Delicious Differences

Here's a function to consider for this math trick:

Y = X + √(X)

Let X be equal to an integer that's a perfect square. If you list consecutive perfect squares in a table like this, then...

X	Y	Differences Between Y Values
0	0	2
1	2	4
4	6	6
9	12	8
16	20	10
25	30	12
36	42	14
49	56	16
64	72	18
81	90	20
100	110	22
121	132	24
144	156

The differences of the Y values are consecutive even numbers!

The next difference in the rightmost column would be 26 if I plugged the next perfect square into X!

In this next table, their additive inverses are plugged into X. Now watch what happens in the rightmost column:

X	Y	Differences Between Y Values
0	0	-1 + i
-1	-1 + i	-3 + i
-4	-4 + 2i	-5 + i
-9	-9 + 3i	-7 + i
-16	-16 + 4i	-9 + i
-25	-25 + 5i	-11 + i
-36	-36 + 6i	-13 + i
-49	-49 + 7i	-15 + i
-64	-64 + 8i	-17 + i
-81	-81 + 9i	-19 + i
-100	-100 + 10i	-21 + i
-121	-121 + 11i	-23 + i
-144	-144 + 12i

The Y values & their differences are complex numbers! Each Y value is the additive inverse of a perfect square plus its square root multiplied by the imaginary unit i! The differences are consecutive negative odd numbers & each one is added to the imaginary unit i! (And of course, 0 + 0i = 0.)

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© Derek Cumberbatch