Frequency of Each Digit of Pi by Eve Andersson

Home : Pi : One Calculation

in first 100 digits of pi

0	8
1	8
2	12
3	12
4	10
5	8
6	9
7	8
8	12
9	13

in first 1,000 digits of pi

0	93
1	116
2	103
3	103
4	93
5	97
6	94
7	95
8	101
9	105

in first 10,000 digits of pi

0	968
1	1026
2	1021
3	975
4	1012
5	1046
6	1021
7	969
8	948
9	1014

in first 100,000 digits of pi

0	9999
1	10137
2	9908
3	10026
4	9971
5	10026
6	10028
7	10025
8	9978
9	9902

in first 1,000,000 digits of pi

0	99959
1	99757
2	100026
3	100230
4	100230
5	100359
6	99548
7	99800
8	99985
9	100106

in first 10,000,000 digits of pi

0	999440
1	999333
2	1000306
3	999965
4	1001093
5	1000466
6	999337
7	1000206
8	999814
9	1000040

Comments

Zero

Why does zero lag behind in the count. It is a perfectly upstanding digit? Just had to ask, its a retorical question. I am sure it has been asked before by greater minds.

sincerely

DMA

-- Douglas Abbott

Why zero lags

Obviously, some digit will lag. Which digit has a reason to lag? None has a reason to lag. Zero is none. Therefore, zero lags.

-- Larry Hosken

Wait a minute

Looking over the graph of the first 10,000,000 digits, now I think that the question was wrong. Zero isn't the worst lagger. Behold--
0: 999440
1: 999333.

Therefore, I would like to amend my answer to "Will some digit lag? Empirical evidence suggests that one must. Therefore, one lags."

-- Larry Hosken

Zero Is not at all a lacker

If you look closly enough you see that zero is never a slacker that it is eather ahead of a number or it is tied with another slacking number...GO ZERO!!!!!!!!!

-- Michael Quagliano

Lagging digits

Assume we take Pi out to an infinite amount of decimal places. We cannot determine the last digit of Pi, because there is no last digit, the string of random numbers goes to infinity. However, because there is no repeating pattern in the decimal portion of Pi we can assume that all numbers are equally likely to be the next number in the sequence as the length of the decimal portion goes to infinity. This in effect defines the next number in the infinite sequence as a random event. This means that each number is equally likely to be the next number so each has a 1/10 chance. Therefore, the occurence of each digit should be equal once we reach an infinite number of decimal places.

This is supported by looking at the differences between the occurence of each number over time as compared to the total number of occurences. The percentages shrink rather rapidly as the order of magnitude of the number of occurences increases.

The implications of this on the upcoming digits in the sequence are rather interesting, as this would mean that zero is actually more likely to occur than the other numbers, something of a paradox considering the random nature of the decimal place string.

-- chris campbell

Re: Lagging digits

Chris,

Your statement, "However, because there is no repeating pattern in the decimal portion of Pi we can assume that all numbers are equally likely to be the next number in the sequence ..." isn't actually correct.

Here is a counterexample. Take the number 1.01001000100001000001... It has no repeating pattern, yet not all digits are equally likely.

It has been conjectured that pi is "normal" which, mathematically speaking, means that all digits are equally likely and that other properties of randomness are adhered to. However, this has yet to be proven.

-- Eve Andersson

My penny's worth

Surely Chris's comment above that "zero is more likely to occur" is just an example of the innate problem with probability people seem to have. If I flip a coin ten times and it lands tails-up seven times, that does not mean that my next coin flip is more likely to be heads to "even the score". If you bet on a horse ten times and it lost every time, you wouldn't bet on it again as it was now due a win...

Were pi to be calculated to infinity, then - assuming it's a normal number - then working from the fists 10,000,000 digits we would expect an infinite number of fours and infinity-minus-1653 zeroes.

BTW Eve I think you need to bring back the green antenna photos. I was just leafing through a 1995 copy of Mac Format magazine and saw you beaming out at me as 'one of the coolest sites on the web'.

