Your new favorite pi approximation.
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Nah, mine is 355/113 forever. It has a pleasing 113355 pattern in it, and it's so crazy close that if you used it to calculate the circumference of the Earth, you'd be off by just 11 feet!
And the rational approximation to pi with error less than 10^-10 and smallest denominator is 312689 / 99532
Feet 💀💀💀
How close is the Earth to spherical, even?
Like we know there are mountains and trenches. We could use sea level to get something smoother, but we know the tides change, though I suppose not by 11 feet. But the error is probably on the order of several inches.
I am doing some guesswork here and I don't know where you would learn fully accurate science about it.
12:21 This was those rational numbers' moments. The sun was shining down on them in their moment of glory. You took that away from them. You monster.
Such compassion!
355/113 is one I worked out myself as being the best approximation that isn't too long and difficult to remember, so I really doubt I will ever have a pi approximation that I like more than that one.
I came to same conclusion long time ago like it better than the old 22/7 Well done.
And as far as I know, it's the only one whose accuracy (in d.p.) trumps the number of digits in its fractional representation!
pi=3. Good to
Yep. 3 is what I use 90% of the time. If I need better than that I use the pi key on my calculator.
And if you *do* need to be closer, without a calculator (for whatever reason), 3.1 is good to
@@mytriumphOfc 22/7 (n=1 integral) is more accurate, and 355/113 more accurate still!
When you said error
I think his maths degree should be revoked
pi being irrational is not trivial!
"his maths degree should be revoked"
(proceeds to forget the +C on some integral)
Strangely, the number in the thumbnail doesn't appear in the presentation.
I too was looking for that. He has a whole class of such approximate fractions outlined at the end, but it looks like that thumbnail fraction doesn't occur in any of them, either.
For the n=1 case, the bracketing fractions are in 4-digit terms, the approximation being just 1- and 2-digit numbers, arrived at by using common denominators on those bounding fractions. For the n=3 case, the common denominator will be pretty big. But does the average of the bounds reduce to what was in the thumbnail? I haven't worked that out.
To me, the interest is in the ability to generate ever-closer rational approximations to π with increasing n, albeit at enormous computational expense.
Fred
Maybe it's the fraction that drops out of the n=2 case?
I'm just gonna say it. 22/7 is overrated.
When you say half the interval size is the maximum error, that's not right. It would be right if the approximation was exactly half way between the ends, but that's not what you have. Pi could be close to one end of your interval, and the approximation at the other. It means the errors you calculate from your bounds should be twice as big.
Hmmm, integrating powers of a function that is less than 1 to make the integral small, and using the fixed denominator to guarantee an arctan term in the antiderivative. Moderately slick.
Very slick!
Thanks, this makes it clearer why the generalization is possible; since the polynomial in the numerator has an exponent which is a multiple of 4, presumably one can show that the remainder of the long division always works out to 4.
It's better than 4/1-4/3+4/5-...
It looks cool on paper, and is easy to derive with integrals, but takes 100 years to converge to something resembling π, I swear.
Yes, I like to point out that this, 4x the infinite alternating odd harmonic series, is both the most beautiful and the most computationally useless way to calculate π.
Fred
The average of subsequent partial sums of this series converges a lot faster though.
using the related 4arctan(1/5)-arctan(1/239) taylor expansion is quite efficient though... xP
@@jay_sensz Yes, that's a common method of convergence-acceleration. In this case it amounts to changing the final (& only the final) term in each partial sum to half its value.
This mars its "beauty" but improves its "performance." Even so, it's still pretty slow, compared with some other series.
One improvement is the Taylor series for 6sin⁻¹(½) = π.
And of course, even faster is the Machin formula given in @benhur2806's comment, which gives ¼π.
Then there are ridiculously fast, but more difficult-to-compute series, starting with one given by Ramanujan...
