Complex Number Kookiness 2: Adding Integers to the Imaginary Unit!

Observe the table below. Can you find something interesting in any 1 of the 3 columns?

Complex Numbers	Absolute Values of Complex Numbers Squared	Differences
0+i	1	1
1+i	2	3
2+i	5	5
3+i	10	7
4+i	17	9
5+i	26	11
6+i	37	13
7+i	50

In the left column, consecutive integers are added to the imaginary unit i. In the middle column, the absolute values of the complex numbers are squared. In the right column are the differences of the squares of those absolute values...and have you noticed that they're consecutive odd numbers? Cool, isn't it?

The trick still works if you include negative integers, but...

Complex Numbers	Absolute Values of Complex Numbers Squared	Differences
-3+i	10	-5
-2+i	5	-3
-1+i	2	-1
0+i	1	1
1+i	2	3
2+i	5	5
3+i	10

You can get a palindrome in the middle column!

This time in the right column, we got negative differences in the top half. Of course 0+i = i, but I printed "0+i" anyway because zero(0) is an integer.

By the way, |a+bi| = |a-bi| = |-a+bi| = |-a-bi|; all 4 of those are exactly the same!(In absolute value) You can have the variables a & b swap places without changing the absolute value!

The absolute value of a complex number:

The absolute value of a + bi = the square root of the sum of a^2 and b^2

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