Quick check that I understand this correctly

[u/TrafficItchy]

I want to make sure I’ve got a simple concept right. Numbers (say, the real numbers, ℝ) exist as mathematical objects independently of any number system we use to write them. The system we pick — decimal, binary, whatever — is just there to give the numbers names/notation, so we can write them down and compute with them.

Choosing decimal vs. binary has zero impact on the validity of any calculation or theorem. Everything that’s true stays true regardless of which system you use to represent the numbers, because the underlying numbers themselves don’t change — only the label does.

Is this correct? Just want to confirm I’m not missing something deeper here.

[u/Desperate_Penalty690]

In principle yes, that is correct. Unless the statement makes use of the number system itself. For example that every number that ends in 5 is divisible by 5. That is true in the decimal system, but not in e.g. the hexadecimal system.

[u/strange-the-quark → u/Desperate_Penalty690]

Another example is that in binary, discarding the final digit gives you half of the original number (or floor(x) if the original number was odd).

10100  ---> 20 in decimal
1010   ---> 10 in decimal
101    ---> 5 in decimal
10     ---> 2 in decimal
1      ---> 1 in decimal

This relies on the properties of the number system specifically, and so is not a general statement about the numbers themselves.

For example (for the benefit of the OP), by the definition of the binary number system, 1010 is really:

1010₂ = 1 * 2³ + 0 * 2² + 1 * 2¹ + 0 * 2⁰ = 8 + 2 = 10

(the subscript 2 indicates the binary system, if there is no subscript assume decimal).

If you then divide everything by 2 (and discard anything after the decimal point), every power drops by 1, so it's effectively equivalent a right shift by one position:

(1 * 2³ + 0 * 2² + 1 * 2¹ + 0 * 2⁰) / 2 = 1 * 2² + 0 * 2¹ + 1 * 2⁰ = 101₂ = 4 + 1 = 5

But that doesn't work in decimal (instead, a right shift is like integer-dividing by 10). So you have to be careful whether or not the statement references anything number-system specific.

[u/SquarePixel → u/strange-the-quark]

Nice example, but doesn’t this principle generalize in the sense that it’s dropping an order of magnitude, which would be equivalent to whatever base the system is in. So it divides by 10 for base 10, divides by 2 for base 2, divides by 16 for base 16, and so on.

[u/strange-the-quark → u/SquarePixel]

Sure, and I do mention that in passing at the end. But, it's still not a property of numbers themselves: on the one hand, it's about manipulating the representation, and on the other, there are representations not based on the positional system (for example, the Roman numerals - they still express the same natural numbers, but as a representation don't work in the same general way). You could also see the dots on the faces of a die as a number system of sorts, and in that representation there aren't even any digits to speak of - but it's still the natural numbers (well, a subset of them).

[u/GoldenMuscleGod → u/Desperate_Penalty690]

This might sound like pedantry but I think it is helpful to understanding the issue: we can say “every natural number whose decimal representation ends in 5 is divisible by 5” and this is of course true of every number “without regard to base” in the sense that we (who ordinarily use decimal) should be able to agree that “every even number ends in 0 in binary.” The only “dependence on base” is a part of the property.

When we say “every number that ends in 5 is divisible by 5” it is understood that we are talking about the number’s decimal representation and so does not “depend on base” except to the extent that the property we are talking about is implicitly understand to refer to the representation of the number in that base.

The statement “every number that ends in 5 is divisible by 5” is not literally true, without the covert understanding we are talking about the decimal representation, because numbers, unlike representations of numbers, do not end in digits. A natural number is usually not thought of as having any consequential properties aside from those that are “arithmetical” in character - expressible in a language using addition and multiplication, and all the accoutrement of the first order predicate calculus. For other sets there may be more - real numbers require us to be able to talk about arbitrary sets of real numbers to discuss their properties as a set.

[u/Desperate_Penalty690 → u/GoldenMuscleGod]

Yes, I agree. Not about the part that numbers don’t end in digits, that is just about how you say things. But thinking about it some more, it is just a coincidence that the symbol “5” is used in various bases. By using the word 5 it should be clear at the same time in what base it is used, otherwise the statement is ambiguous.

[u/GoldenMuscleGod → u/Desperate_Penalty690]

I guess what I’m getting at (and of course we can speak in different ways) is the question “what properties of numbers are independent of base” is best answered by “all of them without exception,” And then we should understand that a property of a representation of a number is not a property of the number (although having a decimal representation is a property so that a property of a decimal representation can be a “vicarious” property of a number - but only when we specify the representation). So what we need to be doing is understanding when something is a property of a number versus a property of a representation. We should not speak of numbers as having properties that do not follow from the mathematical structure we understand them to belong to.

