# Complex Number Kookiness 4

## (a + b*i*)(b + a*i*)= c^{2}*i*

## c = The absolute value of both complex numbers (They both have the exact same absolute value!)

__Examples:__

(7 + 8*i*)(8 + 7*i*) = 113*i*

(√(113) = the absolute value of both of these complex numbers!)

(3 - 2*i*)(-2 + 3*i*) = 13*i*

(√(13) = the absolute value of both of these complex numbers!)

(-3 - 2*i*)(-2 - 3*i*) = 13*i*

(√(13) = the absolute value of both of these complex numbers!)

(3 + 4*i*)(4 + 3*i*) = 25*i*

(5 = the absolute value of both of these complex numbers!)

(1 + *i*)(1 + *i*) = (1 + *i*)^{2}= 2*i*

(√(2) = the absolute value of both of these complex numbers! Wait a minute, they're the exact same complex number!)

Whenever you multiply 2 complex numbers with the real & imaginary values swapped, their product will always be equal to the square of their shared absolute value multiplied by the imaginary unit *i*, even if either the value of *a* or *b* is *negative!*

If *a* = *b*, then the absolute value of a complex number will be equal to a multiple of *the square root of *2.

(5 + 5*i*)(5 + 5*i*) = (5 + 5*i*)^{2}= 50*i*

(5√(2) = the absolute value of both of these complex numbers! Wait a minute, they're the exact same complex number!)

(5 - 5*i*)(-5 + 5*i*) = 50*i*

(5√(2) = the absolute value of both of these complex numbers!)

(-5 - 5*i*)(-5 - 5*i*) = (-5 - 5*i*)^{2}= 50*i*

(5√(2) = the absolute value of both of these complex numbers! Wait a minute, they're the exact same complex number!)

## By the way *the square root of* 50 = 5 *square roots of* 2!

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© Derek Cumberbatch