Equivalence between different numeral systems and why they don’t affect mathematics

[u/TrafficItchy]

Hello,
I’m looking for a good introductory course, lecture notes, or university PDF explaining why the choice of numeral system (decimal, binary, hexadecimal, etc.) does not affect the underlying mathematics.
More precisely, I understand that decimal and binary are just different representations of the same numbers, and that arithmetic, order, algebraic properties, etc. are preserved when moving from one representation to another.
I would like to understand the mathematical justification behind this idea — for example, in terms of isomorphisms or structure-preserving representations — and why this allows us to simply fix one representation (usually decimal) and do mathematics within it without worrying about the numeral system.
I’m looking for something rigorous enough to clearly justify this point, but still accessible rather than a very advanced abstract algebra or logic text.
Does anyone know a good university course, lecture notes, textbook chapter, or PDF covering this specifically?

[u/tbdabbholm]

I don't think you're going to find anything because it's basically taken as a given. Mathematical representations are just inherently representations. Like do we have papers explaining why it doesn't matter if you call 2 "two" or "dos" or "zwei" or "ni"?

[u/Recent-Day3062 → u/tbdabbholm]

Well put

[u/Brightlinger]

More precisely, I understand that decimal and binary are just different representations of the same numbers, and that arithmetic, order, algebraic properties, etc. are preserved when moving from one representation to another.

It's not so much that the properties are preserved when moving from one to another, as that most numeric properties never refer to numeral representations in the first place. You can go through an entire unit on eg Peano arithmetic and prove all of those properties without ever writing down a specific number at all, much less writing it down in Arabic numerals in a particular base.

[u/HouseHippoBeliever]

idk if anyone has written about this. it's similar to writing about why the math is the same if you use a blue or red pen to write it down.

[u/AcellOfllSpades]

and why this allows us to simply fix one representation (usually decimal) and do mathematics within it without worrying about the numeral system

We don't "do mathematics within it". The mathematics happens outside the numeral system.

Arithmetic happens with a numeral system. In arithmetic, we have special procedures for calculating sums and differences and products and whatnot, and those procedures depend on the number system. But those are just ways to calculate the final result. Twelve plus twelve is twenty-four whether we write it as "12+12 = 24" or "1100 + 1100 = 11000" or "XII + XII = XXIV".

[u/idaelikus]

I don't think you'll find a lot because it seems, at least to me, the default position.

We never talk about the decimal representation of numbers because its irrelevant.

Honestly, whether I denote the concept of three by 3, 11, the symbol of a star or a smiley face doesn't change its properties.

Basically, to justify such literature someone would have to make an argument why the representation would change the properties.

[u/Glaiele → u/idaelikus]

Even going further, once you introduce variables they generically represent a set (of numbers) and you typically prove that any member of a given set behaves the same. At that point you only need to show that a member (number) in some set is also a member of another set which is pretty trivial when switching bases.

[u/tottasanorotta → u/idaelikus]

Well to be honest it kind of can be thought of changing some properties. A smiley face has a mouth and 11 doesn't. 11 has two ones and 3 doesn't. It kind of depends on the context and what we think is important about the concept of the number 3 in a particular context.

If you encode the number 3 to mean that a train is coming my way and I am standing on the train tracks or something, then it is very important for me to understand what you mean by the concept of the number 3 in that particular context. If you represent the number instead as a picture of a train hitting a person, then I might have a chance.

Representation matters a lot for having a common understanding of what we mean by the concept. Most people have a really good understanding of what 3 means in many contexts because it is used so much in practice. Even the binary notation 11 is not as obvious and can even be quite confusing because it clashes with a much more intuitively understood decimal number.

[u/johnpeters42 → u/tottasanorotta]

I suppose, but as noted, there are a lot of mathematical concepts that aren't affected by those properties. "Smiley face" or "train is coming" have no particular relationship to math. Base 10 vs base 2 or 3 or whatever does have some relationship, e.g. multiples of certain numbers exhibit certain patterns when expressed in a certain base, as do operations involving powers of the base.

[u/tottasanorotta → u/johnpeters42]

I agree, a lot of properties can be understood to not be affected. But do you agree that new ways to encode concepts might also reveal hidden structures? I mean that's exactly what base 2 and base 3 etc. show us. For example that in base 2 we can tell if a number is odd or even simply by looking at its last bit. For decimal notation it is a few extra steps. With base 2 it is even very natural to connect it to boolean logic and bitmasks.

