All this talk of imaginary units and finite numbers, I have to ask something I've never understood: how are imaginary numbers useful in the real world? Are they actually used in the equations for calculating the position the wizard magnets need to be on Cern's LHC to create evaporating black holes? Do engineers use i to solve real mechanical problems at any given point? Do accountants use i to calculate the maximum profit point on a graph of price*volume sold vs. total profit or something?
It has a use in electronics with alternating current and capacitors and stuff. I'm not entirely sure what, though - I could explain it after digging in a textbook for a bit, but that sounds like it'd be a rather terrible experience for all involved 
As for most other things, I've yet to run into someone using that definition of i for practical purposes. Of course, I'm still only halfway through my bachelor's degree, so I'm probably not the most credible source on such things.
Also quaternions. Just about every rotation in 3 dimensional space can be done faster/easier with quaternions, which are themselves 4 dimensions consisting of the normal numbers and imaginary number dimensions i, j and k.
Imaginary numbers also feature heavily in anything related to wave phenomena: A wave consists of a feedback system between 2 elements, be it pressure and velocity (sound), electricity and magnetism (light, electrical components), or similar. Even at the very forefront of theoretical physics, the concept of 'imaginary time' has come up, as is mentioned by Hawking in some of his books, as it becoems an incredibly useful tool for examining otherwise unexaminable phenomena (like the event horizons of black holes and The Big Bang itself). These typically behave like they are 2 dimensions, both perpendicular to one another. Through systems similar to those in quaternions, you can represent those with imaginary numbers in order to treat numbers geometrically.
Imaginary numbers are perpendicular to real numbers, as well as perpendicular to any additionally defined sets of imaginary numbers. And this relationship shares the qualities of geometric perpendicularity which is immensely useful in a huge varieties of things.
So yeah, your modern life kinda depends on them.
So, think through the math involved in that image. A multiplication of complex numbers is exactly the equivalent of performing a rotation in space. A simple multiplication, and you've suddenly removed the need for any sort of complex sin/cos/tan operations (which for computers are uber-slow, and for humans are just annoying). Multiplying any unit-vector of these complex numbers gives you the equivalent of rotating one by an amount equal to the second.
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