B Is "ten" independent of the chosen number base?

[etotheipi]

This topic is quite confusing. For instance, if I write $14_6$, do I pronounce this as "ten", even though we'd probably just say "one-four, base six"? That is to say that we'd treat "ten" as a number of things that we could count out (i.e. corresponding to a certain number of sticks).

Another point, if we again consider the number $14_6$, then this would correspond to one lot of six sticks and 4 lots of single sticks, where we've defined one, two, three, ..., nine to be known numbers of things. This is often written as

$14_6 = 1 \times 6 + 4 \times 1$

but isn't this construction preemptively implying that we are using base ten? Another example might be

$B3_{16} = 11 \times 16 + 3 \times 1$

Do we only write it like this since we've memorised a certain amount of computations in base ten (e.g. times tables) so it is convenient to use base ten as a middle-man for all calculations?

Is it perhaps because our language system is built around base ten, i.e. twenty two $\equiv$ 22, thirty six $\equiv$ 36, etc.?

 

[HallsofIvy]

This topic is quite confusing. For instance, if I write $14_6$, do I pronounce this as "ten", even though we'd probably just say "one-four, base six"?

I can't speak for you but if I were working in base 6 I would NEVER call that "ten"!

That is to say that we'd treat "ten" as a number of things that we could count out (i.e. corresponding to a certain number of sticks).

Another point, if we again consider the number $14_6$, then this would correspond to one lot of six sticks and 4 lots of single sticks, where we've defined one, two, three, ..., nine to be known numbers of things. This is often written as

$14_6 = 1 \times 6 + 4 \times 1$

but isn't this construction preemptively implying that we are using base ten?

I don't see where this is "implying that we are using base ten". "6" and "4" are symbols representing, say, ******and ****. They do not have any thing, directly to do with base 10!

Another example might be

$B3_{16} = 11 \times 16 + 3 \times 1$

Do we only write it like this since we've memorised a certain amount of computations in base ten (e.g. times tables) so it is convenient to use base ten as a middle-man for all calculations?

If that is how you have learned them, yes, fine. But if you have learned that "B" means "***********" and 3 means "***" there is no reason at all to refer to base 10.

Is it perhaps because our language system is built around base ten, i.e. twenty two $\equiv$ 22, thirty six $\equiv$ 36, etc.?

I would not say it has anything to do with language, rather that it has to do with the fact that, as you say, you are used to using base 10.

 

[Merlin3189]

I'd be very cautious about saying "ten" when talking about $14_6 ,\, 10_2 ,\, 10_8 ,\, 12_8 ,\, 0xA$ or anything except $10_{10}$ Anything else can cause confusion.
We have enough confusion with people using kilo- and mega- prefixes to mean different values.
When I worked a lot with hexadecimal, I think everyone stuck pretty much to "Oh Dee Oh Aye" sort of speak. If you're working in hex, why refer to the decimal values at all? And when you do need to convert, you'd better be pretty sure everyone knows which is which.

We use the number names that we are familiar with. If you like you can invent your own. I guess some people might have done so if they worked a lot in a different base, but I can't recall any. The digit by digit naming that we use for bases other than ten, seems pretty efficient to me.
I notice that it is becoming quite common in adverts for prices to be said in such a way, eg. "only four nine nine". Their motivation is probably more psychology than efficiency, avoiding "hundred" and "ninety", which sound big and avoiding the unit, which might remind people it's money.

Do we only write it like this since we've memorised a certain amount of computations in base ten (e.g. times tables) so it is convenient to use base ten as a middle-man for all calculations?

Surely that's true, if not obvious. Why would you use anything other than the number base you're familiar with?

 

[etotheipi]

I can't speak for you but if I were working in base 6 I would NEVER call that "ten"!

Agreed; I don't think I would either, but at the end of the day $14_6$ still refers to **********, which we denote as ten.

I don't see where this is "implying that we are using base ten". "6" and "4" are symbols representing, say, ******and ****. They do not have any thing, directly to do with base 10!

Like in the example $B3_{16}$, if we wrote $16 \times 11 + 3 \times 1$ this would not make sense in, for instance, base three.

