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The Necessity of Imaginary Numbers

Today’s mathy thing is about simple factoring.  When you take algebra in high school they tell you about how a² – b² can be factored to equal (a + b)(a – b), but that a² + b² is prime and cannot be factored.  This is a blatant fallacy.  It just requires that imaginary numbers be used.

if

i = √-1

then

i² = -1 right?

because √-1 * √-1 = -1 in the same way that √4 * √4 = 4

So if i² = -1 then what if we factored a² + b² to equal (a – bi)(a + bi)

(a – bi)(a + bi) = a² + abi – abi – b²i²

= a² – b²i²

= a² – b²(-1) because i² = -1, remember?

= a² + b²

The point is that imaginary numbers are falsely eponymous.  They are real, they’re just not Real. They can be measured electrically, in the same way that we use the real number set to measure other phenomena.  This fact seems to suggest that the imaginary number set can describe natural phenomena just like the Real number set can.  In the same way that one can have 3 apples, one could have 3i something elses. hahahaha.  Confused?  Me too.

Either way the the math itself requires that imaginary numbers exist to create the symmetry of being able to factor both a² – b² as well as a² + b² equitably.

 

07. September 2011 by Kiv
Categories: Math | Tags: complex numbers, factoring, imaginary numbers, math | 3 comments