Calculating π by hand: bonus k=1 working out
ฝัง
- เผยแพร่เมื่อ 13 มี.ค. 2018
- Chudnovsky algorithm by hand bonus: k=1
This is all the Chudnovsky algorithm set-up for k=1. Including a better approximation of sqrt(10,005).
Watch the main video:
• Calculating π by hand:...
This video was filmed at Queen Mary University of London.
CORRECTIONS
- None yet. Let me know if you spot anything!
Thanks to my Patreon supporters who enable me to spend a day doing a lot of maths by hand. Here is a random subset:
Christopher Samples
Sean Dempsey-Gregory
Emily Dingwell
Kenny Hutchings
Rick de Bruijne
Support my channel and I can make more videos:
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Music by Howard Carter
Filming and editing by Trunkman Productions
Audio mastering by Peter Doggart
Design by Simon Wright
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
Maths book: makeanddo4D.com/
Nerdy maths toys: mathsgear.co.uk/
"just wasn't that well attached to the wall if i'm being honest"; even the second time, that made me laugh my ass off
Matt: the semi bald man who steals whiteboards
Matt Parker: The bald phantom board filcher.
15:39 “that 2 on the end is going to become a 1”... :(
"That's easy enough"
Get on it Matt! The description still says 'no errors yet'!
Was about to say. That should have been reduced to 0 :(
Matt found out the mistake on the main video at the end.
I don't get it, why is that wrong?
This really brings in to perspective just how remarkable modern computing power is.
As my physics professor recently remarked, "It's no wonder so many statisticians commit suicide."
origami katakana
Perfect
And what is the statistics of that?
Binary actually makes these exact calculations much easier. There's no tables or anything for long division. I'd say this entire series is composed of bits of maths that computers are ESPECIALLY good at. ... clearly no coincidence.
Tristan Ridley It has nothing to do with binary, just the power of cpu's.
Having done the math by hand in binary and in decimal, I beg to disagree. The power of a CPU can either be enhanced or impeded by exactly which calculations you need to do, because it works in binary.
I think you have stumbled upon a a new mathematical axiom...
Parker's Pi Series Postulate :
Convergent series for Pi which are easy to calculate by hand do not converge quickly.
Convergent series for Pi which do converge quickly are not easy to calculate by hand.
Is it driving anyone else nuts that he doesn't just divide the 720 by 6 straight away?
Not only that, but the resulting quotient of 120 can be factored as 3x4x10, and each of these factors is also a factor of the denominator.
Yes, but what makes me more nuts is at 15:35 where he subtracts the correction term incorrectly number ending in 72 - 1.333666 doesn't give a number ending in 71
Yeah man, 760/6 is just 5!
7:56 obligatory parker square
13:53 look how well the cup and the table border matches, I've honestly thought that the cup was transparent
Don't you just love when the mathematical reasoning has "-ish" at the end of the sentence! my favorite way of doing maths!
You ever visited a numeric course?
Patrick Abraham i did numerical analysis, we used epsilon and delta for errors.
Anything I can throw weighs one pound. One pound is one kilogram... I did not tell you you could do maths this way.
You're clearly an engineer, welcome!
Basically physics tbh.
13:47 love how the line on the coffee cup lines up with the table edge.
yeah, it could be a brownish glass cup with orange juice in it
That was some wonderful working out. theres really something satisfying about doing it all by hand and actually changing the numbers yourself
Parker pi day, dear Matt.
The fact that this has still only got 52629 view is a travesty.
Literally cannot tell you how helpful it is to have Matt mutter about numbers on one screen while I write on another :D
(14:05) The invention of the "division spiral" :D
The more i watch his Pi calculation videos, the more i am reminded of fractals :D
It makes me very happy that in about one day, 17,831 people (at the time of this comment) spent 15 minutes or so watching someone painstakingly work out a series of multiplication and division problems. :-)
Have you heard of cancelling the six with the 720 and a factor of ten - not sure you've done much hand calculation recently LOL
John Warner he’s all about the brute force
If you can't brute force it why even bother.
