Complex Number Kookiness 5

(X + Xi)^-1 = 1/(2X) - 1/(2X)i

AND

(X - Xi)^-1 = 1/(2X) + 1/(2X)i

Notice how the plus & minus signs switch places in respect to the equal sign!

Observe the table below:

Complex Number A	Complex Number A^-1	Complex Number B	Complex Number B^-1
1 + i	1/2 - 1/2i	1 - i	1/2 + 1/2i
2 + 2i	1/4 - 1/4i	2 - 2i	1/4 + 1/4i
3 + 3i	1/6 - 1/6i	3 - 3i	1/6 + 1/6i
4 + 4i	1/8 - 1/8i	4 - 4i	1/8 + 1/8i
5 + 5i	1/10 - 1/10i	5 - 5i	1/10 + 1/10i
6 + 6i	1/12 - 1/12i	6 - 6i	1/12 + 1/12i
7 + 7i	1/14 - 1/14i	7 - 7i	1/14 + 1/14i

This trick will still work even if X is a non-integer!

(½ + ½i)^-1 = 1 - i

(½ - ½i)^-1 = 1 + i

(1/3 + 1/3i)^-1 = 1½ - 1½i

(1/3 - 1/3i)^-1 = 1½ + 1½i

(√(2) + √(2)i)^-1 = 1/(2√(2)) - 1/(2√(2))i

(√(2) - √(2)i)^-1 = 1/(2√(2)) + 1/(2√(2))i

(π + πi)^-1 = 1/(2π) - 1/(2π)i

(π - πi)^-1 = 1/(2π) + 1/(2π)i

Just a warning: the variable X must be a positive number; otherwise, the plus or minus sign to the right of the equal sign WON'T change & you might add another minus sign to the real part of the dependent complex number, not to mention that addition & subtraction work differently when you involve negative numbers! Also, X can't be equal to zero(0) or you'll get a "division by zero" error on your calculator! Zero is the only number that has no other number as a reciprocal; it's infinity(∞), which ISN'T a number at all!

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© Derek Cumberbatch