean by the "norm"? Is it the magnitude or something? Like the imaginary number is on the y-axis (I don't get why the y-axis but for some reason internet always uses it? Buuut they're likely just ignoring the y-axis and replacing it with a separate imaginary axis?) and then there's a point, say (3, 4i) and the magnitude is the same as sqrt(9 + 16) which is 5?
But I don't understand how the absolute value fits in..

So what is an imaginary root?

Uhh, so what indeed? All I know about it is that i is the square root of -1, so I suppose it's all the multiples of the square root of -1?

Well, complex numbers work just fine and dandy in polynomials. You can multiply and add complex numbers just fine as long as you remember the identity i * i = -1. So if a real root is a solution to p(x) = 0 where x = a + 0i, then an imaginary root is a solution to p(x) = 0 where x = 0 + bi.

When I have imaginary roots, and I make a graph, it it that I won't even see the imaginary root graphed?


As for the second post, I'm currently wondering what it means.
First, imaginary roots exist...
Real numbers are one dimensional because they're on a number line? And then having two number lines as the x and y axis they're...plotted as two dimensional??
Also, if real numbers are one dimensional and are plotted in two dimensions, then imaginary numbers should be one dimensional and plotted in two dimensions too! Maybe..?
Complex numbers are a combo of real numbers and imaginary numbers put together. How that works, I'm guessing, is by having a figure with four axis, two for imaginary and two for reals and somehow one point can be made from that? But how that works gets into the difficult visualization part.

But you CAN think of it as a FIELD.

It's possible to think of it as a "field." What is a field..?

For complex numbers, imagine a plane. If you consider f(x + yi) -- that is, the real part of the number along the x axis, and the imaginary part along the y axis -- then you can evaluate the function at any point on the plane, and the result is a two-dimensional vector. For example, this is a graph of the function f(x) = (x + 2)(x - 2), if you allow x to take on complex values:

From this point forward I'm confused.
What is f(x + yi) and what is the vector referring to?
In the graph picture, I guess the x-axis turned into an axis that for complex numbers because of the problem with how difficult it is to visualize three dimensions? But that can't be it, surely? I'm definitely guessing here. And besides, why have just one axis take on complex values? Why not both? Is that even possible?
As for the last bit I still need to follow it on pen and paper, but essentially it's possible for something to just have complex roots!


Not math question: What is a "pacifying issue"? I read somewhere that someone thought that, for sure, the government was thinking (1970) that Earth Day or something environmental was a "pacifying issue." Does it mean an issue the government wants to just ignore..? Thinks will just die after a while?