-- Aidan Merritt

And another thing...

If you really want to take this to it's logical extreme, work out the frequencies of 1 and 0 in the binary expression of pi. Intuition tells me that 1 would be way ahead, but I can't explain why I think this.

-- Aidan Merritt

Binary Digits

A quirky article by Mike Keith talks about drawing binary digits of pi on a hexagonal grid and then staring at the result until you can convince yourself you see a picture there.

Towards the end, he points out that in the first 1000 binary digits, zero appears more frequently than one. Around bit 164, it's much more common than you would expect.

-- Larry Hosken

binary

is anyone aware of a site where the pattern of binary is used to draw a "diagram"

ta

-- scotty pi

Can you put pi up on the comment board?!
Image: Abstract.bmp

-- Emily Harte

Pi R square

Any discussion of Pi, aside from its esoteric side, always brings a smile to my face. As best I can remember, it was on Beverly Hillbillies where I first heard:

"Pi R square" ..."no, it's not"... "It certainly IS" ..."No, any fool knows: CORNBREAD are square, Pie are ROUND!"

Maybe it's even older than that... anybody know fer shure? Charles Husbands

-- Charles Husbands

Memorizing PI

I got started being interested in pi when I visited my brother last month and my 14 year old nephew told me he would get an extra 30 points in math if he could memorize pi to the 74th decimal place. I helped him figure out ways to remember the numbers and am happy to say that my nephew called me a couple of weeks ago to tell me that he did get the extra points.

I never memorized numbers longer than 16 digits before (my credit card numbers) or rather 19 numbers if you can really count my checking number which includes the routing number. I was curious to find out how far I can memorize pi. I have adopted the credit card system, by separating the numbers in pi in groups of 4s. So far I am only up to the 30th group of 4's. That's 120 numbers. Wow!!! Not being a genius, and being a grandma, I am rather impressed with myself, LOL!!!

So I have issued a challenge to my 3 children, my 4 grandchildren that I would pay $1 for each group of 4's they could remember both in sequence and at random. Thus each would be able to earn up to $60.

I love to play blackjack, therefore I remember pi by associating the numbers with the game.

For instance: group 2 is 9265 = 2 double down hands, group 4 is 7932 = 21, group 5 is 3846 = 21, group 8 is 7950 = 21, group 15 is 4944 = 21, group 21 is 8628, happy that the dealer has 3 eights instead of 3 sevens! :)

Group 16 = 5923, my ex husband's work extension, not that I care because I don't have his phone number (I deleted them from my cell phone as well as from my own memory). :))

I was born in the 105th group of 4's in pi after the decimal point. I told my 3 kids I would give them $200 each if they tell me theirs...well I'm just relieved that I have only 3 children. I would be broke otherwise, heh! I had to tell my nieces and youngest son's fiancee the offer was good only for my 3 children, much to their dismay.

Well, I've just memorized 3 more groups. Maybe I will let the kids to up to 50, depending on how far I can go by Feb. 16 weekend, which is when I will see all of the grand/kids. First I will demonstrate that even I can recite the numbers, sequentially or at random. So I am not asking them to memorize anything that their old mother/grandma cannot do herself. Oh, I'm up to group 36 now.

-- Lydia Shinohara

other bases

Pi is obviously going to look different if we calculate it in base 8 or base 12 or any base other than 10. Do we have any workups available of what the digit frequency looks like if we use some other base? Do we still have 'laggers', and -- if so -- are they the same ones? I doubt that they would be the same, but it would be an interesting experiment.

-- Jonathan Elliott

Because...

If (assumption) the digits of pi are pseudorandom, zero will tend to lag, depending on how you average things out. We choose by tradition not to write in a trailing zero after the decimal point, so any 'lengths' of pi that would otherwise terminate with a zero, don't. The non-zero digit that precedes the zero gets a double score!

-- Julian P