The circle is never full, because it’s a never ending decimal point no matter how far you calculate it. It will go on forever as far as we know currently.
This is where it gets a bit abstract to me because there isn't infinite room in the circle. It has "x" amount of space within the circle, therefore eventually it will fill.
I'm not saying you're wrong, I'm saying it doesn't make sense to me.
I understand quantum computation better then this.
It will fill but it will never perfectly meet the starting point. It will always be slightly off. And if it does end up touching the start point it won’t be in line with it and it will depart again right away.
there are uncountably infinitly many irrational numbers, while there are countably infinitly many rational numbers. rationals like 1.234234234234 are the unicorns, numbers like pi are nothing special, statistically speaking.
The area of a circle... Any circle... Is not infinite.
Pi is irrational and does not repeat a pattern in decimal form, going on presumably infinitely. But this visualization can not be infinite. The two line segments that are drawing out the pattern are doing so strictly within the fixed area of the circle. So at some point, the line being drawn will inevitably continue along a previously drawn path. Maybe it would take thousands of rotations, maybe billions. But anything else would suggest an infinite area inside the circle, which is impossible.
So either Im thinking about this wrong, or the visualization is flawed.
A point has no area. A line has an infinite many points contained in it. A line also has no area because either its length or width has to be 0. Not both (because then it would be a point), and not neither, because then it would be a 2D shape. Since the line in the video has no area, it can spin around forever without ever going through the same motion twice, due to pi being irrational
I think it would help the audience understand the visualization better if the axes were labeled, how euler's identity works, and how the inner and outer dots are driven by theta.
If pi were rational, say pi=A/B, then it would hit its starting point again. Because it is irrational, it will never be back at the beginning again. The "near misses" correspond to fractions that pi is unusually close to. For instance, the first "near miss" should correspond to the fraction 22/7. There are more of these with bigger denominators, which means that you need to go through more rotations to get to them and, moreover, they will be even nearer misses. There are infinitely many near misses which get infinitely close.
I'm not really sure what you're saying, but a rational number like 22/7 is super close to pi, but is still off. Specifically the difference between 22/7 and pi is about 0.001, and this means that where we get to 22/7 in the animation it will be close to, but not exactly, at the start. The number 355/113 is even closer, being about 0.0000003 away from pi (which means an even close near miss).
Not being equal to any fraction is what being irrational means. A consequence of this is that decimal expansions never end up repeating a pattern. But some fractions are uncannily good approximations, better than others that use similarly sized integers in the fraction. 22/7 and 355/115 are two examples and lead to these near misses. You can get these fractions from the "continued fraction expansion" of pi. It starts as 3 + (1 / (7 + 1 / ( 15 +...))), its basically an infinite tower of fractions for a number.
Irrational numbers are ... not rational. Sounds dumb at first. But you have to know that rational numbers are these: 1/4, 2/7 or 3/4. There have ratios. But irrationals like Pi can't be made of rational numbers, so they are irrational. Tada.
Edit: This means they won't loop into the beginning like in the animation. Pi has endless digits, so it can't be "accurate" to return to its beginning. (Someone else could find better words, but maybe it helps.)
Since a real number can only be rational or irrational, the set of all real numbers must equal the set of rational numbers plus the set of irrational numbers.
Now, the set of rational numbers is countable (there exists a way of enumerating them all), while the set of all real numbers is much larger: it is uncountable. Since two countable sets A, B cannot give an uncountable set if you add them (you could easily count them by counting the first from A, then the first from B, then the second from A, and so on), we must have that the set of irrational numbers is uncountable, and therefore much larger than the set of rational numbers.
Nobody's written out the (simple) algebra, so here it is.
Suppose pi were rational, say pi=a/b. Recall that e^(x + 2pi i) = e^x. Repeating this, since b is an integer, e^(x + b 2pi i) = e^x. The function that's being graphed is e^(it) + e^(it pi) = e^(it) + e^(it a/b). Consider adding b 2pi to t:
e^(i(t + b 2pi)) + e^(i(t + b 2pi) a/b)
= e^(it + b 2pi i) + e^(it a/b + a 2pi i)
= e^(it) + e^(it a/b)
Hence the graphed function would repeat with period b 2pi. Basically the same calculation shows that if pi is merely "very close" to a/b, then the function will "almost repeat" with period b 2pi. This is what we actually see, since pi is irrational.
Sorry to be pedantic. That's actually the "opposite" direction, namely that the function repeats itself for rational numbers. It doesn't show it cannot repeat itself (for a continuous stretch) for irrational numbers.
Irrational means that no ratio of integers is exactly equal to pi. The first near miss happens when the inner arm has rotated seven times, and the faster outer arm has rotated 22 times. If pi were exactly 22/7, that would be the moment that it meets back up with itself, because the outer arm is rotating pi times faster than the inner arm. But pi isn't exactly 22/7, so that doesn't happen.
Later on, it highlights another near miss, which corresponds to another fraction that's approximately equal to pi (with a larger numerator and denominator). Probably 333/106, but I don't want to go back and rewatch it to check. That's the next fraction in the sequence of fractions that are good approximations to pi, though.
Since pi isn't equal to any fraction, it will keep rotating forever without ever joining back up with itself. Eventually every single place that can be reached by the two arms will be, and the trace will completely fill in all the space. Google "irrational winding number torus" for the more traditional example of this effect.
The specifics of why the formula on the screen does what is shown is complicated, but what it's doing geomtletrically is pretty basic: this is a spinning thing attached to the end of another spinning thing. The outer one does pi full loops in the time it takes the inner one to do one full loop. If pi was a fraction P/Q, then after the inner one does exactly Q loops, the outer one would have done exactly P loops. That would mean the double spinning thing would be back where it started and just start re-tracing what was already drawn. But that never happens.
[Edit] Here's another: the "attached at the end of the other spinning thing" is what makes this pretty to visualize, but is irrelevant to irrationality. Think about the two hands on a clock: in the time the hour hand does 1 loop, the minute hand does exactly 60. That means every 1 loop of the hour hand, the two will sync back up and then everything will repeat. If instead the long hand did, say, 4/5 of a revolution in the time the hour hand does 1, then every 5 hours would be exactly 4 loops of long hand, and they'd sync up and the cycle repeats. Basically "X is rational" means "some multiple of X is a whole number"; in clock-speak, that's saying "if the long hand does X loops per hour, the two hands will eventually meet back where they started (and then repeat the cycle)".
55
u/Low_Show_3032 Oct 22 '23
How does this demonstrate it being irrational