I used Euler's transformation of arctan(x) series and I have got
def mysum(n):
s = 0
p = 1
for k in range(n+1):
s += p
p *= (k+1)/(2*k+3)
return 2*s
It should set one correct bit per iteration
I love this, so simple but reasonably fast convergence.
So if Matt Parker wanted to do this for the next Pi Day and get up to 600 decimal places, how big would he have to make x? And while that is harder for a computer to do, would that be less error prone for a group of humans than the method he did this year, allowing for a faster solve time by the humans?
One of my favourites is sqrt(g), where g is the gravitational pull of the earth ~9,81
But did you know that it's not a coincidence? The initial definitions of the meter and second were related so that a pendulum of length 1 meter would have a period of 2 seconds, which gives g=pi^2 m/s^2.
This is so much cooler then using Continued fractions!!
I was thinking of taking those 2 fractions and seeing how the continued fraction of each looks like
Top tier approximation of Pi is:
Pi ~= 31^(1/3)
The accuracy of calculation is dependant on how many digits the number you're multiplying with.
I had never heard about these wheels. Thanks.
all this talk about rational no. caused me to check the lotteries ...
I guess the n=2 integral is "left... as an exercise for the viewer?" He did jump from n=1 (22/7 w/error < 1/2520) to n=3 (error less than 0.0000000001- yes, that's 10 d.p.).
But when I did the math for n=2, the approximation was "pi is 9637/3003 with absolute error of magnitude 853/5,250,960 or less."
22/7-pi is approx 0.012, which is about 1/84, so, the error is by far more than 1/2520
A moment of rationality of Mr. PENN
This identity is wonderful! What is its origin, and how was it derived?
My favorite approximation of pi is sqrt(2) + sqrt(3).
sqrt(10) isn't far off either
Ahhh yes replace the irrational number with a sum of two other irrational numbers,might as well use π/2+π/2 and get maximum precision shall we??
@@di-risoWhat about tau/2?
Good one! Not as crazy as the near inequality pi^4 + pi^5 = e^6 though!
At 11:30, you are mistaking "maximum error" with "round-off error." Any epsilon>0 from either endpoint leaves effectively the entire interval on the other side. The probable error is about 1/2e of this interval.
If epsilon is meant to be in the denominator, then this needs grouping symbols: 1/(2e).
@@robertveith6383 Not epsilon; e=2.718...
@@robertveith6383 Parentheses are implied; otherwise it would be e/2. Also, it is e=2.718... not epsilon.
The error is less than half the width if you average the upper and lower estimates, so I think you should have written out the average for good measure.
This would be better if it explained at the end why the generalization is possible. It's clear that values of n > 1 produce tighter bounds, but not why those bounds still include pi.
I guess, when you perform the Euclidean division of the numerator by (x^2+1), you end up with some 1/(x^2+1), which integrate to arctan, hence pi
The form of the polynomial long division always produces 4/(x^2+1) as the remainder, because all you're doing is multiplying all of the components of the equation by x^4 except the x^0 component.
Lim n -> oo [ 2^n * sqrt( 2 - sqrt( 2 + sqrt( 2 + sqrt ( 2 + ...)))) with n many total roots. Use a large natural number for n, and that should give a good approximation.
The best thing about the 22/7 approximation is that the final decimal point matches pi.
0:26 is the integral a good approximation of pi because of of just because it gives a good enough value?
The bit I don't get is the equation around 6:44. If it actually equals 22/7 - pi then the given integral doesn't approximate pi it approximates zero.
yes, the integeral would be _exactly_ zero had the value of pi been 22/7 the answer to the given equation basically gives us the value of pi. the vid tries to approximate the answer to 22/7 to find a good approximation for pi. you can of course increase the values of n like shown in the vid and try to find the another equation and approx pi (left as an exercise to you, if you need help try re-watching the vid)
@@yuvrajguglani821Point missed, I think. Your claim that "the answer to the given equation basically gives us the value of pi" is incorrect. At the timestamp given it approximates zero.