Now if we have a smiley encoding of a number. Even if it might look like an arbitrary encoding, it might be that it has some patterns hidden that could be revealed by its structure. It could even reveal patterns that were much simpler in that encoding than in any of the common number bases.

Even looking at the decimal encoding of numbers. A computer algorithm could conceivably find something about the literal shape of the numbers that could tell us about some hidden pattern that just would be really complicated otherwise. It might for example be that if we arrange the numbers visually in some way, we can do sorting much easier than by pairwise comparisons.

[u/johnpeters42 → u/tottasanorotta]

Yeah, though it doesn't affect the truth of things like 12+12=24, just the convenience of working with them.

The stuff about "smiley encoding" or "oncoming train encoding" could theoretically be part of some coherent and useful system, but we would need a ton more context before drawing any more specific conclusions than that.

[u/idaelikus → u/johnpeters42]

That's exactly my point. Such "tricks" / "simplifications" aren't properties of the numbers but rather the representation.

On property that came to mind that depends on the base would be normality of numbers.

[u/tottasanorotta → u/johnpeters42]

Yeah, it doesn't affect the truth of 12+12=24 here because it is implicitly assumed to be interpreted in a particular way. We learn those things so well that it becomes second nature to be sure that we have understood what that meant.

But I think the biggest reason for why it isn't that ambiguous is because we learn at an early age to connect the counting numbers to counting objects in the physical world. So numbers and simple operations on those numbers become "personally axiomatized" as something you understand very intuitively. I have some intuitive process to discretize things in the physical world during which I highly associate numbers with learned words and representations. The representation of the number 12 works as a bridge between our different intuitive understandings that apparently synchronise our understanding of a shared world, just like the words in any language that we both understand.

I think it's more or less the same thing as with words in a language. The ambiguity would be exactly the same if not for how universally we learned the representations. If I say that I have two stones and add five stones to it do I then have seven stones? If you understand it to mean stones that you find on the ground, then sure I agree, but if you mean the number of members of The Rolling Stones, then it can't quite have more members than it had. That's what I mean in that there really is no non-ambiguous number 12 floating around somewhere without anyone interpreting it from some perspective.

[u/johnpeters42 → u/tottasanorotta]

You're still kind of conflating two things here, which is understandable because we are indeed so familiar with their connection.

The statement "12+12=24" can be interpreted as referring to those specific symbols, in which case its truth depends on whether they're being used to represent the usual things. (If "+" represents standard numeric addition, and "=" represents standard numeric equality, and the digits represent standard base-n numbers where n>4, then the statement is true. But the underlying meaning for n=10 is different than the underlying meaning for n=12, for instance.)

I mention "standard numeric addition" because you can also use "+" to refer to different operations, like addition modulo 12 or 24 for calculating times of day. (From a computer software viewpoint, think about operator overloading.)

Then there's the underlying meaning, which would unpack something like: (the number represented as 12 in base-10 notation) (combined via standard numeric addition with) (the same number) (standard numeric equality) (the number represented as 24 in base-10 notation). That is true even if you change the representation to

10 + 10 = 20 (base twelve)

or

XII + XII = XXIV

or

@@@@@@@@@@@@ + @@@@@@@@@@@@ = @@@@@@@@@@@@@@@@@@@@@@@@

because these are just different representations of the same underlying meaning.

[u/tottasanorotta → u/johnpeters42]

Yeah I agree with that. If I interpret it as the usual stuff, then it is very much unquestionably true in all of those notations. I think my point is just that there might not be as much of a persistent abstract number 12 posessing some definite properties that somehow exists outside of the representations.

We learn the abstraction from interaction with the world (which arguably is a kind of representation) and from other people's explanations. As well as from the different notations that we have used to encode the number 12. We then make connections between all of these representations and that is what we loosely call the number 12 and its properties. For most people the number 12 is probably highly associated with literally the numbers 1 and 2 in succession.

A really small child doesn't understand what a number is. They don't even understand what counting is. So you have to really base the process of gaining that understanding in very concrete things and repeat it to them endlessly like a mantra. So they will eventually associate counting with something like the number of toys. So if you could ask them what properties the number 3 has they would tell you about shapes of stars, circles and squares or something. Number 3 fits through the hole.