I'd be very cautious about saying "ten" when talking about $14_6 ,\, 10_2 ,\, 10_8 ,\, 12_8 ,\, 0xA$ or anything except $10_{10}$ Anything else can cause confusion.
We have enough confusion with people using kilo- and mega- prefixes to mean different values.
When I worked a lot with hexadecimal, I think everyone stuck pretty much to "Oh Dee Oh Aye" sort of speak. If you're working in hex, why refer to the decimal values at all? And when you do need to convert, you'd better be pretty sure everyone knows which is which.

I think that's a smart way to go, no point adding extra confusion!

 

[HallsofIvy]

Agreed; I don't think I would either, but at the end of the day it $14_6$ still refers to **********, which we denote as ten.

That's one way to denote it. Another is "$14_6$" and another is "$13_7$". Yet another is "X".

Like in the example $B3_{16}$, if we wrote $16 \times 11 + 3 \times 1$ this would not make sense in, for instance, base three.

"B" would not make sense in base three!
I think that's a smart way to go, no point adding extra confusion!
[/QUOTE]

 

[Mark44]

I guess some people might have done so if they worked a lot in a different base, but I can't recall any.

Base 20, sort of. For example, in Lincoln's Gettysburg Address, he speaks of 87 as "four score and seven." Numbering in French follows a similar pattern, with the same number being written as quatre vingt et sept, or four twenties and seven.

We also have words in English that are based on a different base: twelve; e.g., dozen and gross (a dozen dozen).

 

[etotheipi]

It might be a little bit like the concept of vectors. A vector doesn't depend on the basis you use to describe it, and you can only assign numbers to it once you have chosen a basis. A vector is fundamentally just a geometric object.

For numbers, we have to distinguish between the actual concept of a number (i.e. an amount of stuff, a distance on a number line etc.) and its representation.

Like when we say $35_{10}$, we really mean three lots of ********** and 5 lots of *.

That said, it's still really tricky to try and get away from thinking of representations of numbers as atomic objects, since that's what one becomes so used to. Even descriptions of place value fall victim to circular logic, since if we say that

$352 = 3 \times 100 + 5 \times 10 + 3 \times 1$, we're stuck with the problem of e.g. what is 100? If we break this down with place value, we get the exact same thing. To get out of the loop, we need to associate the $100$ with the concept of an amount of stuff!

 

[Merlin3189]

All column values are multiples of the previous one:
Th H T U are $10^0 \, 10^1 \, 10^2 \, 10^3$ (only the other way round.) So You only need to know up to ten.
6352 = 6x10x10x10 + 3x10x10 +5x10 + 2x1 the 100, 1000, etc are just shorthand for the row of 10's

Same is true in any number base
0x3456 = 3 x 0x1000 + 4x 0x100 + 5 x 0x10 + 6x1 = 3 x 0x10 x 0x10 x 0x10 + 4 x 0x10 x 0x10 +5 x 0x10 + 6 x 1
(or 3x16x16x16 + 4x16x16 + 5x16 + 6x1 )

================
Edit: NB the 3,4,5 and 6 in the penultimate line are all base 16 digits, not base 10.
I should have written 0x3, 0x4, 0x5 and 0x6 to be consistent.

 

[etotheipi]

All column values are multiples of the previous one:
Th H T U are $10^0 \, 10^1 \, 10^2 \, 10^3$ (only the other way round.) So You only need to know up to ten.
6352 = 6x10x10x10 + 3x10x10 +5x10 + 2x1 the 100, 1000, etc are just shorthand for the row of 10's

Same is true in any number base
0x3456 = 3 x 0x1000 + 4x 0x100 + 5 x 0x10 + 6x1 = 3 x 0x10 x 0x10 x 0x10 + 4 x 0x10 x 0x10 +5 x 0x10 + 6 x 1
(or 3x16x16x16 + 4x16x16 + 5x16 + 6x1 )

I guess if we construct a "side length" consisting of 10 sticks in base $n$, then 100 corresponds to the number of sticks in a square, 1000 in a cube, 10000 in a 4D hypercube...

You're right though, we only need consider the powers of $n$, and treat these as our "basis numbers". If we also accept that the algorithms for addition/multiplication etc. aren't specific to any particular base, that also helps to separate the concepts of numbers and representations in different systems.

 

[Mark44]

That said, it's still really tricky to try and get away from thinking of representations of numbers as atomic objects, since that's what one becomes so used to.