The question was actually essentially "how close to 0 is the integral?" The closer it was, the closer the Q in question would be to pi- in fact, said error would be about abs(22/7-pi).
@@wyattstevens8574 That is the point.
"So what I'd like to do today is work through all of the details evaluating this integral and seeing why it's a good approximation of pi". It isn't.
From this family of integrals, it’s a way to demonstrate that pi is irrational as these integrals are strictly positive. Thank you, Michael, for the idea.
How do you get that it’s irrational?
What if it’s some really large numerator over some really large denominator that this method is approaching?
My favourite approximation is 3, if 3 is not precise enough, I'm usually going to need at least 7 digits....
And this was the value legislated by Indiana so good enough for me
3.000000
@@peterkron3861 with the caviat that setting pi to exactly 3 is simply wrong and using 3 as an approximation means being aware of the error. Some legislators only demonstrate that they are completely unaware of any of their errors....
@@peterkron3861I shall forever now call three the Locally-Yokelly number and you have reminded me of that song: 'Indiana Wants Me' (obviously 'Me' is probably not a mathematician.)ua-cam.com/video/2p3OfHP5Hmo/v-deo.htmlsi=3I4uiHQK7vN0LcbO
Here's a good philosophical question for you guys. Could another universe exist in which pi had a different value?
It's crazy to me how precisely pi is known, to something like 100 trillion digits, when there is no known practical reason to know more than 40 or so, other than testing software and looking for patterns in its digits or continued fraction, which don't seem to exist.
The best approximation is 355/113 because all you have to remember is 113355
Isn't the whole idea of 22/7 that it's easier to do mentally and by hand?
I guess, using the same logic of integrating p(x)/(1+x) it's possible to find rational approximations of ln(2)
10:33 22/7-π~1/791 and most of these rational approximations seem give one more decimal in the approximated value of pi than there is decimals in nominator denominator together. So there is no point to rationalise it if the number of decimals on the ratio is not much lower than in rationalised pi...
Why does that integral give values near to the pi ?
Why is the '0' the digit that appears last in π at 33rd place (after others [1...9])?
Program can give easily values such as 312689/99532.
Or integrating Taylor series of 4*sqrt(1-x^2) from 0 to 1...
I seem to recall seeing a proof of the irrationality of pi involving integrals like these. Can you prove that pi is irrational in this way?
@12:48 ""Error < 10^-10" Wouldn't that be expected given that the four integers all have at least 10 digits?
In other words, it'd be a sad result if the error was (for example) "...< 10^-6" given all the ten digit numbers.
Does this mean that as n goes to infinity the error is zero?
The integral is from 0 to 1. At the endpoints, x^(4n)(1-x)^(4n) will equal 0.
In between, for 0
I believe this is all correct. The long division and antiderivative should grow in complexity, and to be 100% sure that we get an estimate for pi, we need to prove that the C/(1+x^2) term is nonzero for all n, or at least infinitely many n, but I suspect all n.
@@mtaur4113 x^(4n)(1-x)^(4n), as a real polynomial, will be able to be expressed as a product of linear terms and irreducible quadratics. It's already factored, into a product of just linear terms, so no irreducible quadratic will divide it. That includes the irreducible quadratic 1+x².
@@xinpingdonohoe3978 This could still produce Ax/(1+x^2), unless I missed something.
@@mtaur4113 I guess, look at the example he did. The minimum power for the thing we're dividing is x^(4n). And since we've only got 1 and x^2, then the division chain at the end will go x^(4n), x^(4n-2), x^(4n-4), etc. because each one is what is needed to multiply x^2 with to get the above one, as the long division asks. So you won't see odd powers of x below x^(4n+1), so there won't be an x^1 in the remainder since it won't appear at all in the division.