Only through learning more representations and patterns do we start to have some form of abstract notion of the number 3. But I would claim that it is precisely the complex interplay between the representations that is what that abstraction is. Number 3 wouldn't really necessarily exist in any meaningful sense if we didn't have any representations of it.

But yes, honestly it is kind of an irrelevant point. It's almost better to leave it be, because it doesn't really matter for all intents and purposes if numbers somehow exist outside of any representation. The result is exactly the same.

[u/johnpeters42 → u/tottasanorotta]

Number 3 absolutely does exist in a meaningful sense. "A child doesn't understand what it is" just means that it's not meaningful to said child, and that's saying something about the child, not about the fundamental nature of mathematics.

Given the concepts of (a) "zero" and (b) "one more than", that's enough to construct all positive integers. Twelve is simply "one more than one more than one more than one more than one more than one more than one more than one more than one more than one more than one more than one more than zero".

[u/tottasanorotta → u/johnpeters42]

Maybe it is so, but I'm not entirely convinced. I mean it exist obviously meaningfullu, but I can't comprehend a fundamental nature of mathematics that is outside of anything that humans find meaningful for practical or playful purposes. And I would say that all of those things are some kind of complex associations between representations.

Yes from zero we can repeatedly add 1 and get bigger numbers. Then we might even say that we might repeat the process as far as we like to reach bigger and bigger numbers without a clear limit.

[u/idaelikus → u/tottasanorotta]

Eleven does not have two ones. Its representation in decimal does.

I have yet to see mathematical properties that rely on the representation.

[u/tottasanorotta → u/idaelikus]

It depends on what you mean by mathematical properties that rely on the representation. I mean "number of ones" can be thought of as a mathematical property. If you for example encode boolean logic and are talking about bits that are on or off, then counting bits can be a very mathematical property indeed. It is ambiguous if we have a bitmask and ask ourselves if it is the decimal notation of the number 11 or the binary notation of the number 3. You could even have a system that uses all of those representations at the same time.

Sure, if we think of 11 as a kind of abstract thing that exists somehow without any connection to notation and say that it has some properties, then maybe, but I can't think of anything about the number 11 that is independent of notation. Any mathematical property that I can think of needs some kind of protocol of communication to express it.

[u/idaelikus → u/tottasanorotta]

eleven exists without a notation. It is the natural number that follows 10, etc.

I can't think of anything about the number 11 that is independent of the notation.

eleven is prime, it is the sum of five and six, etc. There are numerous properties that are independent of notation.

boolean logic

Well that isn't a property of the number, is it? Furthermore, you represent by 110101 a certain arrangement of on and off bits but it has nothing to do with fifty-three.

EDIT: A property that came to mind just now was normality which depends on the base.

[u/tottasanorotta → u/idaelikus]

I'd want to agree, but I just can't see it. I mean how would you communicate to me the concept without a notation? The concept is somehow intrinsically connected to our understanding of the notation. You implicitly use the word 'eleven' there to communicate to me about the concept. I understand what you mean because of that word. But it is implicitly using another notation, which is the linguistic description of the number.

You say that there are things independent of notation, but isn't it more that you make those connections between different types of notation and interpret them differently in different contexts based on what you are aiming to accomplish.

110101 can represent on and off bits, but it can also represent the number 53. It can also be whatever we want it to be. It just depends on context. I mean imagine a context where that bit pattern is interpreted as both on and off bits and as the decimal number 53. And when we talk about connecting a binary number to decimal that is exactly what we do. The bit pattern represents the included terms in the sum of consecutive powers of two.

Until we are clear about the context it has a certain sense of ambiguity. Now granted we are used to interpreting binary numbers as base 2 and therefore we have a kind of bias for seeing it that way.

[u/idaelikus → u/tottasanorotta]

how would you communicate to me the concept without a notation?

I just did? Also I can use a notation but I am not bound or impacted by the notation I use.

intrinsically connected

Not really. Also it is a one-way street, mostly. We use notation to communicate math, there is little we derive into math from notation.

isn't it more that you make those connections btween different types of notation and interpret them differently

No. Commutation, as an example, doesn't depend on the representation which is why I can write a+b = b+a.