As you said, it depends on what you're used to. Because I've been involved in computer programming for many years, I'm very comfortable working in binary (base-2) and hexadecimal (base-16) and even octal (base-8). I'm perfectly fine working with 16₁₀ or 10₁₆ (usually written as 0x10) or 20₈. They're all just different representations of the same underlying number. Just as vectors can have different representations in different bases, so can numbers.

 

[sysprog]

A gentle joke (that I didn't originate): there are 10 kinds of people in the world; those who understand binary, and those who don't . . .

 

[hutchphd]

This topic is quite confusing. For instance, if I write $14_6$, do I pronounce this as "ten", even though we'd probably just say "one-four, base six"? That is to say that we'd treat "ten" as a number of things that we could count out (i.e. corresponding to a certain number of sticks).

Another point, if we again consider the number $14_6$, then this would correspond to one lot of six sticks and 4 lots of single sticks, where we've defined one, two, three, ..., nine to be known numbers of things. This is often written as

$14_6 = 1 \times 6 + 4 \times 1$

but isn't this construction preemptively implying that we are using base ten? Another example might be

$B3_{16} = 11 \times 16 + 3 \times 1$

Do we only write it like this since we've memorised a certain amount of computations in base ten (e.g. times tables) so it is convenient to use base ten as a middle-man for all calculations?

Is it perhaps because our language system is built around base ten, i.e. twenty two $\equiv$ 22, thirty six $\equiv$ 36, etc.?

I am mystified by this question (usually I am fascinated by your questions). We have 10 fingers and 10 toes: by custom and maximal use we have agreed that our default system is base 10. If we use another we need to specify. Good system, works fine, why discuss it?

Am I missing some nuance here??

 

[etotheipi]

I am mystified by this question (usually I am fascinated by your questions). We have 10 fingers and 10 toes: by custom and maximal use we have agreed that our default system is base 10. If we use another we need to specify. Good system, works fine, why discuss it?

Am I missing some nuance here??

No you're quite right, my confusion was just about the separation of the concepts of numbers (e.g. ten/**********) that exist independently of base, and then their representation in different bases.

We might say 43 = 4 x 10 + 3 x 1, but that begs the question what is 10? Well

10 = 1 x 10 + 0 x 1

and we could keep going down this rabbit hole forever. The only resolution is to relate the 1 in the first digit of 10 to “one lot of ten *’s”, which would be a physical number.

 

[jbriggs444]

No you're quite right, my confusion was just about the separation of the concepts of numbers (e.g. ten/**********) that exist independently of base, and then their representation in different bases.

The nomenclature I was taught in school was "number" for the thing that is independent of presentation and "numeral" for the representation thereof.

If one steps into the world of computers and computer languages then "numeric literal" is a term for the syntactic element that represents a number as a sequence of characters within a program. e.g. 10, 0x0a, #2#1010 or 1e+1

Or one could consider an arbitrary coding system in which numbers are represented as binary strings. That opens up fertile ground for discussion. Do we use a fixed width representation or variable width? If variable width, can we optimize the coding scheme for average code size? This leads toward Huffman coding or, perhaps, Lempel Ziv Welch. What if our transmission channel is noisy? That way leads to the work of Shannon.

 

[Mark44]

We might say 43 = 4 x 10 + 3 x 1, but that begs the question what is 10? Well

10 = 1 x 10 + 0 x 1

and we could keep going down this rabbit hole forever.

No, not at all.
Any numbering system newer than Roman numerals has a base. The decimal (base-10) system is so prevalent, we almost never explicitly show the base.

In any numbering system with base B, there are B digits: 0, 1, 2, ..., B-1. A number 10_B is $1 X B^1 + 0 X B^0$.

I found an interesting wiki article about a variety of numbering systems used by different cultures around the world - https://en.wikipedia.org/wiki/Numeral_(linguistics). Not everyone uses a decimal system.

 

[etotheipi]

In any numbering system with base B, there are B digits: 0, 1, 2, ..., B-1. A number 10_B is $1 X B^1 + 0 X B^0$.

The thing is, how can one write down ‘B’ without implicitly using another base?

I can’t write base-10, since this could imply base-two, base-three, base-fourteen depending on which base the 10 is written in!

Perhaps I let B = sixteen, since these names are independent of the numeral,

this might give $N = a \times (\text{sixteen})^{2} + b \times (\text{sixteen})^{1} + c \times (\text{sixteen})^{0}$. But I can’t write down the sixteen without referring to a different base.

