### Author Topic: Math Quirks:...Anybody (else) experienced strange Date coincidences ?  (Read 11352 times)

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#### SiliconWizard

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #50 on: July 11, 2023, 09:39:11 pm »
I very often have a peek at the clock and catch 12:12, it must be a Final Destiny thing.

That, or OCD.

#### DavidAlfa

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #51 on: July 11, 2023, 09:52:16 pm »
Absolutely not, i'm doing my stuff at work, when I finish something... Hmm how many time did it take?
10:45... 2hours - note down in the repair form.

Start  with the next one,  initial test / diagnose / repair / more tests...
Huh, I might take a break, I'm dead! Time for coffee!
What time is... 12:12 ...it...?

Else than that I rarely watch the clock. Except when the hunger comes in at the last hour or so, then it's way worse than OCD! (But that's about 14h/2pm).
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#### MIS42N

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #52 on: July 11, 2023, 11:02:36 pm »
edit: "relevant results"? You do realize that the expansion of pi contains every possible subsequence?
Are you sure?
Take pi to any arbitrary length. Change one digit. A subsequence that does not occur in that expansion.

The problem with infinity. Pi may have an infinitely long string of digits, but there will be a bigger infinity of substrings that are not in that string.

Aaaargh (disappearing up own ***)

#### Andrew LB

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #53 on: July 12, 2023, 11:15:40 pm »
I very often have a peek at the clock and catch 12:12, it must be a Final Destiny thing.

I always look at the clock when it says 11:34. It's really annoying.

And no, it's not OCD. It always happens when i haven't looked at a clock in hours.

#### MIS42N

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #54 on: July 12, 2023, 11:52:43 pm »
I very often have a peek at the clock and catch 12:12, it must be a Final Destiny thing.

I always look at the clock when it says 11:34. It's really annoying.

And no, it's not OCD. It always happens when i haven't looked at a clock in hours.
Maybe it's not the clock you are looking at. 1134 (kHz) is the frequency of the local radio in Armidale NSW. The car radio shows the time when the radio is not on, and the frequency when it is on. Caught me a couple of times.

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #55 on: July 13, 2023, 07:30:01 pm »
Are you sure?
Take pi to any arbitrary length. Change one digit. A subsequence that does not occur in that expansion.

I don't think so. You have to be careful when reasoning about "infinity", since it is not something that actually exists. It's a concept that serves as a shorthand for "behavior under limit condition(s)". When we speak of "infinitely many" of something, i.e. an infinite set, we have a criterion for how the members of the set are generated, and the limiting condition is what the criterion generates when allowed to proceed without end.

In particular when you say "bigger infinity" there is only one plausible meaning for that phrase, and it's not what you think it is. The cardinality of an infinite set is a "number" that specifies how the members of that set can be mapped onto any other set. So even though intuitively there are "more" rationals than integers—since between every two integers there are infinitely many fractions—the set of all rationals and the set of all integers have the same cardinality: there are "equally many" of them, because we can devise a method of mapping one set onto the other. In Cantor's language, they both have the same cardinality, $\aleph_0$. Any infinite set that can be mapped onto the integers is enumerable: there is a way to order the set such that you can cover all of the elements by counting them.

In particular, note that the infinite set of all decimal substrings is enumerable. Proof:
I. Assemble all of the strings of length 1 in lexicographic order and count them.
0 1 2 3 4 5 6 7 8 9
II. Then append all of the strings of length 2 in lexicographic order.
00 01 02 ... 10 11 12 13 14 ... 96 97 98 99
III. Then append all of the length 3 strings, etc up to length N strings as the limit of N goes to ∞. This sequence includes all decimal substrings of any length whatsoever.
Therefore the set of all decimal substrings is enumerable and its cardinality is aleph-null.

The problem with infinity. Pi may have an infinitely long string of digits, but there will be a bigger infinity of substrings that are not in that string.
As I have shown, the set of all substrings cannot be a "bigger infinity" than the sequence of digits in pi's expansion.

∞+1 is not "bigger" than ∞. Neither is ∞+∞. To keep things clear, better to use $\aleph_0$ instead.

