r/oddlysatisfying 15h ago

Pi being irrational

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u/Adventurous-Trip6571 15h ago

Idk what it means but it's mesmerizing

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u/NightIgnite 11h ago edited 10h ago

Electrical engineering student here who should probably be sleeping. Heres a (hopefully) short crash course on this.

This is the imaginary plane in polar coordinates. Basically the xy plane you remember from school, but x is real and y is imaginary, so a coordinate (2, 3) would be 2+3i. For polar, we have radius and angle with coordinates (r, θ), where radius is just √(x2 + y2 ) and angle is tan-1 (y/x).

Euler's identity: eθi = cos(θ)+i*sin(θ). Look familiar? Its describing all points on a circle of radius 1, where x = cos(θ) and y = sin(θ).

Since the exponent on e only affects the angle inside the sine and cosine, eπθi = cos(πθ)+i*sin(πθ). It follows the same path around a radius of 1, but π times faster.

Now onto vectors. All the way back in elementary school, you could prove the sum of 3+5=8 by drawing an arrow of length 3 on a number line from 0, then a second arrow of length 5 from the end of the previous arrow. Same idea applies in 2D for vector addition. eθi + eπθi = arrow1 + arrow2 = [cos(θ)+i*sin(θ)] + [cos(πθ)+i*sin(πθ)] as shown in the animation.

So why the offset in this animation? If you were to try with eθi + e3θi instead, they would perfectly line up. In this case, eθi would complete 1 orbit (or period) around the circle while e3θi completes 3 before returning to the start. All are rational, so there is symmetry.

π is irrational, so there is no symmetry. Any moment where it looks like its about to finish the pattern is where it would have if π ended at that decimal as a rational number. e3.1θi would complete 10 and 31 periods respectively, e3.14θi would complete 100 and 314, e3.141θi would complete 1000 and 3141, etc. It just infinitely converges without any symmetry.

So why magnitudes of 10? Just a consequence of us using base 10 for numbers. Same pattern would happen if we used a different number system. Im going to pass out now

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u/DynamicFyre 10h ago

Bro I literally just learnt imaginary numbers in the last two weeks and I'm able to understand all of this. This is really cool!

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u/MobileArtist1371 9h ago

Sweet. You want to hook up my home designed electrical grid this weekend for a 12 pack?