Dividing by variable when solving algebraic equation

[Alex K]

This kind of game of wandering around the complex plane to pick up to correct solution to a complex square root or a logarithm is super important in physics, by the way. For example, there's a theorem central to scattering theory and quantum theory called the "optical theorem" which tells you that the imaginary part gives you the total scattering cross section (basically the probability of two particles doing something when they meet), and the sign of that imaginary part tells you whether causality is preserved or not, because if you got it wrong, you'd have negative scattering cross section, which would mean the probability for something scattering is negative. The original prescription how to properly deal with paths through the complex plane in particle scattering is by Feynman himself, and is called the Feynman prescription accordingly.

[robvalue]

Very nice description, thanks Alex I'm rusty with a lot of this stuff.

I've always found complex numbers fascinating. Even though they don't map directly to the real world (you can't have i apples), you can map the real world into complex numbers, manipulate them, then extract meaningful results.

I remember one instance where you introduce complex numbers to make an integration easier. I don't recall the specifics, but it was something to do with turning a trig function into the real part of a complex exponential function. I think. Like:

SinA = real[SinA + iCosA] = real[e^(iA)]

[Alex K]

yup, except the i is with the Sin though and Cos is the real part as you can verify if you plug in 0 into the exponential

I simply think of complex numbers as *the* way to do calculations with pairs of real numbers. The laws we use for them is basically the only way they can be if we want the usual rules of adding and multiplications to apply. I find that that makes them appear much less mysterious.

One mystery is that complex numbers seem to be intricately woven into quantum physics. Quantum amplitudes and wave functions are always complex, there doesn't seem to be a way around it...

[robvalue]

Ah yeah of course.

Wow, how strange! I wish I still had the brain power to learn such complex things. Heh. Pun not intended.

[mihoda]

1=sqrt(1)=sqrt(-1 * -1)=sqrt(-1)*sqrt(-1)= i*i=-1

The error is in the first equality.
sqrt(1) = +-1 

When you later split the factors you are explicitly forcing one branch of the solution by again, reducing a choice:
sqrt(-1) = +-i

So in full:

+-1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = +-i * +-i = +-1

[GrandizerII]

1=sqrt(1)=sqrt(-1 * -1)=sqrt(-1)*sqrt(-1)= i*i=-1

The error is in the first equality.
sqrt(1) = +-1 

When you later split the factors you are explicitly forcing one branch of the solution by again, reducing a choice:
sqrt(-1) = +-i

So in full:

+-1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) =  +-i * +-i = +-1

Square root of a number, by convention, is always positive.

[Kernel Sohcahtoa]

yup, except the i is with the Sin though and Cos is the real part as you can verify if you plug in 0 into the exponential

I simply think of complex numbers as *the* way to do calculations with pairs of real numbers. The laws we use for them is basically the only way they can be if we want the usual rules of adding and multiplications to apply. I find that that makes them appear much less mysterious.

One mystery is that complex numbers seem to be intricately woven into quantum physics. Quantum amplitudes and wave functions are always complex, there doesn't seem to be a way around it...

I must say that when I learned about complex numbers from Trigonometry, 7th edition by Charles P. McKeague and Mark D. Turner, I was very fascinated with their coverage of them.  In particular, I enjoyed the following areas: trigonometric form for complex numbers; products and quotients in trigonometric form (I loved De Moivre's Theorem); roots of a complex number (this was awesome).  Hence, IMO, I found complex numbers to be quite wonderful.