Surely you should know that 26253741264076800 is divisible by 12?
Easy enough to spot it's divisible by 12 (two 0s at the end, digit sum is a multiple of 3)
Matt Parker
You really made a Parker square of that division...
I had to pause the video at “the board... just wasn’t attached to the wall very well, if I’m being honest,” because I was laughing too hard.
When doing the final subtraction, shouldn’t it have been a zero in the ones place, as you were taking 1.3... away from 2?
Edit: just watched the rest of the main video, where you address it. Nevermind 😄
My Maths teacher used to say, believe in division and not multiplication... if you had in the part when dividing the numerator and denominator, 720(562731543)/6(262537412640768000) if you cancelled the 720 and the 6, you would have gotten 120(562731543)/(262537412640768000).... simplifying alot of the work BTW AWSOME VIDEO AND AMAZING MATHS !!! LOVE IT
You can also divide out a ten, and a three , and then take a quarter.
You could also chop a 0 from the top and bottom to get 12(562731543)/(26253741264076800)
Aka dividing out a ten
and then you can divide by 12 before multiplying to get 562731543/2187811772006400
Dividing the denominator by 12 first does not save any time. Your final division would be one digit shorter, but you will have to do an initial division by 12 instead of multiplication by 12, which takes substantially longer. However, cancelling out the initial factor of 60 would have made a lot of sense.
At 1:20 you did wrong yhe sum. 5+3=8, not 12?!?!?!
He caught the error later in the video.
Just Saw it too.
Always show your working, Matt. Do it again and see me after class.
where is the end of the video?
Tapash Alister yeah
It's in the main video: this is just the tedious bit spliced out of that to save time and pain ;-)
When matt works out pi with k being 1000000
TH-cam is too small to contain
I've always done digit-wise multiplication, then you get a grid, then you sum accross the diagonals, and add the correct number of zeroes per diagonal, then a final summation. it seems convoluted, but it reduces the chance of making an error from trying to manipulate large numbers in your head.
I like it when it's the maths professor suffering at the whiteboard instead of the student. « And... we have... another complicated bit of division. » :-D
*casually steals whiteboard*
15:45 - that was the mistake :/
yep, should be 0.......
Derek Zoigt description says no errors yet tho
Lol
15:08 Fun fact: "this" and "mess" are spaced pretty much exactly one second apart. (my microwave oven beeped at the same time and lined up with both words)
I love the language used throughout... All the ish, techniques stuff, and yeah I might stuff this up. Making math normal.
I, once calculated Pi, very inefficiently, summing the length of sides of regular polygons, I started with the hexagon and progresively double the number of sides. I used the trigonometric identity : Sin(2x) = 2SinxCox and started with sin30° = 0.5
I had to solve quadratic equations in every iteration. It took quite a while to get a 3.141592
1:14 5+3=13, carry the one... hmmmmm.
I don’t get it.
12?
Does this man ever factorise out of his fractions? Goodness Matt!
Fractions are your friend; the calculation for y would have been easier (albeit only a bit easier) if you had used 3/8000 instead of 0.0003125
Matt: "How much harder can the second term be?"
* I check the algorithm*
Me: "Now that we've done the first coin flip, how much harder can the other few hours be?"
You did the second term on the second channel please tell me your not going to do each term on their own channel. 🤣
That would actually be kind of funny but no, he isn't... Just did it hear because he didn't want to add the same thing as he did in the first video again basically... Plus he forgot how to add almost at the end of 545140134 + 13591409
Just as the workload in the calculation dramatically increases and the effect of the correction plummets, I suspect the view count on any additional channels would quickly approach zero.
andymcl92 Honestly, I think I'd find them more interesting as time goes on and Matt gradually descends into madness.
Now I don't feel so bad at long division. Thanks!
U are damn patient!!!grt
11:30 "the board just wasn't that well attached to the wall if I'm being honest"
The lecture room in the University Museum of Natural History in Oxford used to have three huge rolling chalk boards - might have been replaced by white boards now - but you need that sort of space for this calculation.