Quote
Take pi to any arbitrary length. Change one digit. A subsequence that does not occur in that expansion.
This is a mutilation of Cantor's diagonal proof. Cantor proved that the cardinality of the reals, which he called the continuum or c, must be greater than the cardinality of the integers, aleph-null. This amounted to a proof that the reals cannot be enumerable:

I. Consider only the reals between 0 and 1, exclusive. They have the form ".XXXXXXX"... where each X is some decimal digit. If the number is a fraction, then the expansion will be all 0 or some periodic pattern after some point, but the reals also include transcendental numbers which never repeat.

II. Suppose we have enumerated all these numbers in an (infinitely long) list. So we have a kind of grid of digits. The first row of the grid is the first number in the list, the second row is the second number, and so forth. The first column in the grid is the tenths digit of all of the numbers; the second column is the hundredths digit, and so forth.

III. So making a diagonal from the upper left, take the tenths digit of the first row, the hundredths digit of the second row, etc., to construct a number b with infinitely many digits.

IV. Now change every digit of b to produce d (for example by adding 1 modulo 10), each digit of which differs from the corresponding digit in the grid. Because d differs at at least one digit from every number in the entire grid, it cannot be in the list. But d is between 0 and 1, since it has the form ".XXXXXXX", so it should have been in the list. Therefore the assumption in (II) that we had an enumerated list of reals between 0 and 1 is false. So the reals are not enumerable and c > aleph-null.

Note that the only way this argument works is if the grid is infinitely long in both directions: both infinitely many numbers in the list, and infinitely many digits. Otherwise it can't prove anything, since there could be some position at which d does not differ from an element of the list and might indeed be in it.

Cantor also famously hypothesized that there is no other cardinality between aleph-null and c, so that c represents the next set larger than the integers, or $\aleph_1$. This is called the continuum hypothesis. While most everyone believes it to be true, it has so far not been proved.

Quote
Take pi to any arbitrary length. Change one digit. A subsequence that does not occur in that expansion.
One thing you have to be careful of is arguments that prove more than intended, in other words overgeneral arguments. Since you invited me to, I take "arbitrary length" to be 1.

I. Pi to 1 digit is 3.

II. "Change one digit": okay, now I have 4.

III. "A subsequence that does not occur in that expansion": while it's true that the first digit of pi does not contain 4, it is irrelevant to the topic because 4 is contained someplace else within the expansion, which is all that I claimed (in fact, infinitely many places). And the same is true for substrings of any length, not just 1.

The property of containing every possible sequence is called normality: the uniform distribution of digits. It has been known for a long time that nearly all real numbers are normal, but proving that specific numbers are normal is difficult. The known decimal digits of pi are very uniform, and those digits are known to around 10 trillion places, but uniformity of all digits is unproven. We can, however, easily construct normal irrational numbers, such as Champerowne's constant.
« Last Edit: July 13, 2023, 07:32:13 pm by helius »

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#### MIS42N

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #56 on: July 14, 2023, 12:44:45 am »
Quote
Take pi to any arbitrary length. Change one digit. A subsequence that does not occur in that expansion.
One thing you have to be careful of is arguments that prove more than intended, in other words overgeneral arguments. Since you invited me to, I take "arbitrary length" to be 1.

I. Pi to 1 digit is 3.

II. "Change one digit": okay, now I have 4.

III. "A subsequence that does not occur in that expansion": while it's true that the first digit of pi does not contain 4, it is irrelevant to the topic because 4 is contained someplace else within the expansion, which is all that I claimed (in fact, infinitely many places). And the same is true for substrings of any length, not just 1.

The property of containing every possible sequence is called normality: the uniform distribution of digits. It has been known for a long time that nearly all real numbers are normal, but proving that specific numbers are normal is difficult. The known decimal digits of pi are very uniform, and those digits are known to around 10 trillion places, but uniformity of all digits is unproven. We can, however, easily construct normal irrational numbers, such as Champerowne's constant.

I think I was not clear. I was arguing that take pi with x digits. Change the first digit to create a different number with x digits. Those x digits are not contained in the original x digits. There was no intent to extend pi to more digits (in which case I alter the first digit of the more digits and the argument still holds). However, I was assuming a finite string of arbitrary length. Bad assumption. A finite string of arbitrary length is not an infinite string. Different beast.

I agree with your original statement. A persuasive argument. It relies on pi being a truly random sequence of digits and I assume there is a proof of that somewhere.

