Brother Of Jerry
Date: March 15, 2019 07:37PM
In digital signal processing, complex numbers are the very life blood of the field. Neither voice recognition, nor cell phones, nor MP3s would be possible without them.
Correction from upstream post by LW: it's not irrational numbers that you square twice to get back to Kansas. It is i (= SQRT(-1) ) that you square twice to get back to Kansas. The reason that works is that in the complex plane, multiplying a number by i rotates it 90 degrees (pi/2 radians would be the preferred nomenclature, but hey). So squaring i rotates by 90 degrees twice, reversing the direction of a number. That is the technical reason why multiplying a positive number by a negative number gives a negative result. The "direction" of the original positive number is twisted by 180 degrees in the complex plane, reversing its direction. It is also why a negative times a negative yields a positive result. Again, the "direction" of the original negative number gets turned 180 degrees in the complex plane, from the negative X axis to the positive X axis.
But back to squaring i twice: that is equivalent to multiplying a value by i four times, which is four 90 degree rotations, and you are right back where you started, after a little spin around the block.
A formula that codifies all this, and BTW impressed the hell out of Richard Feynman when he was 15 years old, is e^(i*pi) + 1 = 0, or alternatively, e^(i*pi) = -1. That combines pretty much all of the big guns of mathematics into a single identity: e, i, pi, 0, and 1, as well as the operations of addition, multiplication, and exponentiation. That is kind of jaw-dropping. (See: https://en.wikipedia.org/wiki/Euler%27s_identity
There are some irrational numbers that become rational when you square them once - the square root of 2 being a simple example. However, in general, for most irrational numbers, squaring them just gives you another irrational number. pi squared is irrational. I assume pi squared squared is irrational. So, in general, squaring an irrational number twice does not get you back to Kansas. Points for a clever analogy, though, even if misapplied. :)
Yes, nerdy. but it was handy for getting a paycheck at one point in life.
Edited 1 time(s). Last edit at 03/15/2019 07:44PM by Brother Of Jerry.