Where did halving the max error come from?
he stated that the max no. on the integral was 1(set x=1); thus 1/(1+ x^2) = 1/(1 +1) = 1/2
put another way, he bounded the value of pi to within a range of size X. if you find the midpoint of the range defined by the inequalities and set that as your approximation of pi, then the farthest the true value can be is one half of X
@@alquinn8576 thanks
22/7 - π is a very good approximation of zero
Suggestion for everybody, Veritasium video on the calculation of PI : ua-cam.com/video/gMlf1ELvRzc/v-deo.html The Discovery That Transformed Pi
I don't understand why you can half the error to 1/2520... Someone help plz
he bounded the value of pi to within a range of size X. if you find the midpoint of the range defined by the inequalities and set that as your approximation of pi, then the farthest the true value can be is one half of X (+X/2 or -X/2 from the midpoint)
355/113
Might as well just memorized pi lol.
I'm on the "pi = 22/7" team...as an aside, 19/7 is an excellent approximation of e.
Could be worse...the engineer in me is tempted to use 3 for both 😅
3 is a reasonable value. It all depends on your margin of error.
I'm still partial to 355/113. It's not meaningfully slower than 22/7, but instead of being only accurate to two decimal places, it's accurate to six--and it takes much, much larger approximations to get even small increases. For anything but aerospace work and quantum physics papers, 355/113 is genuinely indistinguishable from the real thing
wao pi
π ≈ π-0.00000000000000000000000000000000000000001 🥶
That'll do in a pinch, but I find it easier to simply remember the first 32 decimal digits of pi (I do know these first 32 off by heart.)
[Request]: What is the last non-zero digit of $(\dots((2018\underset{! \text{occurs}1009\text{times}}{\underbrace{!)!)!\dots)!}}?$
Why is pi so easily estimated by roots of integers? 31^1/3 is accurate to 3 places. 306^1/5 is to 4. 29809^1/9 is to 5. 93648^1/10 is to 6. 294204^1/11 is to 7. Is there a reason there are so many cases of this? I mean, there's got to be a pattern.
There's nothing special about pi. Given the density of a^(1/n) you can approximate any real number with any precision.
All of your fractional exponents must be written inside grouping symbols!
@@robertveith6383Well yes, if I was presenting this rigorously I would use superscript notation. (1) This is a UA-cam post. (2) You figured it out. (3) Satisfied.
311/99
Smart❤😂👍🏆💪
nah, mine is π/1
Thanks sir for help we I am from """indiaa"""
"""Shreeniwash ramanujan """
Country
Srinivasa*
Yes
Ramanujan
Aaryabattt
Aacharay madhav who discover calculas and infinite series
Bro you are suffering from crippling inferiority complex. Get some love from your parents.
My favorite approximation is actually just pi.
Heh. But that's the exact value, it's not an approximation. It's like if the abbreviation for May was "May".
pi^4+pi^5=e^6
Pithagorean Eorem
Tell me you are an engineer without telling me you are an engineer:
If that were true, what would the consequences be? Could we no longer conclude that one of π+e and πe was transcendental, for example?
@@xinpingdonohoe3978 hard to say. They aren't equal, but if they simultaneously are equal, subtract to get 0.000000034=0 or whatever, then multiply both sides by whatever and then every number is equal to zero. Contradictions cut deep.
Maybe there is some other mathematical structure where something else plays the roles of e and pi. You would have to go outside of looking at Euclidean geometry and metrics or something, and I don't know anything off the top of my head. I saw something trending about computing "pi" for metrics where an exponent different from 2 is used in the Pythagorean theorem. pi is the minimum possible, and I don't remember if the e^6 is the bigger or smaller number. If it's bigger, then you could choose a p, but if smaller, you can't. But this probably isn't the only weird thing you could do with it.
My favourite is pi=3.
3/1 works for me. But I’m a simple guy.
No. Pi is an "irrational" number and cannot be described/represented by a string of numbers (incl ratios) in ANY numerical base. Did not watch the vid
Nah, I'll just use 3 and call it a day
Nah mine favourite approximation is π≈π
en.wikipedia.org/wiki/Continued_fraction#Continued_fraction_expansion_of_%CF%80_and_its_convergents
Nice thumbnail