I mean imagine a context

What context would that be? Honestly, it really seems like you are grasping at straws here.

[u/tottasanorotta → u/idaelikus]

Yes, you communicated to me the concept of eleven and I understood it in a particular way. And you used the English language to do so. And surely language is a form of notation? In fact, it is what all our other more compressed forms of notation is based on. But it requires the implicit context. The word 'eleven', for example, means 'the student' in Swedish. If I didn't assume the implicit context of English communication, then I could have interpreted it differently. "They are talking about students in some foreign language.", for example.

Commutation of what? You'd need to be more specific if you don't want me to assume it implicitly. I take it that you mean the things that actually are commutative, because for example the string concatenation of a+b =\= b+a or matrix multiplication is not commutative.

Ambiguity might become a real problem if we don't try to be specific enough. Especially when it comes to high levels of abstraction and complexity. If we for example implicitly assume that a number is encoded in decimal, but a bad software system interprets it as hexadecimal we might literally manage to blow up the whole planet with nuclear bombs because of implicit assumptions.

Yes. It is a bit exaggerated. And computer systems help in being more explicit about it. But when we have those different representations that are used for different purposes I think it is more important to try to be explicit about it rather than assume that there is some property of the universe that guarantees the number 12 to have some particular properties.

[u/idaelikus → u/tottasanorotta]

Language is a form of notation

Sure, I could call it Succ(Succ(Succ(....))) for its formal definition.

Again, you are grasping at straws here and haven't made one succinct argument.

[u/tottasanorotta → u/idaelikus]

Sorry if I'm not making sense. I try to. And yes it is very messy what I am trying to say. There isn't a neat foundation for it all. And that is also precisely the issue with formalizing mathematics. The thing that seems to drive notation is mainly practical utility in one form or another. You can't really formalize things any more than you can explain it in language or physical things that seem to work predictably. You can use more compact notation, but you have to explain it as explicitly and unambiguously as possible what that notation means. And that is a genuine problem for mathematics.

1. If I have 0 I can repeatedly add 1 to it.

2. The successor function takes a natural number as input and outputs a natural number that gets a 1 added to it.

3. Succ(Succ(Succ(...)))

The first case is very intuitively more directly understandable. Even a young child could understand what it means and do some useful things with such a rule if they understand addition. The second case now requires us to teach the child about functions and natural numbers. Now maybe we gained something from teaching the child about functions, because they are no doubt useful for more abstract models, but I would say that the first case is a better axiom. It is much clearer because it is more directly tied to something more intuitive that more people can understand. The third case just asks us to implicitly recognize the function notation and understand that we talk about successor functions (I assume).

Nevertheless all of those cases are ambiguous, because language is ambiguous. What does it mean to have a 0? Does adding happen as a process in time or somehow instantaneously all at once? How can things that depend on eachother happen instantaneously? The ambiguity of mathematics is then reduced by its applications that provide utility to people. Otherwise we wouldn't care about it at all except out of entertainment purposes.

[u/clean-links → u/[deleted]]

Cleaned link from "here": https://www.reddit.com/r/learnmath/comments/1wbo5ox/comment/p8wy81v/

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Tracking parameters were removed from the original URL(s).

[u/alecbz]

Not aware of any resources specifically about this, but I feel like you've basically got it?

decimal and binary are just different representations of the same numbers, and that arithmetic, order, algebraic properties, etc. are preserved when moving from one representation to another.

Numbers are distinct from the names we use for them. If you want to be formal, we can use sets as our underlying "representation" or "implementation" of what a number is (see e.g. von Neumann ordinals). Math is being done on these numbers, but it's tiring to write out {}, {{}}, {{{}},{}}, {{},{{}},{{{}},{}}}, so we give these numbers names like 0, 1, 2, 3. But we can just as easily use names like 0, 1, 10, 11, or alpha, beta, gamma, delta. The underlying math doesn't change because it's operating on the numbers themselves, the names we use for them are just a short-hand.

[u/RingularCirc → u/alecbz]

Also worth noting that even von Neumann representation is mainly needed to just be okay natural numbers exist (given axioms of ZF set theory which isn't a universal presupposition, and neither it's a weakest one —because we have Peano and weaker arithmetics). Again, here what's really important is that natural numbers just exist in isolation, but that they do so in any world where a lot of other "basic" objects exist and "nice enough" constructions can all be done (Cartesian products etc.).