Weirdly fun to watch xD
At 12:54 after you subtract the 1 mulitple (the step where you get your second 1 digit) the subtraction should equal 89475, but you got 89675. This does mistake propegates down and in reality after the 9 digit you should get a 4 digit.
Generalizing the minimum possible steps to accurately calculate pi to X digits by hand using various methods sounds like the making of a graduate thesis to me. Or perhaps the 2019 pi day video...
wow😃 this man doesn't give up easily
This surely increase his mental math
1:17 your addition went wrong, it should be 558,731,543
that's what I get for doing the working along with you, nevermind you found it a minute later
That is great...
still waiting for k = 2 on channel 3 =D
1:45 why not divide 720 by 6 to get rid of the 6 at the bottom and then just get -(120*558731543)/262537412640768000?
exactly what I was thinking... lol
262537412640768000/40=
----6563435316019200/3=
----2187611772006400
That part only took a few minutes. Definately a time saver.
Is this what a college math lecture like?
I am amazed by how quickly this converges... I tried it by hand myself, working to an 18 digit approximation for the numerator and a 19 digit estimate for the denominator based on k=1, and I got 13 accurate digits by hand. I made some mistakes somewhere, and plugging the numerator and denominator I derived into a calculator, I should have gotten 16 accurate digits!
I wish I understood how the hell the Chudnovsky algorithm works... It makes no sense to me.
Great
Hey, a really great formula I like to use for approximating square roots
,/x~,/p + x-p/2,/p
Where x is the number you are trying to get the square root of and p is the nearest perfect square to x
PS ( ,/ - square root sign) ( / - normal division sign)
I don't care 1234567890 What does the tilde stand for?
Am I missing something or is the rest coming out at some other time.
On the main channel. This is just the tedious part cut from the main video
12:54 The calculation for the 6th digit(second '1') is a little off. 500 - 025 /= 675. This will affect the answer only if more digits of precision were needed.
8:05 seriously, why has NOBODY ever taught me this. I have had highschool and university level maths, and the square root was always so theoretical to me. This is mind blowing.
I think a binomial expansion would have been easier in this case
I realised as soon as he did it that he did 5+3=12
you look good bald :D also that 3/4 sleeve black t-shirt was nice choice
and have to watch it again with more focus to understand
14:10 That's a parker square of a digit placement.
So that's a Parker Wall-Mounted Whiteboard?
i want to see k=2 now...
You madman!
I want to see k=100 now...
"Fudge factor". That got me
I understand so much better now why early mechanical calculators, electromechanical calculators and early computers worked the way they did. This is insanely tedious. I really feel Babbage now, I won't make fun of him for not thinking of the more modern generalist approach to computers any more! Making math crunching less tedious was already doing a huge service to the scientific community and mankind at large!
"A classic 80", like a classic parker square.
10:06 bit of a parker square, that.
In a previous video did you not state that you are unwilling to concede that 0!=1? Seems like I remember you saying that.
The MacLaurin series for √(10,000 + x) is a much easier way to estimate √(10,005). Taking just the first four terms gives √(10,005) ≈ 1/(0!) * 100¹ * 5⁰ + (1/2)/(1!) * 100⁻¹ * 5¹ + (-1/4)/(2!) * 100⁻² * 5² + (3/8)/(3!) * 100⁻³ * 5³ = 100 + 5/200 - 25/8,000,000 + 125/160,000,000 = 100 + .025 - .000,003,125 + .000,000,781,25 = 100.024,997,656,25. That's far more precise than your value, and you can calculate it by hand in a fraction of the time.
The absolute error is less than the next term, which is 1/4,096,000,000,000 ≈ 2 × 10⁻¹³
One word "Dedication"
I think the first sum was 558731543. When you added 5 and 3 you wrote 2 instead of 8.
Who de hell made this up that the formula even works? That guy must be mad man
What happened to the end of the video??