Unless I use the natural correspondence between the English naming system to provide $16_{10} \equiv$ sixteen? My thinking is that we “memorise” how a certain number (name) translates to a base ten numeral.

 

[jbriggs444]

The thing is, how can one write down ‘B’ without implicitly using another base?

Conventionally we ground the recursion immediately and write down the base implicitly using decimal.

A writer would not write $10_{10_{10_{10}}}$. There is no point in adding notational complexity. And certainly no point in driving that notation into infinite regress.

In the Ada programming language:

The base and the exponent, if any, are in decimal notation.

 

[Mark44]

The thing is, how can one write down ‘B’ without implicitly using another base?

B is $10_B$
If B = 2, or binary, we have $2_{10} = 10_2$.
If B = 8, or octal, we have $8_{10} = 10_8$.
If B = 64, $64_{10} = 10_{64}$.

I can’t write base-10, since this could imply base-two, base-three, base-fourteen depending on which base the 10 is written in!

No! Base-10 (or decimal) implies that each digit is a successively higher power of 10, as you move from right to left. The decimal numeral 243 means $2 X 10^2 + 4 X 10^1 + 3 X 10^0$
I don't know why this is so difficult for you.

One thing that you might be confused on is that whatever the base is (I'm calling it B), the digits in that base never include B. They always go up to one less than B.

In binary, there are two digits: 0 and 1.
In octal, there are eight digits, 0, 1, 2, 3, 4, 5, 6, and 7.
In decimal, there are ten digits: 0 through 9.
In hex, there are sixteen digists: 0 through 9, A, B, C, D, E, and F.
And so on.

Perhaps I let B = sixteen, since these names are independent of the numeral,

this might give $N = a \times (\text{sixteen})^{2} + b \times (\text{sixteen})^{1} + c \times (\text{sixteen})^{0}$. But I can’t write down the sixteen without referring to a different base.

Of course you can!
$16_{10} = 10_{16}$, which is usually written as 0x10. The prefix "0x" implies that we're using base-16 or hexadecimal.

Unless I use the natural correspondence between the English naming system to provide $16_{10} \equiv$ sixteen? My thinking is that we “memorise” how a certain number (name) translates to a base 10 numeral.

You can always convert a numeral in one base to its representation in any other base. I won't go into the details, but you can find algorithms for doing this using a web search.

 

[etotheipi]

No! Base-10 (or decimal) implies that each digit is a successively higher power of 10, as you move from right to left. The decimal numeral 243 means $2 X 10^2 + 4 X 10^1 + 3 X 10^0$
I don't know why this is so difficult for you.

I meant that in writing base-10, we've explicitly assumed that the 10 is written in base ten. If I grew up with 8 fingers so used the octal numbering system, I'd interpret 10 to mean ********, and not **********.

 

[Mark44]

I meant that in writing base-10, we've explicitly assumed that the 10 is written in base ten. If I grew up with 8 fingers so used the octal numbering system, I'd interpret 10 to mean ********, and not **********.

Well, of course.

 

[jbriggs444]

I meant that in writing base-10, we've explicitly assumed that the 10 is written in base ten. If I grew up with 8 fingers so used the octal numbering system, I'd interpret 10 to mean ********, and not **********.

As a rule, we do not identify the language we are using before we start speaking. It is an impossibility. Instead we just begin speaking.

Even if we preface our remarks with "I will be giving this presentation in English" we do not also state that "This is not the dialect of Klingon-1432 in which the syllables 'I will be giving this presentation in English' encode an intention to eat the firstborn male child of all who are present". It is both inefficient and unnecessary to make such a disclaimer.

 

[etotheipi]

I think this is perhaps like defining a unit vector.

$\vec{e_{1}} = 1\vec{e_{1}} + 0\vec{e_{2}} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

We could keep going but will just end up with the same thing .

Except now our "components" are the digits and the unit vectors are the powers of the base.

I find I can do the necessary manipulations/conversions etc. fine, but just struggle to really understand the underlying logic without any circular reasoning / relying on intuition.

 

[etotheipi]

"The base and the exponent, if any, are in decimal notation."

Actually, so long as this is a universally agreed convention, I think this clears up most of the confusion. But the base numbers/powers of 10 must be thought of as abstract numbers and not representations.