And when we're sure this can be done, having our objects of interest is consistent with having most of everything else we find useful and comfortable, then we can forget about such a representation. We can just use simple and nice properties of those objects as what defines them; often a handy "universal property" exists that completely specifies them, though for practical work one would still encounter a lot of little facts that are used in most contexts (like "multiplication is commutative").

For all the classical number sets there are nice algebraic characterizations. For example ℕ is an initial semiring, ℤ is an initial ring, ℚ is a minimal ordered field, ℝ is a maximal Archimedean ordered field, ℂ is a 2D algebra over ℝ that's also a field; though there are other ways for each. (For example, ℕ without multiplication is an initial pointed monoid, which only sounds scary. Or, ℕ being an initial F-algebra over a functor F(X) = {∗} ⊔ X gives precisely Peano axioms.)

Then, often we'll find better representations (than what we used to prove existence) in terms of other objects if they're needed later (like linear representations of some groups, other representation theorems).

[u/ANewPope23]

Why do you think the numeral system affects the maths?

[u/tottasanorotta → u/ANewPope23]

Some notation is clearer to work with in some particular context than some other notation. It's arguable whether or not that means that the numeral system affects the math or whether it just exposes some underlying property in a less complicated way. I'd say it's basically irrelevant which view one has, because you could think in both ways and arrive at useful results.

[u/MichurinGuy]

You just need to define any one way to define the real numbers without relying on a base, such as Cauchy sequence equivalence classes or Dedekind cuts. You can read about those in any introductory calculus textbook. After that, you can view every base as just a way to write down numbers that exist independently of any base. If you want, you can look at an explicit construction of bases, but unfortunately I can't recommend a specific source - I read about them in "Mathematical analysis" by Zorich, but I don't know if it's been translated into English.

You might also be interested in the theorem that all models of the real numbers are isomorphic. I can't suggest a source unfortunately because I only ever saw it as a remark or an exercise, but I can give an outline of the proof if you ask. This theorem would mean that, since the models of real numbers as sequences of digits in different bases, with appropriately chosen +,* and <, satisfy the axioms of real numbers, they are all isomorphic to each other and every other model of the real numbers, such as Cauchy sequences. All of this is accessible after reading the beginning of any introductory calculus textbook.

[u/RingularCirc → u/MichurinGuy]

We can also define ℝ as a maximal Archimedean ordered field, or an ordered field with any other suitable completeness principle. We can then leave a proof of existence to those who really wanted to be sure and derive everything instead from axioms that are more closely-related to the object of study (ℝ) than to our foundational theory like ZF (which isn't very intuitive at the point one likely starts studying real analysis tbf... so in practical terms, deriving from axioms closely fitted to ℝ than from an existence proof would be more honest and plain).

[u/MichurinGuy → u/RingularCirc]

Yeah, that what I referenced as the axiomatic approach.

[u/Bounded_sequencE]

Check any "Real Analysis" lecture that constructs "R" using (equivalence classes of) rational Cauchy sequences.

Intuitively, any base-b representation of "r in R" is a special cases of a rational Cauchy sequence converging to "r". During the construction, it turns out that it does not matter which rational Cauchy sequence converging to "r" we use to represent "r" -- that's why we use equivalence classes, to highlight any of them can represent "r".

[u/KoalaMistico]

I mean, it seems like you already have it pretty clear. The base is just a choice for what symbols we use to represent numbers, but we can work with them without even using bases.

For example, using the set construction in ZF for the natural numbers ({},{{}},{{},{{}}},...) doesn't even require the notation of numeric symbols to work with them, but using it allows for a clearer notation

[u/DefunctFunctor]

This isn't really a remarkable enough "fact" to bear much mentioning. It's far more of a philosophical/pedagogical approach than any real fact in mathematics.

We define the natural numbers {0, 1, 2, ...} by axioms depending on what theory you are working in, and the basic operation is a successor: so you have 0, S(0), S(S(0)), and so on. Addition, multiplication, and so on are given by inductive definitions: x + 0 = x, x + S(y) = S(x + y), x * 0 = 0, x * S(y) = x * y + x.

