Friscolating Dusk Light
Night landscape for Chris and Katie Aldridge. 36x36
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.
The Book of Proof
A very bright, young intern at my office named Zac turned me onto this book. It’s free in .pdf format for all to enjoy. So far it seems to be helping me speak the language of math more. Whenever I have tried to go about “proving” ideas mathematically in the past, I get stumped on the taxonomy of math and wonder exactly how much I have to declare as assumed before I can proceed to “prove” whatever I’m postulating. This is a canonization of mathematical “proof” theory.
Find it here.