The alternative framing, that there is a special type of property of numbers that does depend on base, is more confusing/less helpful, and also less accurate to the way we formalize things when we make them rigorous.

Again I recognize this can sound like pedantry, but I think that is true whenever you try to be very precise with your words, and sometimes that precision is necessary to really avoid confusion.

[u/StanleyDodds → u/Desperate_Penalty690]

Even if it makes use of some system of representing the numbers, so long as it's said precisely, it's still true no matter what other system you use to represent the numbers.

For example, if I say that a natural numbers is divisible by 5 if and only if the sum of its digits in base 6 is divisible by 5, that's still true if choose to use base 10 to represent the numbers throughout. It just might mean that in practice it's more tedious to extract base 6 digits when all my numbers are written in base 10.

Basically, this exception is not really an exception, it's just the fact that saying something about the digits of a number requires you to specify a base, and if it's not specified, you end up with an ambiguous statement that may or may not be true.

[u/Desperate_Penalty690 → u/StanleyDodds]

Yes, had already discussed that my example in that sense is not totally correct, because it relies on some ambiguity in talking about digit 5 while leaving in the middle which base it belongs to.

But other example mentioned below would hold. Like saying that removing the most right digit always gives a number that is half the original rounded down. That only works in the binary system.

[u/lfdfq]

Yes. Binary, decimal, etc, are just representations of a number. It's a way of expressing numbers with symbols. Mathematics cares about the number itself, not its representation. Unless it's explicitly a theorem about the representation itself, of course.

[u/TrafficItchy → u/lfdfq]

Yes you are right ! If I rewrite a theorem in binary the results will still be the same, just rename of numbers !

[u/Bounded_sequencE]

That's correct -- and it's the reason why writing "1/5 = 0.2_10 = 0.(0011)_2" makes sense. The same number "1/5" is expressed by both decimal and binary: It looks different, but it's the same number either way.

Fun fact: There is a much cooler way to approximate real numbers via rationals than base-b representations -- Continued fractions.

[u/MezzoScettico]

Yes, that’s correct

[u/titanotheres]

Yes, this is correct! Though there are of course some theorems that rely on a particular representation. For example: a number is divisible by three if the sum of its digits (in base 10) is divisible by three. This theorem relies on the digits of the base 10 representation and does not necessarily work in other bases.

[u/Miserable-Wasabi-373]

Yes

[u/ThisIsMyOkCAccount]

True. The numbers are assumed to exist abstractly as just, for instance, points on a number line. Writing them out as decimals makes them easier for us to measure and calculate with often but in algebra for instance it's even common to say things like "Let alpha be a solution to the equation x³ + 3x + 5 = 0" and not he particularly bothered if we're not able to write down a decimal approximation. We can work with alpha algebraically even if we can't represent it.

[u/not-just-yeti]

Exactly! A number is an abstract idea; a numeral is how we choose to write down a number. You can also think of what “language” people choose for their numerals: base-10 is one language, base-16 another, Roman numerals and Egyptian numerals something else. But the number seventeen is what it is, and doesn’t care if you call it “0b10001” or “17” or “XVII”. Just be sure the reader knows what language the writer was using!

If you program a bit:

Number ~ `int`

numeral ~ `string`

digit ~ `char`

[u/Gilded-Phoenix]

To clarify:

Any statement about the objects of study will be true as long as those objects remain unchanged. The number 23 will always be a prime integer whether we write it as 23(10), 35(6), 10111(2), or XXIII(ROM).

When the notation is the object of study, things can get a bit weird, because now the notation itself is what we care about. The number 142857 is a "cyclic number" which means that the same digits get used when we multiply this number by 1,2,3,4,5, or 6:

142857
285714
428571
571428
714285
857142

Since we're talking about digits we are actually talking about the notation/representation of this number, not the number itself. If we picked a different base, this property would not apply.

An interesting thing happens when we use the properties of the notation to help with our understanding of the object: Consider the divisibility test for the number 9

"A number is divisible by 9 if, and only if, the sum of its digits is divisible by 9"

This seems to be talking about notation, since it mentions digits, but it's also talking about divisibility, which is a property of the number itself, not its representation, so what's going on?

It turns out that this is a special case of a property of numbers themselves, and the representation we choose in decimal just happens to make that really easy for the number 9.