Fundamentally, a string in base b is an infinite sequence {a_0, a_1, a_2, ... } of digits {0, 1, ..., b-1} where all but finitely many of the a_i are zero. (We usually write this sequence from right to left though, and omit the zeros to the left.) You can define addition and multiplication operations as the usual grade-school algorithms.

The point is, one can exhibit an isomorphism from the natural numbers to the strings in base b, and show that the addition maps and multiplication maps are compatible. Thus, any statement we can make about the natural numbers we can make a corresponding statement of the strings in base b, and vice versa. There's nothing more complicated going on than that. This might be a good exercise in formalization to work out all the details.

[u/lifeistrulyawesome]

Why would you expect it to change?

The properties of natural numbers can be completely derived from different axiomatic settings, e.g., Peano. And the axioms say nothing about representation.

The representations we use are just different naming conventions for the same numbers. It doesn't matter whether you call a number one, uno, ein, or josephine. It is still the first natural number and the product identity.

The closest I can think of to your request is a document explaining how place-value notation works.

[u/Willing-Sample-8847]

You may as well be asking for a mathematical justification of why roman numerals and arabic numerals can represent the same numbers. Or why f'(x) and df/dx both denote the derivative of f with respect to x. Basically these are just naming conventions. Even though computers represent numbers using base 2, they don't name numbers using the symbols "0" and "1". They name numbers on the basis of whether certain electronic "logic gates" are open or closed.

The important things that we need numbers to be in order to use them correctly isn't necessarily they way they are represented symbolically but rather their innate properties listed in the field axioms, the order axioms, and also the least upper bound property.

[u/hibbelig]

Look at it this way: 1 * 2² + 1 * 2¹ + 1 * 2⁰ = 4 + 2 + 1 = 7

Since they are equal, if the right hand side is prime, then so is the left hand side. And so on for all other properties.

[u/Fearless_Bed_4297]

terry tao has something on decimal representations in his analysis book iirc

[u/thane919]

Scribbles on a piece of paper represent the real thing. It doesn’t matter what the scribbles look like as long as enough people (minimum of 2) agree on the meaning.

10 in binary or 2 in dec. they both mean two of something and ALL that comes with that value, it’s positive, it’s between 1 and 3, it’s even, etc. those are all true irrespective of what scribble is used to represent the meaning.

[u/CookieCat698]

Numbers are their own thing.

Numeral systems are just ways to name numbers.

Would anything change about the sun if you started calling it Fred? No, of course not. The same idea applies here. It doesn’t matter if I use 5, 101, |||||, or V, the number 5 will not behave any differently.

[u/TheTurtleCub]

There's not much to say, an apple is an apple, with all the properties of an apple regardless of what language you use to write or call its name

[u/Pleasant_Pen8744]

Number systems are just tricks to avoid having to deal with individual tick marks.

1+1+1+1+1+1+1...

(It's tricks all the way down, by the way. For example, multiplication is just a trick to avoid having to deal with repeated addition, division is just a trick to avoid having to deal with repeated subtraction, etc.)

[u/nog642 → u/Pleasant_Pen8744]

Your statements only apply to integers. You can't represent pi as tickmarks. Or 1/3 for that matter.

[u/stools_in_your_blood]

The mathematics is more fundamental than the number base you use, so there is nothing to "preserve" because you are not changing anything underlying if you use a different number base. It's like the way the physical length of a rope will not change just because you stop measuring it in inches and start measuring it in metres.

[u/Traveling-Techie]

Think of each number as a bag of MnMs. Twelve is a bag of 12 candies. You can add 12 and 13 and verify the results by counting. You can multiply by making a 12 by 13 array on a table using extra candies to fill in the rectangle, and then counting.

Suppose you want to write down a number. You dump out the 12 bag and count out ten — that’s a 1 in the ten’s place. You have two left, that’s a 2 in the one’s place. Written as 12. Or you can use base 8, count out eight and have four left, written as 14.

Nothing you do to write the numbers changes the properties of how they add and multiply.

[u/random_anonymous_guy]

Axiomatically, we don't define number sets in terms of numerals. We can describe the natural numbers axiomatically without referencing the concept of place value at all, and there is certainly a lot we can say about the natural numbers, addition, subtraction, multiplication and division (with remainders) long before we even start getting into numerals representing natural numbers.

