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.
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.
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
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
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.
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 😄
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
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. :-)
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.
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.
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
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 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
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)
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?"
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.
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.
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.
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...
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 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.
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.
"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 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...
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!
[8:47] The overestimation is approximately 0.000003124218994. You were over by an amount that was presented as a Notch, [10:00]. Multiply the top and bottom by 3200 to simplify the division at [11:13]. The variable y written as 1/320080 is prettier. You welcome. Note: Completing the Square for the error propagation at the 2nd iteration, k = 1, of Sqrt(10005) produces the same amount as 1/320080. This means that the numerator of the Chudnovsky Algorithm is quadratically converging to Pi. What is the rate of convergence for the denominator found in the Chudnovsky Algorithm? Is the numerator converging too slowly?
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.
Why dont you try calculating pi by calculating the number of points in an arbitrarily large square, and in its inscribed circle, as delineated using Bresenham's Circle Drawing Algorithm.
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.
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?
"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
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.
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?
Matt: the semi bald man who steals whiteboards
Matt Parker: The bald phantom board filcher.
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.
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.
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!
13:53 look how well the cup and the table border matches, I've honestly thought that the cup was transparent
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
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.
7:56 obligatory parker square
Parker pi day, dear Matt.
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 😄
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
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...
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. :-)
Always show your working, Matt. Do it again and see me after class.
*casually steals whiteboard*
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
RUclips is too small to contain
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.
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 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.
15:45 - that was the mistake :/
yep, should be 0.......
Derek Zoigt description says no errors yet tho
Lol
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.
(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
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
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?
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
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.
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.
11:30 "the board just wasn't that well attached to the wall if I'm being honest"
still waiting for k = 2 on channel 3 =D
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?"
I could use a 6-hour live stream of Matt narrating arithmetic
Now I don't feel so bad at long division. Thanks!
1:14 5+3=13, carry the one... hmmmmm.
I don’t get it.
12?
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.
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.
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
So that's a Parker Wall-Mounted Whiteboard?
This surely increase his mental math
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.
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...
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
"Fudge factor". That got me
Is this what a college math lecture like?
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
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.
I realised as soon as he did it that he did 5+3=12
Who de hell made this up that the formula even works? That guy must be mad man
wow😃 this man doesn't give up easily
i want to see k=2 now...
You madman!
I want to see k=100 now...
"A classic 80", like a classic parker square.
Top 10 TV cliffhangers
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
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
14:10 That's a parker square of a digit placement.
10:06 bit of a parker square, that.
I think the first sum was 558731543. When you added 5 and 3 you wrote 2 instead of 8.
Now show us how that equation was derived, and how they know it gives an increasingly accurate approximation of Pi.
One word "Dedication"
That is great...
What happened to the end of the video??
why don't you use hexadecimal to make calculations faster and easier and than convert back to decimal?
"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?
U are damn patient!!!grt
You could have divided 720 by 6, would make it slightly easier 😊
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...
In a previous video did you not state that you are unwilling to concede that 0!=1? Seems like I remember you saying that.
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.
The first sum went wrong by 4million
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!
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
[8:47] The overestimation is approximately 0.000003124218994. You were over by an amount that was presented as a Notch, [10:00]. Multiply the top and bottom by 3200 to simplify the division at [11:13]. The variable y written as 1/320080 is prettier. You welcome.
Note: Completing the Square for the error propagation at the 2nd iteration, k = 1, of Sqrt(10005) produces the same amount as 1/320080. This means that the numerator of the Chudnovsky Algorithm is quadratically converging to Pi.
What is the rate of convergence for the denominator found in the Chudnovsky Algorithm? Is the numerator converging too slowly?
Weirdly fun to watch xD
Where is the end of the video
1:41 you could just simplify 720/6 to 120/1
I saw that mistake (2-1.3...=1...) while watching, so sad.
Wouldnt it be fun to calculate the accuracy of Matt's calculations? [Correct calculations] / [Total calculations]
Can anyone please tell me,where did the '"-"(minus) go?😅
Would it be easier in binary?
Correction: 13:04 '189500 - 100925 = 89675' it should be '89475'
How when the small number was negative you just subtract it from k0 . Please explain
He added it, but the number is negative. So he simply subtracted it.
RWBHere but why did he subtract it from k0 isn't that completely different part
why would you not cancel the 6 from the 720, you mad man!
Great
It's interesting to see big calculations done by hand. You have to get creative with the methods.
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.
13:50 super easy, barely an inconvenience.
The video is funny when you fast forward it in 10 second chunks.
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.
Why dont you try calculating pi by calculating the number of points in an arbitrarily large square, and in its inscribed circle, as delineated using Bresenham's Circle Drawing Algorithm.
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.
you could cancel the 6 with the 720, and make this a bit easier.
You can also divide out a ten, and a three , and then take a quarter.
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?
It's 'fell swoop' Matt
Parker Square of a Pi Estimation
I love that he could have made it a bit easier for himself since 720 is divisible by 6