Wouldnt it be fun to calculate the accuracy of Matt's calculations? [Correct calculations] / [Total calculations]
It's interesting to see big calculations done by hand. You have to get creative with the methods.
You could have divided 720 by 6, would make it slightly easier 😊
Matt_Parker_2 CORRECTIONS to add:
- At 0:15:44 at i say "zero" i meant to do "42,698,670.6663334359680" that should be and instead saw "42,698,671.6663334359680" damn it.
Shouldn't be in description be 42,698,672 - 1.3336665640320 will be 42,698,670.6663334359680 instead of 42,698,671.6663334359680?
Why didn't you add it to the k=0 term from part 1?
Effectively he did add it, it's just that the k=1 term has a negative.
Top 10 TV cliffhangers
"Multiply this one by 720, multiply this one by 6, and divide one into the other" why wouldn't you just simplify the 720/6 to 120, to just have the one massive number in the denominator? Or even better, simplify the 120 into 12 and lob off a zero from the massive number in the denominator?
At 15:40 shouldn't the unit digit be 0?
At 1:17 there is an addition error...
545140134+13591409 will be 558731543... Which you corrected later of course...
Loving your new look though...
Edit: In the last subtraction at 15:40 the unit place should change from 2 to 0 and not 1...
13:50 super easy, barely an inconvenience.
Now show us how that equation was derived, and how they know it gives an increasingly accurate approximation of Pi.
1:41 you could just simplify 720/6 to 120/1
Hey Matt, please insert a Parker square around your mistake at 1:27. I spotted your mistake only to find you resolve it 2 minutes later.
Would it be easier in binary?
That’s exactly how I multiply large numbers 😂 x*720=x*700+x*20=x*7*100+x*2*10
Can anyone please tell me,where did the '"-"(minus) go?😅
One. *Fell.* Swoop.
Shakespeare wrote, “at one fell swoop”.
Mostly I’m complaining because you didn’t factor the 6 out of the 720 before dividing.
I love your videos, by the way.
Where is the end of the video
can you give us an estimate of the actual amount of time K=0 took, K=1 took, and what you would expect K=2 to take?
I did k=2 after this and used his values for k=0 and 1. It took me about 30 minutes of constant writing. Not sure how long it took for him to make k=0 and k=1, but since there are quite a few cuts, it can't take much more for each. I'd guess it takes roughly 30 minutes per digit. Now, I'm fairly slow at making hand calculations so I might be wrong though haha.
I love that he could have made it a bit easier for himself since 720 is divisible by 6
Correction: 13:04 '189500 - 100925 = 89675' it should be '89475'
why don't you use hexadecimal to make calculations faster and easier and than convert back to decimal?
1:42 You could have used that 720/6=120 and saved some time ^^
Is their any chance that permanently in the fumes of whiteboard markers or sharpies for years of your life has affected your mental capacity? I know too long and my focus wanes.
Great work though, always love your entheusiasm.
why would you not cancel the 6 from the 720, you mad man!
can someone expalin the correction with y (using some smaller numebers perhaps=. I get the x^2 overestimation but he lost me there
He's evaluating the next term in the Taylor Series around the point x=10,000. He's giving the geometric justification instead of the calculus justification. The Taylor Series for square root around the point 10,000 is 100+(x-10,000)/200-(x-10,000)^2/8,000,000. Replace x with 10,005, and you get 100+5/200-25/8,000,000=32,007,999/320,000 approximately 100.024996875. That's good to 12 decimal places.
At the beginning you could just cancel out the 720 with 6, and you would make your life much easier :P Only one multiplication, not two
The first sum went wrong by 4million
you could have simplified it with fraction maths, making the final division give you pi, simplify the sum into a single fraction for k=1 + k=0
Did I miss something? Or did the video end before the maths were finished?
EZMawloc full video on main channel
This is an edit from the main video, 'rescued' from the mythical 'cutting room floor', so that those with ADD don't have to.....
Thanks for the videos, Matt.
Hey, Python can make this simpler ;) j/k this was fun to watch