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Happy Pi Day, Crazies!
Well, at least if you live in one of these countries.
Anyway, I was thinking recently:
Does the number pi actually exist?
Let’s find out!
Of course, we can’t really answer this without a pinch of definition.
The number pi is defined as the ratio of a circle’s circumference to its diameter.
Yes, there are bunch of different ways to calculate it, but as 3blue1brown would remind us:
All of them are somehow related to circles.
So, if we’re going to ask if the number pi exists,
it might be helpful to first ask if a circle exists.
A circle is the set of all points equidistant from the same point.
But that’s just math. It’s abstract stuff.
Can we actually draw a shape with that property?
Well, let’s give it a shot.
Find a piece of paper and a compass.
The ends on the compass have a consistent distance, so it should be perfect.
I can put it on my paper, twirl it around, and
BAM! I’ve got a circle.
Or do I?
Let us take a closer look.
You can quickly see places I was a bit wobbly with the compass.
Sometimes it’s so bad my curve doesn’t even reconnect with itself.
But, even if I didn’t suck at drawing, I’m still limited by the width of the line,
the graphite material in the pencil, and imperfections in the paper.
So, let’s try something manufactured a little better like this pan.
If I measure the diameter all the way around, it seems consistent,
at least to the precision of my measuring device.
Which brings us to our first problem with physical objects.
If you measure the circumference and the diameter and then divide those two numbers,
you’ll get something close to pi, but not exactly pi.
You can’t know for sure that it’s a perfect circle.
But what if it is a perfect circle? Then does pi exist?
Eh, not so fast.
This brings us to our second problem with physical objects.
Even if this pan is as round as physically possible, it’s still not a perfect circle.
Super Zoom!
If you get close enough, you can see atoms.
There are sides to this shape.
They’re just really tiny ones and curved ones at that.
Also, we’re now zoomed in so much that size and shape start to lose meaning.
Even this digital circle I showed you earlier is made of pixels on your screen.
If you put your eye ball right up against the screen, you’d see something like this.
That’s not a smooth edge.
So I don’t know if this whole circle thing is going to pan out.
Anyway, all this talk about circles reminds me of this experiment
that Physics Girl did a while back.
She and Veritasium threw a bunch of darts at a board with circles drawn on it.
The ratio of darts that landed inside the circles compared to the total should be proportional to pi.
Eh, roughly.
You know what?
I’m kind of in the mood to check the limits of that experiment.
Computer simulation time!
Here is a board hit with 1000 darts, 10 thousand darts, 100 thousand, and a million.
I ran each simulation 100 times and there’s a definite decrease in the average error
the more darts you have.
Unfortunately, sometimes randomness comes in lumps.
If I ran this simulation enough times, there would eventually be one
where all the darts landed inside the circle
and I’d get a little over 27% error.
The point is, there will always be some error with this experiment.
That being said, Dianna and Derek got really great results. You should go check it out.
Hold up. I think I hear comments incoming.
All computers do is work with numbers,
so your simulation is assuming pi exist in order to prove pi exists.
Check. Mate.
Let’s make a few things clear.
One!
I haven’t come to any conclusions, so you shouldn’t either.
Two!
Oh, you want to play the pedantic game?
You’re going up against the master!
Technically speaking, computers don’t work with numbers.
In a computer, there are a bunch of transistors controlling the flow of electricity.
The electricity is either flowing or it isn’t.
That’s the reality.
We simply interpret a flow as a 1 and a lack of flow as a 0,
because it allows us to interpret complex flows of electricity as binary math.
There's no actual numbers there. It’s just electricity.
But, you know what, let’s pretend they are numbers for a minute.
Even if we do that, pi is not one of those numbers.
A typical processor these days is 64-bit, which means each register is 64 bits long.
Each bit can either be a zero or a one, so we’re working with binary math.
Unfortunately, that doesn’t mean we get up to 64 binary digits.
We’d like to work with negative numbers as well as positive ones.
Also, like most numbers, pi has decimal places.
Remember, it looks like this in decimal form.
There’s a decimal point.
In binary, it looks like this.
There’s still a point, but this is binary, so we call it a binary point instead.
A computer needs to handle numbers like this, so we let the point float around the 64-bit register.
That’s why it’s called a floating point number.
OK, so we sacrifice one bit is for the sign, then 11 more bits for an exponent.
The actual digits of the number only get the remaining 52 bits.
In binary scientific notation, a number looks like this.
So, the best a computer can do with pi is this:
Converting that only gets us 18 decimal places.
What if it’s, like, a super mega advanced processor?
There will still be a limit.
They do make highly specialized 256-bit processors, which means more digits,
but there will always be a finite number of them.
The number pi has an infinite number of digits.
Computers can only ever store an approximation of pi, not pi itself.
We don’t call it an irrational number for nothing.
I mean, there is no discernible pattern whatsoever in those digits.
That’s crazy, right?!
Actually, this might be a better way to handle our original question.
Does the number pi actually exist?
Let’s imagine that all numbers are on this hypothetical line.
It’s got all the natural numbers: 1, 2, 3, 4, 5, etc.
Zero is on there too as well as all the negative integers.
It’s also pretty easy to imagine rational numbers.
You just make ratios of the integers:
1/2, -33/4, 10/3, -1/12,
each with a corresponding decimal version that either ends or repeats.
Irrational numbers are bit more difficult.
More ratios just gives you more rational numbers.
But, let’s say you’ve got a triangle with two equal sides.
We’ll give them a length of one.
That’s no big deal. One is an integer.
We’re pretty comfortable with that.
Unfortunately, the length of the remaining side is root two.
Root two is irrational.
You can’t write it as a ratio of integers.
Its decimal places go on forever with no pattern.
There are lots of other root examples, but pi is even weirder.
It’s transcendental, meaning it isn’t the solution to any algebraic equation
with rational coefficients.
It transcends algebra.
But numbers like root two and pi have to be on the number line if it’s going to be continuous.
Without them, there would be gaps.
For that reason, we call this line the real number line
because it contains all real numbers.
So the real question here is this:
Do space and time work like the number line? Is spacetime continuous?
Well, general relativity would have you think so.
It models space as being like the real number line in all 3 dimensions
and also models time the same way.
But this fact about general relativity is what makes it
fundamentally inconsistent with quantum mechanics.
For gravity to work like the other interactions in quantum mechanics,
space and time must be discrete.
They must come in little pieces.
That’s the exact opposite of continuous.
In a model like that, having gaps in the number line would be perfectly acceptable.
It might be that no irrational numbers exist, let alone pi.
Unfortunately, we don’t have a model like this yet, at least one that works.
Until then, we won’t a definitive answer.
So what do you think?
Are space and time continuous or discrete?
Does pi exist?
Duke it out in the comments.
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And until next time, remember, it’s OK to be a little crazy.
JohnFx was wondering: What are black holes spinning relative to?
Right, motion is relative, so you have to assign the coordinates to someone.
In the case of black holes, that’s usually a distant inertial observer.
It’s not always explicitly stated, but it is the standard.
Anyway, thanks for watching!