The Easiest Way to Calculate Pi

Forget the complicated formulas! Here is a super-easy and fast way to calculate many digits of Pi.

Twitter: @TPointMath

Music:

Licensed under Creative Commons: By Attribution 3.0

[depenthene]

Even easier: press π button on your calculator.

_DISCLAIMER: I do understand the point of the video._

[Narbris]

For 9 it approaches 3*pi.

It will always bring you to the closest odd multiple of pi. This is because n+sin(n)=n is true any time sin(n)=0.
sin(n)=0 any time n is a multiple of pi.

It is only odd multiples because anything between sin(pi) and sin(2*pi) is negative, bringing you closer to the previous odd multiple of pi. So starting a 6 brings you down to pi and 7 brings you up to 3 pi.

[pupnoomann]

But I have to know how many digits are correct to evaluate more of them. I basically have to know pi to approximate pi.

[hoopshank]

Surely using your calculator to calculate sin (3) and then the sin of further decimal expansions is not avoiding those 'sophisticated mind-busting formulas', it's just getting your calculator to do them a bit at a time. As @debenthene says below, you might as well just press the 'pi' button.

Damn if only those early mathematicians working out an approximate value of pi had known this method.... and... er.... had a modern calculator.

[ChandravijayAgrawal]

if we want to calculate value of pi it means we don't know correct digits of pi then how could value of pie is calculated by adding sin(3)

[fablungo]

But this method requires a calculator, at which point I can click the pi button anyway? And as a method of calculating pi, computationally, if you expand the sin operation to the computation required to calculate it, it's just as complicated as the more "well known" methods.

[Drwurstable]

But how do I know how many digits are correct in each iteration?

[sufm]

Good video !
I learnt something new today thank you

[concepthub]

Sir, my question is how we can know the correct decimal places in pie in the first place ? I am confused.

[thejonymyster]

to (if im understanding it right) restate Chandravijay Agrawal's point, how are we supposed to know which ones are the correct digits without already having calculated pi up to that point? like lets say i only know that pi is about 3. how would i know that 3.141 are the correct digits of pi? or are calculating and finding pi two different things? no offense

[imagineexistance]

Its because sin(x) is e^ix-e^-ix/2i
If x is a multiple of pi, it would be -m+m/2i
So its zero, and adding up to that is the value of radians decreases as it converges to 0 and so it slowly gets closer to pi, if you always want pi, try x+sin(x)*(x/2)

[pockx]

You do understand that the sine function uses pi right?

[DallyLama]

wait so how do i know which last digit is the correct one?

[wobbly]

Why for 9 it aproaches to 9, 42... (3pi)

First lets answer, why does it aproach pi when starting with 3? At the point x=pi => sinx=0 if you zoom a lot it seems like a line, that in fact has slope -1. You can see this if you take its derivative (cosx), which around x=pi is very close to -1 (and exactly -1 at x=pi).

So, we have an aproximation of a line with the following properties: f(pi)=0, slope=-1. This comes out to be f(x)=pi-x. That means, that when you do x+sinx when x is near to pi, sinx≈pi-x, so x+sinx≈x+pi-x=pi. When you repeat that with the new aproximation, because you are near to the point x=pi, the aproximation of sinx≈pi-x is even better, so you get a better result than before.

Now, it's easy to see what happens to 9, as it's near to the point x=3pi, which also has a slope of -1, so the new linear aproximation near this point is g(x)=3pi-x. With the same thought process as before, we can see it'll approach 3pi.

[ffggddss]

On my 10-digit calculator, it took all of *2* iterations to get all 10 digits' accuracy!

Of course, it's no good as a practical way to compute π, because if you're doing it by hand, you then have to compute the sine of numbers that are far from 0, so that Taylor's series won't help reduce your computational burden;
while if you're using a scientific calculator that has a sine function, but for some reason, no "π" key, it's quicker (immediate, in fact!) to just take cos⁻¹(-1).

Nevertheless, there's something irresistibly "cute" about this.
Thanks for posting it!

[guillermo]

I was playing around with angles and distances on geogebra and discovered an easy way to calculate pi, not as easy as the method displayed on this video of course, but easy indeed,

Basically this is the equation i stumbled upon

α: Some angle between 0º and 180º


Lim cos( (90º - α/2) ) * (360º/α) = π
α→0

So you basically substitute greek letter α for a very narrow/ small angle close to 0, like 0.5 or 0.05 and you solve and it should be pretty easy.

[lukedope]

how about do it one time and then do ans+sin ans and spam the = button like idk 2 times

[Gawaboumgah]

It's theory of the fixed-points ?

[JoelGarcia-mljx]

What if my Sin function is of low precision?

[CANC]

Wait. But how do we know what number of initial digits to choose for successive sine computations?