- winthroptsmith
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Here is a Pi Day problem even though the day (3/14/15 in the U.S.) is long past. I posted it to my blog on March 14 but no one has solved it yet.I had been thinking about how the sequence 3.14159... is not just a reflection of pi, but also of the arbitrary choice of base ten. This led to a new little puzzle:What is 30.12120111??

- Traden4Alpha
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LOL!It's a useful constant for those with pi fingers who are visiting Earth.

- winthroptsmith
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Ha! Glad there could be such a practical application.How did you get it?

- Traden4Alpha
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My first line of reasoning was that the presence of no digits greater than three implied the base must be greater than three but probably not much greater than 3. I tried looking at the decimal value of this string interpreted as base-4, base-5, base-6, base-7 but nothing seemed especially pi-related.After that fruitless search I almost posted a flippant reply about 10 being equal to pi in base pi. For grins, I tried this string in base-pi, got 9.9999903587, and got quite a chuckle!P.S. Base-pi has some weird properties such as non-uniqueness. For example pi in base-pi is equal to both 10.00000... and 3.0110211100202211.....

- winthroptsmith
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Thanks. I wondered how anyone would solve it.I was going to say that an irrational base would also be strange in that all rational numbers, and almost all irrational numbers, would require a non-repeating infinite sequence of digits. But even rational bases are limited: all irrational numbers, and most rational numbers, demand an infinite number of digits with a rational base. It's just that rational numbers with a rational base will have a repeating pattern.

- Traden4Alpha
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Yes, the issue of repeating vs. non-repeating patterns is quite fascinating.Yet non-integer bases have another trick up their sleeve -- the digit strings that equate to a given value are not unique. In constructing the digit stream, choice points occur in which two possible digit patterns can lead to a valid stream.For example, pi in base pi can be written as:10.00000000... or3.011021110020221130001... or 3.003333310120012000222... or2.312200022220201202212... ormany other (potentially infinity other) strings.During construction of the digit stream, there's some rules for whether a "1" digit could be replaced by a "03" digit pattern and for whether a "3" digit could be replaced by a "23" digit pattern. In the case of irrational numbers in base pi, the density of digit choice points seems quite limited in a statistical sense (perhaps about 1 choice per dozen digits) but that does not imply that certain irrational numbers might not offer a certain pattern of choice points such that a carefully chosen pattern of digit choices induces a repeating pattern of digits.

Last edited by Traden4Alpha on May 17th, 2015, 10:00 pm, edited 1 time in total.

- winthroptsmith
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Very interesting.Rational bases can also lead to non-unique representations, such as 1.0 = 0.99999...But here there are only two patterns, not an infinity of them.Bonus question: if there are infinitely many ways to write pi in base pi, is this a countable infinity? (not that I know)

- Traden4Alpha
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QuoteOriginally posted by: winthroptsmithVery interesting.Rational bases can also lead to non-unique representations, such as 1.0 = 0.99999...But here there are only two patterns, not an infinity of them.Bonus question: if there are infinitely many ways to write pi in base pi, is this a countable infinity? (not that I know)Unless something "strange" happens, the set of ways to write pi in base pi should map to the set of real numbers. The set of choices made in constructing each variant is a binary string that has one digit for each dozen or so digits of pi. Thus, each real number encodes a different way to write pi in base pi. Thus, the answer seems to be no but I've got no proof.

- winthroptsmith
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Interesting that not only are pi's digits infinite in base ten, but that pi in base pi might connect to an uncountable infinity.Here's a follow-up blog post, with credit to Traden4Alpha.

- Traden4Alpha
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QuoteOriginally posted by: winthroptsmithInteresting that not only are pi's digits infinite in base ten, but that pi in base pi might connect to an uncountable infinity.Here's a follow-up blog post, with credit to Traden4Alpha.Thanks! BTW, nice blog!

- winthroptsmith
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You're welcome. Glad you like the blog.

QuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: winthroptsmithInteresting that not only are pi's digits infinite in base ten, but that pi in base pi might connect to an uncountable infinity.Here's a follow-up blog post, with credit to Traden4Alpha.Thanks! BTW, nice blog!dbl

believe the non-integer base/radix has appeared more frequently under the name of "beta-expansion", which was introduced first in late 50's, e.g., the golden ratio base has been extensively studied and used in A/D conversion. ExSan should be interested in looking at complex bases, since they are known to generate 2-d fractal geometry (dragon curves). one good thorough reference is Knuth's TAOCP (4.1 in Vol 2).regarding whether the number of representations of e.g., 1 in a certain base q is countable or uncountable, this question has been solved for 1<q<2, where the digits can only be 0 or 1. except for a class of numbers (e.g., Pisot numbers), in almost every base, 1 has a continuum of beta-expansions. but there are quite remarkable exceptions like Komornik-Loreti constant, which is transcendental and 1 has a unique expansion in this base.

Last edited by wileysw on June 26th, 2015, 10:00 pm, edited 1 time in total.

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