#### TimFox

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #57 on: July 14, 2023, 02:42:24 am »
I believe the correct statement is that the sequence of digits in the decimal expansion of pi passes all the tests for randomness.
I don't know if randomness can be "proven" past that statement.

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #58 on: July 14, 2023, 07:10:36 pm »
The number theoreticians just speak of uniform distributions of digits (not "randomness") because it has nothing to do with random probabilities. Every digit of pi is calculable from simple formulas: it has zero entropy.

There are some numbers that are called "uncomputable" because there is no simple formula for finding their approximants. An example is Chaitin's constant.

There are, of course, other ways to represent numbers such as pi which are totally regular: that must be the case since it is computable. An example is
$$\frac 4 \pi = 1 + \cfrac {1^2} { 2 + \cfrac {3^2} { 2 + \cfrac {5^2} { 2 + \cfrac {7^2} { 2 + \dots } } } }$$
« Last Edit: July 14, 2023, 07:16:28 pm by helius »

#### TimFox

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #59 on: July 14, 2023, 07:15:31 pm »
From my early math studies (ca. 1965), I remember an example where the reader could group the digits in the decimal expansion of pi into discrete groups of 5 digits and compare the probabilities to poker hands played with a corresponding deck of only 10 cards.

#### SiliconWizard

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #60 on: July 14, 2023, 09:11:55 pm »
The number theoreticians just speak of uniform distributions of digits (not "randomness") because it has nothing to do with random probabilities. Every digit of pi is calculable from simple formulas: it has zero entropy.

Yes, that's an interesting question actually.
Yes, all decimals of pi are calculable iteratively. So in that regard, they are not "random".
Does the sequence of decimals of pi provide a reasonable pseudo-random generator though?

More generally speaking, if we admit that everything in the universe is a succession of causes and consequences, then "randomness" is just a fiction anyway.
But then comes the trickery of quantum physics and you don't really understand what a cause is anymore. Or randomness. Or just about anything, actually.
Small digression maybe.

Fun exercise: can you find irrational numbers that do not have a uniform distribution of their decimals?

« Last Edit: July 14, 2023, 09:13:38 pm by SiliconWizard »

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #61 on: July 15, 2023, 03:32:01 am »
Fun exercise: can you find irrational numbers that do not have a uniform distribution of their decimals?
We could construct a number that didn't use all 10 digits in its expansion.
For example, along the lines of Champerowne's constant, suppose we have a decimal sequence
1.101001000100001000001...
It's easy to see that this is irrational. Whether such numbers arise in other contexts, I do not know.

#### Nominal Animal

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #62 on: July 15, 2023, 03:33:32 am »
Interestingly, binary (and quaternary, octal, hexadecimal, etc.) digits of Pi can be calculated directly using Bailey–Borwein–Plouffe (discovered in 1995) and Bellard's (discovered in 1997) formulae.

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #63 on: July 15, 2023, 03:37:19 am »
Interestingly, binary (and quaternary, octal, hexadecimal, etc.) digits of Pi can be calculated directly
It's always interesting when there is a "closed form" for a function that is usually defined iteratively or recursively.
The Fibonacci function is a favorite of mine, which is always taught in programming textbooks, but never as a closed form.
Yes, it is possible.

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#### Nominal Animal

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #64 on: July 15, 2023, 06:31:48 am »
Yup:
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \text{where} \quad \varphi = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \psi = \frac{1 - \sqrt{5}}{2}$$
and $\varphi \approx 1.61803$ is the golden ratio, and $\psi = 1 - \varphi = \frac{-1}{\varphi} \approx -0.61803$ is its conjugate.  Funky!
« Last Edit: July 15, 2023, 06:33:48 am by Nominal Animal »

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #65 on: July 15, 2023, 07:53:52 am »
It's even possible using just integers
In fact that's true for any function defined as a recurrence formula, because it has a generating polynomial encompassing all of the terms of the sequence. The Fibonacci sequence just happens to be the most primitive recurrence.

#### SiliconWizard

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #66 on: July 15, 2023, 11:52:24 pm »
Proof there: http://mathonline.wikidot.com/a-closed-form-of-the-fibonacci-sequence

It's even possible using just integers

By that, I suppose you mean a closed form not involving any radicals, or any irrational numbers? Just integers, do you even mean no (non-integer) rational number either?
Would you care to share such a closed form?