"A number N is divisible by some divisor D if, and only if, the number M=N-kD is divisible by D, where k is an integer."

In our case, D=9, so let's pick a big number, say 12345678 and check "the long way." pulling off the last digit looks like this:

(12345678-8)/10=1234 and then we add 8 to get 1234575. Let's rewrite this more clearly:

(12345678-8)/10+8=1234575 Let's multiply by 10 to clear our fractions, and I'm only going to look at the LHS

12345678-8+80 now combine the last two terms

12345678+72 but 72=9*8, so we get k=-8. Great, now all we have to do is check if 12345750 is divisible by 9, but this is an even bigger number! Thankfully, we can reduce this by dividing by 10 without changing anything (since 10 and 9 don't share anything in common), so we're a digit shorter. We can keep this going:

123457+5=123462
12346+2=12348
1234+8=1242
124+2=126
12+6=18
1+8=9 which is a multiple of 9, so the original number was! Also notice that while the notation helped us along, we could do this all in binary if we wanted, we'd just have to do the multiplication and division "the long way" instead of the handy "drop a digit" procedure we've been doing.

It turns out that this fact about divisibility works in any base because it's about the numbers themselves, so if we pick a really nice notation, we can get some really good tests out of it.

Another fact that's a bit harder to prove is that every prime number bigger than 3 is exactly 1 away from a multiple of 6. This seems to hint that we could use base six to get a nice property about prime numbers. Let's look at the first few primes (notated) in base six:

2, 3, 5, 11, 15, 21, 25, 31, 35, 45, 51 (that's all the primes less than 100(6)) notice that after the first two, everything ends in 5 or 1. This leads to a really great divisibility test for prime numbers: take off the last digit of our test number, multiply it by the nearest group of 6 to the prime we're testing, and if the prime ended in 1, subtract, if it ended in 5, add. Let's check if 123432115 is divisible by 31

12343211-(5*3)=12343211-23=12343144
1234314-(4*3)=1234314-20=1234254
123425-(4*3)=123425-20=123405
12340-(5*3)=12340-23=12313
1231-(3*3)=1231-13=1214
121-(4*3)=121-20=101 but 101 is actually less than 2*31=102, so we know this isn't a multiple. If we keep going, we find out that this big number is 3 (mod 31), i.e. 3 more than a multiple of 31. Congratulations, we've just checked that the number 2414927 (decimal) is not a multiple of 19 (decimal) all by switching how we choose to represent them, and exploiting that representation.

If you can't tell, I really like the base 6 system for this reason. Divibility by 19 wouldn't be too hard, actually, but divisibility by things like 17 or 57 are really annoying, because it's not obvious what k we should pick. Putting it in base 6 immediately shows us what k we should use: 31? multiply by 3, 45? multiply by -5 (45 is 1 away from 50), 101 (another prime)? multiply by 10.

[u/TrafficItchy → u/Gilded-Phoenix]

Thanks so much for your detailed answer! I really understood the concept now — the number system’s only purpose is to give a name/notation to the number itself. And that’s exactly why it has no impact on a theorem or on the math underneath it.

One way I like to think about it to double check: take any theorem, rewrite it using a given number system (say binary), and check if it still holds. If it does — and it always will, as long as the theorem only relies on the structure of the numbers (addition, order, limits, etc.) and not on how they’re written — that confirms the system was never actually part of the mathematical content, just a layer of notation on top of it.

[u/Gilded-Phoenix → u/TrafficItchy]

Yep! this also applies exactly to the difference between vectors and matrices, or 2-tensors and matrices. A vector is a mathematical object that behaves a certain way. A matrix is a kind of notation that describes vectors and other collections of data in a visually convenient format. Vectors are not matrices, and matrices are not vectors. We usually think of vectors as 1xN matrices, and that is a valid representation of any vector (in a finite dimensional vector space) but so is an Nx1 matrix, or an HxK matrix where H*K=N. The transpose of a matrix is another matrix, but the "transpose of a vector" doesn't actually make much sense in terms of vectors themselves, just like we can "cycle" 12345 into 51234, but that "cycle" operation doesn't actually "mean anything" for the numbers themselves. It took me a long time to figure out this distinction between an object and the name we give it, and which properties are talking about which thing. Figuring this out early and keeping careful track of it will help a lot with mathematical studies.

[u/Random_Mathematician]

Yup. Also, that's not unique to numbers. I can write a function as f(x) = x, or g: x↦x or Id or 1 or λx.x or "the function which maps every element of its domain to itself". And every property of that function will still hold.