What fractional bases really mean

[SciKidSirish]

About the author

My name is Sirish and I am an author and youtuber. I have been making videos for about 5 years. I am in the 6th grade and I am 11 years old. I am currently writing my second book. I will be going to 7th grade this fall. here is an interview

Project

about fractional bases and what they really mean. By base, I mean a number system like binary or decimal. I am looking for mathematical guidance from an expert. My work is shown earlier in the "about the author" section.

Contact details

The best way for people to contact me is via the email subash.arumugham@yahoo.com.

Note

As a side note, I have Basic editing experience. I don't have experience with animating. Working with an animator too would be great, but not necessary. This summer I will be able to dedicate 7 or 8 hours a week at a minimum to this project. Also, a side note, I am not old enough to use discord according to its term of service, so I cannot be contacted there.

[follymath]

I'd be happy to help. I'm a software engineer, not a mathematician -- but I've been studying alternative bases and other exotic number systems as a hobby for many years. It all started by searching around for answers to exactly the kind of question you're asking. Over the years I've managed to stumble over some surprising insights. One in particular that may be interesting is a way to visually represent numbers in any base.

Here's a simple little desmos graph that will give you the digit coefficients for any number, in any base. This link should open with parameters set to b=2 and c=25, which gives a geometric representation of the number 25 in binary: 11001. Just know that the most significant digit is on the right, so you have to read off digits from right to left. I put the whole coefficient * base ^ place expansion into the labels to show exactly what each term represents to help make this clear.

The black path traced out by this "string" of digits is the numeral. Notice what happens as you change the value of c -- the value to represent. More interesting is what happens when you vary b, the base.

Since you're interested in fractional bases you could set the base's numerator and denominator separately, and set b to this fraction. This will let you vary the numerator and denominator independently, and would make it easier to recover nice, clean rational digits.

One idea worth exploring is what happens to a number when you represent it in base-(1/b) -- that is, the reciprocal of the base. So if you had base b = p / q, what happens to the digits of the same number when represented as b = q / p? Do you how the digits should just be reversed?

There are lots of other interesting numerical phenomena I could point out if you're looking for material.

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Just to give you a little intuition for how this works -- we can recover digit values from that goofy red shape defined by mod(c,x). The value of each digit is determined by the slope of the line between two points. There are a variety of interpretation choices available, different knobs to tweak, each associated with a distinct number system.

One thing to keep in mind though is that this same geometric interpretation applies much more generally, and works for any mixed radix system, and for all kinds of different choices of digit sets (b's complements, diminished complement, bijective bases, hyperbinary, and so on). To say nothing of nega-bases, imaginary/complex bases. I could go on...

There's so much more to say about all this, and what it all means, but I'll leave it at that for now...

Just be warned: when you ask "What do fractional bases really mean", this rabbit hole is deep! This is the reason I've yet to do anything with all the crazy I've found exploring this space of numerals, along with other related spaces.

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If this is useful or interesting to anyone else, get in touch -- I have a half a lifetime of material, including some (possibly) novel results, just collecting dust, or bit rot.

[SciKidSirish]

Thank you very much @follymath! I would love to work with you.

[SciKidSirish]

@follymath, what would be the best way to contact you?

[follymath]

Right here ought to do -- it's as good as email. You can ask any questions on this ticket, which will end up in my email anyway. ~I was also planning on joining the discord~ I joined the discord under this same follymath handle, though it sounds like you may be age-banned, which is ridiculous.

[SciKidSirish]

Thank you very much!

[SciKidSirish]

@follymath, when representing noninteger bases, how does the notation work out?

[follymath]

when representing noninteger bases, how does the notation work out?

Well, let's explore the space a bit. In the most general sense, a numeral in any base is just a list of coefficients, each associated with a specific "place value" -- fully expanded, each term can be thought of as coefficient * base ^ place. So the question of notation really comes down to what set of digits you want to be able to draw from. In particular, how many digits do you want to allow.

If you think of positional bases like a series of wheels on, say, a car odometer, this could be thought of as how many numbers are painted on the faces of these "radix wheels". This usually gets munged together with the separate question of what value you want to use for base in that formula, but these are separate, somewhat related choices.

The point is you have some flexibility in what you want to consider a valid digit in a fractional base. You can restrict your choice of coefficients, to, say, specific integers. But these restrictions will represent different choices on the geometry I pointed out. In the set of choices I'd made in the example I sent above, you might notice that negative digits (digits sloped downward) are allowed -- though only values with coefficients strictly greater than -1. It is not possible to reach these goofy slightly negative digits for integer bases, so I hadn't futzed around with it to change it, but there is a slightly different take where you can just kind of roll the cone of acceptable digits slightly, from the range (0,b-1] up to [0,b). This is probably closer to what people think of when they think of fractional base numbers, so I'll throw something together as soon as I get the chance.

(And sorry for taking so long to reply -- I've been on vacation this week and there's been a lot less downtime than I'd anticipated!)

There are some other choices available -- each choice gives it's own number system entirely. For example, the range (0,b] is a reasonable candidate to represent bijective bases (bases without a 0 digit), but for any real number base. But more than likely, when you're representing a number with a fractional base, you're going to need digits which are also fractions. Which means "notation" can be pretty loose -- a set of digits in a list, for instance. Though if you're just looking at fractional bases, I'm pretty sure you can cheat a little and put all your digits over a common denominator. I did a bunch of explorations in this some years back, and will revisit. But just pick some examples and look at the fractions that comes out. It'll always be a list of "digits". I'll try to adapt some examples to write out the number itself in a nice, clean way somewhere.

Just pick a handful of examples and see what they look like. Pick a large integer and express it in base 3/2, base 2/3, base 10/7, base 7/10, etc. Pick a simple decimal, like 17.8, and try it in the same bases. Start simple and build from there.

[follymath]

Here's a quick example where I at least cleaned up the definition of bases to split it into a numerator and denominator, and fixed it so the representation takes the split up base into account -- so the digits show up as cleaner fractions, rather than messy decimals.

I'll work on an alternative that gets rid of negative digits, but this ought to at least be a lot easier to read results from.