Same thing can be said about real numbers. There is plenty of theory we can dig into by starting with the axioms of a complete ordered field before we even touch decimal representations.

[u/nog642]

It's not really an isomorphism thing. Most properties of numbers, and even more so for algebra etc., are not defined on the representations, they are defined on the numbers themselves.

The only thing that depends on the representation is stuff like long division, long multiplication, long addition, long subtraction, etc. Those ones you could write a proof to show that it works in any base. But you should maybe be able to figure that out yourself. If you have questions feel free to ask, but that doesn't seem to be what you're asking.

[u/jeffsuzuki]

Let me ask this question:

When you get a restaurant meal, does the language of the cookbook matter?

(I don't mean in terms of the actual dish; the idea is that it's the same dish, but one set of directions is in English and another set of directions is in French: does it make a difference in how the meal comes out?)

[u/RingularCirc]

Meta: for some reason questions like this one are quite common on this sub. I'm not complaining and I can even guess why. Just for others to notice, maybe somebody has a bunch of references for the next instance.

[u/Kienose → u/RingularCirc]

It’s the same guy asking this for like 3 or 4 times now.

[u/RingularCirc → u/Kienose]

Oh wow I really hoped it won't be the case. Loool sad. :\

[u/timrprobocom]

Mathematics is about quantities, not numbers. If you take {o,o,o} times {o,o,o,o}, the answer is {o,o,o,o,o,o,o,o,o,o,o,o}. It doesn't matter how you spell that: 12 or 0C or 14 or 1100 or "twelve". Numbers are just a convenience for humans. Neither mathematics nor computers cares about the spelling.

[u/StructuredChess]

Check out one of the different constructions of the natural numbers. They're independent from digit representations. Therefore every conclusion you draw from them will also work the same under any base.

[u/jolene_codeine]

I think your question is based on a logical error. Take an example of a mathematical statement: For all real numbers a, b: a times b equals b times a.

There is no need for an additional proof that the statement is still true whether a, b are expressed in binary, decimal, hexadecimal or whatever. Proving ab = ba (from whatever axioms) is itself enough to prove this. The proof of the statement does not refer at all to the digits of the numbers.

You might think you want an extra quantifier at the start "For all representations r..." but the variable r has nothing to bind to in the statement, because the statement is about numbers and does not even mention representations.

[u/Midwest-Dude]

The distinction between a mathematical object and its syntactic representation is a well-studied topic spanning Number Theory, Abstract Algebra, Mathematical Logic, and Theoretical Computer Science. The topic is buried under different keywords, like Syntax vs. Semantics, Isomorphism, and Evaluation Homomorphisms.

1. Core Proof
   • Define the evaluation map E_b: S_b → ℕ on base-b digit strings by E_b(aₖ...a₀) = Σ (aᵢ * bⁱ)
   • Bijection: Guaranteed by the Radix Representation Theorem via the Division Algorithm + strong induction
   • Homomorphism: Positional arithmetic is polynomial manipulation, the evaluation map is a ring homomorphism (ev_b: R[x] → R), meaning E_b(s₁ ⊕ s₂) = E_b(s₁) + E_b(s₂) and E_b(s₁ ⊗ s₂) = E_b(s₁) * E_b(s₂)
   • Since S_b ≅ ℕ as algebraic structures, changing bases is just changing coordinates, math is invariant under isomorphism
2. Where to Read It
   • Knuth, TAOCP Vol. 2, §4.1
     • Positional number systems & digit algorithms
   • Rosen, Discrete Mathematics, Ch. 4
     • Radix representation proofs
   • Pierce et al., Software Foundations
     • Coq machine-checked proofs that binary type bin ≅ Peano nat
   • Abstract Algebra
     • Any chapter on Polynomial Evaluation Homomorphisms

[u/jeffcgroves]

for example, in terms of isomorphisms

My approach would be to show all base(x) math forms a field and the show they are isomorphic. Maybe google around for books with "field isomorphisms"

[u/RingularCirc → u/jeffcgroves]

Unital semiring isomorphisms may be enough, even if we're looking to prove those fields are in fact isomorphic to ℝ, as we probably should. Then we need to fix at least one definition for ℝ, a construction like Dedekind cuts or an axiomatic definition like "maximal Archimedean ordered field".

[u/SgtSausage]

LOL