#### helius

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #67 on: July 16, 2023, 02:25:46 am »
$$\DeclareMathOperator{\fibonacci}{fibonacci} \fibonacci(n) = \cases{ n & \text{if } n\lt 2\cr \left\lfloor \frac {2^{n(n+1)}}{(2^{2n} - 2^n - 1)} \right\rfloor \bmod 2^n & \text{otherwise}}$$

By the way, there's nothing special about base 2. It will work in any base, because as I said it's derived from the generating polynomial for the recurrence which is $\frac {1}{1 - x - x^2}$. In some sense this polynomial is equal to the entire Fibonacci sequence.
« Last Edit: July 16, 2023, 02:41:57 am by helius »

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#### SiliconWizard

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #68 on: September 04, 2023, 10:39:25 pm »
Sorry about the delay - yes the closed form above appears to be correct. It's rarely shown! The form with the golden ratio is much more common.

Interestingly (or is it?), from the generating function $\displaystyle s(z) = \sum_{k = 0}^{\infty} F_k z^k = \frac{z}{1-z-z^2}$ , with $z = \frac{1}{2}$, we get : $\displaystyle \sum_{k = 0}^{\infty} \frac{F_k}{2^{k+1}} = 1$ , while $\displaystyle \sum_{k = 0}^{\infty} \frac{1}{2^{k+1}}$ also converges to 1.
« Last Edit: September 04, 2023, 10:41:04 pm by SiliconWizard »

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#### Circlotron

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #69 on: September 05, 2023, 12:12:42 am »
Yup:
$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \text{where} \quad \varphi = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \psi = \frac{1 - \sqrt{5}}{2}$$
and $\varphi \approx 1.61803$ is the golden ratio, and $\psi = 1 - \varphi = \frac{-1}{\varphi} \approx -0.61803$ is its conjugate.  Funky!
And of course 1/phi = phi - 1 and phi^2 = phi + 1.
What is there not to like about phi?

#### SiliconWizard

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #70 on: September 05, 2023, 08:58:40 pm »
What is there not to like about phi?

I find it too irrational.

#### RJSV

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #71 on: September 08, 2023, 04:38:37 am »
Thanks DavidAlfa!   Of the '1212' mentions.   My (beloved CAT pet died on that date.
I'm pretty sure, something ELSE happened that dark day...(try looking up December 12,2000.)

This only goes to show, the incredible and coincidental links, with some of the non-mathmatical world.

And thanks, folks, for the detailed coverage of phi.

#### RJSV

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #72 on: September 08, 2023, 05:04:11 am »
Another example, of the linking to supposedly non-mathmatical things is '49' appearing in (various) family members phone numbers;  My family history has the Grass Valley, CA  / Nevada City Gold Rush, (i.e. '49'ers) derived from the 1849 Discovery of gold, that cemented the future fate of California, as a kind of wild ride experience.

Oh and I'm not very experienced in the formal mechanisms, of numerology, but often will 'test' out things, sequences backwards.  So that has connotation that a number like 948 would STILL be a qualified march, (to 49).
Thus I've recognized (phone area code) '494' as a 'double' hit.

A friend's birth date, 4-25, seemed an ironic match, almost '420', and that's a good match, for him.
Dividing that, by five, and you've got '85', divide again, to '17'.  So what, but that's in his address, and if you think about it, I actually just had divided (the 425) by '25', actually.
This game, for that friend, goes on and on, a relatively harmless but also maybe useless fact... But I'd have to say that, the coincidences in that particular person, the birth, and residence numbers have so many coincidental matches with that it's 'hilarious'
Much of it with a '711' slant.   His house at '1711' Blackbourne St.

#### RJSV

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #73 on: September 08, 2023, 05:19:23 am »
It can get a bit silly, I know, but as long as I'm doing study just for curiosity / entertainment.
So...also tried that little process, with 4-20 even though that's not his month-day of birth, but
420/5 = '84'.   A bit comical, as that person was born 1984...
Of course here, I'm not really sure why I've been dividing by five...

#### babysitter

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##### Re: Math Quirks:...Anybody (else) experienced strange Date coincidences ?
« Reply #74 on: September 10, 2023, 02:41:47 am »
How can you discuss this without hitting

This guy found the numbers for some physical/astronomical constants in the size of his bike!

BR
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