imaginary number function that passes through certain points I am trying to figure out how to create a function that accepts as input an imaginary number, and outputs a real number between $0$ and $1$.
Specifically, The function that satisfies
$$
f\left ( a+bi \right ) =  c + 0i
$$where$$
f\left ( \frac{1}{2} + \frac{\sqrt3}{2}\;i \right ) = 1
$$and$$
f\left ( 1 + 0i \right ) = 0
$$
I do not have any real experience with imaginary numbers, and would appreciate even a tip on how to start approaching this.
 A: From wolfram: "A function is a relation that uniquely associates members of one set with members of another set."
All that means is you wish to have a relation between $\mathbb C$ the set of all complex numbers to the real interval $[0,1]$.  (BTW, you meant "complex"; not imaginary. a complex number is of the form $a + bi$ which is a combination of real and imaginary.  An imaginary number of the form $0 + bi$ which is "purely" imaginary.  A real number is of the form $a + i0$ which is "purely" real.)
You want $f(1) = f(1 + i0) = 0$ and you want $f(\frac 12 + i\frac {\sqrt 3}2) = 1$.
The remaining values can be anything.  You could say: $f$ is the function so that $f(1) =0$ and $f(\frac 12 + i\frac {\sqrt 3}2) = 1$ and $f(z) = 0$ for all other $z$.  That is perfectly acceptable and a legitimate and well-defined function.
Or we could have $f(0+ 0i) = .25$; $f(a+bi) = 0$ if $b=0;a ne 0$; $f(a+bi) = .75$ if $a=0; b\ne 0$; $f(a+bi) = 1$ if $a\ne 0; b\ne 0$. 
Or you can have anything you want.
I suspect you have more in mind than just that.  You say $f(a+bi ) = c$.  But what is the relationship between $a, b$ and $c$?
You chose the number $z=\frac 12 + i\frac {\sqrt{3}}2$ which is an interesting number because $z^2 = (\frac 12 + i\frac {\sqrt{3}}2)(\frac 12 + i\frac {\sqrt{3}}2) = \frac 14 + i\frac {\sqrt{3}}2 - \frac 34= -\frac 12 + i\frac {\sqrt{3}}2$.  And so $z^3 = (-\frac 12 + i\frac {\sqrt{3}}2)(\frac 12 + i\frac {\sqrt{3}}2)= (\frac {\sqrt{3}}2i)^2 - (\frac 12)^2 = -\frac 34 - \frac 14 = -1$.  So $z$ is one of the cube roots of $-1$.
I suspect that has something to do with what you want the function to do but I don't know what.
A: Given distinct complex numbers $z_1$ and $z_2$, the function $f:\mathbb C\to[0,1]$ given by $$f(z)=\left(\left|\frac{z-z_1}{z_2-z_1}\right|\vee0\right)\wedge1$$
does the job, where "$\vee$" is the "$\max$" operator and "$\wedge$" is the "$\min$" operator (here, you have $z_1=1$ and $z_2=\frac12+i\frac{\sqrt{3}}{2}$).
This function takes value in $[0,1]$ and has $f(z_1)=0$, $f(z_2)=1$.
Of course, there are many functions with the properties you desire.
A: Turns out that I was looking for was Isomorphism between Roots of Unity under Multiplication and Integers under Modulo Addition.  Thank you all for your input.
