Naked Science Forum
General Science => General Science => Topic started by: chiralSPO on 11/02/2018 21:54:21

The familiar equation for the unit circle centered at the origin of the (x,y) coordinate system is:
x^{2} + y^{2} = 1
Using only real numbers for both x and y only yields useful solutions for –1 ≤ x ≤ 1 and –1 ≤ y ≤ 1. But if we allow for either x or y to have nonzero imaginary components (x or y can be imaginary or complex), then the solution is no longer just a unit circle, but an odd 2dimensional surface in 4dimensional space (x, a, y, b) such that (x + a×i)^{2} + (y + b×i)^{2} = 1.
Try as I might, I am having great difficulties trying to visualize this surface. I can generate a few different 3d slices, like a = 0 for which all of the points have only real components of the x coordinate. This slice contains a unit circle in the x,y plane, and a hyperbola in the x,b plane (x^{2} – b^{2} = 1). Similarly for the b = 0 slice we also see the same unit circle and a hyperbola in the y,a plane (y^{2} – a^{2} = 1).
Which other slices would be useful for me to try to graph to visualize the rest of this shape? (I am tempted to look at x=y or just something like several parallel planes using integer values of a or b.

suggestions? is there an obvious name for this shape?

The function for the unit circle is a difficult one. It is a bit late here, just going to bed, but I will give this some thought tomorrow.

I thought that there was a principle in mathematics that a quadratic equation (one with x^{2}) had 2 zeros (or roots), some of which might be imaginary.
For x^{2} + y^{2} = 1, there are two real roots, at (1,0) and (+1, 0).
But I'm not sure how this applies to a formula where there is a y^{2} as well... :\

I believe Mathematica shows plots of such functions free trial versions are available.

Thinking about it some more...
x^{2} + y^{2} = 1 can also be written as: y = SQRT(1  x^{2})
The part inside the square root is a parabola z = 1  x^{2} = (1x)(1+x)
...which has two zeroes, at x = +1 and x= 1.
When you take the square root of a number, SQRT(0) = (0), so these two zeroes remain.
 When you take the square root of a positive number, the answer is real (both positive and negative), so no new zeroes appear.
 When you take the square root of a negative number, the answer is imaginary (both positive and negative), so no new zeroes appear.
 When you take the square root of an imaginary number (apart from the origin): I have a vague recollection that you can work it out in polar coordinates: The radius changes (but not to 0), and there is a rotation about the origin. If correct, this would mean that no new zeroes appear.
So I suggest that this equation has just two zeroes, for all values of x in the imaginary plane. In the case givem both zeroes occur for real x.
PS: I heard about an ancient Persian mathematician who worked out how to solve quadratic equations.
If I recall correctly, he had to represent 6 classes of quadratic equations, because he couldn't grasp the notion that "something" could equal "nothing".
While today we often represent the solution of all these quadratic equations as "=0".
See: http://www.bbc.co.uk/programmes/b006qykl/topics/Medieval_Persian_mathematicians

The equations are: y = SQRT(1  x^2), iy = SQRT(x^2  1), x = SQRT(1  y^2) and ix = SQRT(y^2  1). I may be wrong. Please correct if so.

When you take the square root of an imaginary number
I worked it out overnight...
If you take a number on the complex plane, you can represent it in polar coordinates as (r, θ).
The two square roots of (r,θ) lie on a circle of radius SQRT(r), centered on the origin.
One root lies on the circle at angle θ/2, and the other lies 180° around the circle.
And the only way the square root of a number can be zero is if the original number has r=0 (ie the origin).
So the square root function does not produce any more zeroes in the formula of a circle, when calculated on the complex plane.
...but when I looked it up in Wikipedia, this method was already described (well, in the meantime it gave a few grey cells a workout :) )
See: https://en.wikipedia.org/wiki/Square_root#Principal_square_root_of_a_complex_number