Why can bases of number systems not be rational numbers?

Question

This is a question that keeps bothering me: much of mathematics was created by extending rules to new domains e.g. negative numbers were created by consistently extending arithmetic beyond positive integers. However, with number bases it seems impossible to do this. Number bases are integers, but could they be rational numbers? I've had previous discussions about this, see here. One problem can be seen with this example: in base 2.5, 0.22 is greater than one. That's not what we want! What, if anything, can be done about this?

EDIT: To put it another way - if every positional number is less than any digit to its left, why doesn't this apply to fractional base positional numbers?

Answer 1

You could "fix" the problem (partially) by further limiting the legal digits. For example, you might say that base $a$ is not permitted to use any digits greater than $a - 1$. So base $2.5$ can only make use of $0$ and $1$, and $0.22$ is therefore not a valid base-$2.5$ number. Unfortunately, this has a side effect: numbers like $0.8$ (in base $10$) have no base-$2.5$ representation at all.

We could try to strike a middle ground instead, by limiting combinations of digits - for example, in base $2.5$, we could say that $2$ must always be followed by $0$ or $1$. Then the largest number obtainable without crossing the decimal point is $0.212121\ldots$, which (after some quick infinite summation) evaluates to exactly $1$. In bases between $2$ and $2.5$ we would need to alter the rule; as we get close to $2$, we must demand longer and longer strings of $0$'s following every $2$. But a rule should exist for every base (don't quote me on that, I haven't worked out the math) and it shouldn't be too hard to establish general definitions for it.

Answer 2

Not only rational but even irrational numbers can be number bases if one so wishes. Bergman published a paper called Number System with an Irrational Base in Mathematics Magazine back in 1957. It represents numbers as sums $\sum_{i=-\infty}^\infty a_i\varphi^i$, where $\varphi=\frac{1+\sqrt{5}}2$ is the golden ratio, and $a_i$ are $0$ or $1$ (only finitely many $a_i$ with positive $i$ are non-zero). Bergman denotes $\varphi$ as $\tau$ and calls it the "Tau System".

This presentation is non-unique, but Bergman derives nice transcription rules (related to the Zeckendorf decomposition of integers into Fibonacci numbers) which make conversions tractable, e.g. $2=10.01=1.11$. He also developed paper and pencil algorithms for addition, multiplication and division. Even so, already expressing $1/2$ is not so straightforward, and "with 1/10 I had to work it out 5 or 10 times before I got the correct answer, as there is much room for error", as he confessed.

Obviously, you have not heard of this at school, and there is a reason, which Bergman himself mentioned:

"The Tau System has a good many other interesting and unusual characteristics, and investigation by the readers of some, such as the frequency, occurrence, and nature of numbers with a $1$ in the units column (when in simplest form) might prove interesting. I do not know of any useful application for systems such as this, except as a mental exercise and pastime, though it may be of some service in algebraic number theory. For instance, the numbers expressible in the Tau System in terminating form consist of all the algebraic integers in $R(\sqrt{5})$ and some of the properties of numbers in this and other systems might correspond to facts about associated fields." [$R(\sqrt{5})$ is $\mathbb{Q}(\sqrt{5})$ in modern notation.]

In the 60 years since then such number systems did find some enthusiasts. Stakhov's book Mathematics of Harmony has a whole chapter 9.1 called Numeral Systems with Irrational Bases, which promotes them (mostly using quadratic surds for bases) with great pomp characteristic of this author.

In fairness, Bergman's and similar systems did find some applications beyond pastime and number theory, like designing self-correcting analog to digit converters and noise tolerant processors. But overall they are underwhelming, and it remains a niche subject with few people working on it.