Also, a given algebraic base $b$ is does not depend on a specific number base until we decide to substitute in a value.

 

[jbriggs444]

Actually, so long as this is a universally agreed convention, I think this clears up most of the confusion. But the base numbers/powers of 10 must be thought of as abstract numbers and not representations.

Also, a given algebraic base $b$ is does not depend on a specific number base until we decide to substitute in a value.

When you write $b$ do you mean to refer to a numeral or to a number? The usual interpretation is that symbols have values that are numbers, not values that are names of numbers.

[Insert Niklaus Wirth joke about "call by name" or "call by value"]

 

[etotheipi]

When you write $b$ do you mean to refer to a numeral or to a number? The usual interpretation is that symbols have values that are numbers, not values that are names of numbers.

[Insert Niklaus Wirth joke about "call by name" or "call by value"]

Yes, sorry I meant that $b$ refers to a number, but if we were to "solve for it" we would obtain a representation in some base depending on the bases of the numbers we put in for the calculation.

On the bright side, I think I finally got over the mental block that was causing the confusion, and it seems completely obvious. It was that the actual symbol for ten, in base ten, is $10$ - and like you say we terminate the recursion here. I was getting stuck thinking that we'd need to keep breaking down $10$ into a more fundamental representation, though we actually just mean "one ten", and there is nothing more fundamental to it. I.e. we could relabel $10 \equiv t$ and treat $t$ as the symbol for **********.

 

[pbuk]

Actually, so long as this is a universally agreed convention, I think this clears up most of the confusion. But the base numbers/powers of 10 must be thought of as abstract numbers and not representations.

Absolutely not. I think you should speculate a bit less and study a lot more. You will discover that all number bases (except possibly unary) are representations, and place value is a human artifice.

Peano Axioms

 

[jbriggs444]

Yes, sorry I meant that $b$ refers to a number, but if we were to "solve for it" we would obtain a representation in some base depending on the bases of the numbers we put in for the calculation.

It is not clear what you mean here. If b is a number then when we "solve for it" we will obtain a number.

How we might choose to present that number as a character string is not terribly relevant.

 

[etotheipi]

It is not clear what you mean here. If b is a number then when we "solve for it" we will obtain a number.

How we might choose to present that number as a character string is not terribly relevant.

You're right, I'm doing a terrible job of explaining this!

I meant that if we say $N = w b^3 + x b^2 + y b^1 + z b^0$, then if we want to represent - for instance - the base 6 number system, we need to let $b = 6_{10}$. It isn't too relevant though, as you say.

 

[etotheipi]

Yes, OK, I think I found the problem and I think @Mark44 made a mention of it earlier.

It is easy to understand how $34_{six} = 6 \times 3 + 4 \times 1$ because we have a symbol which means $6$.

But in base ten, it is not so easy to see because we have no special symbol for ten, only $10$ (whose recursive sort of nature has already been mentioned). It helps if we introduce the symbol $A$, because then $34_{ten} = 3 \times A + 4 \times 1$, since this avoids the recursion. Essentially, $A$ becomes our base, and we have no circular referencing.

The alternative which I posted earlier would be to substitute $A$ for $10 \equiv \text{one-ten, zero-one}$, but this seems quite weird.

And I do apologise if I'm boring anyone, I feel bad because this topic must be completely trivial for someone in-the-know, but it's pretty tricky to pick up!

 

[pbuk]

And I do apologise if I'm boring anyone, I feel bad because this topic must be completely trivial for someone in-the-know, but it's pretty tricky to pick up!

It's not that this is a trivial topic, it's not even a topic at all. In mathematics the symbols we use in the everyday world to represent numbers when we communicate things to each other simply don't exist. So worrying about any perceived difficulties in interpreting these symbols is a problem of language and convention, not science.

 

[hutchphd]

I believe there is nothing profound (nor very interesting ,...sorry!) here, other than the notion that we need to have agreed, a priori, about the symbols we will use to count. Of necessity they must be at least as many as our "base". If they are greater in number (i.e hex symbols in decimal) no ambiguity results, but the manipulations are a bit strange and the representations are not unique. If I write a number
2C1_A
there is no doubt that this is equivalent to what normal humans would write as
321 (did I do that right?200+120+1 yes),
but wouldn't it be better to limit ourselves for uniqueness' sake...

 

[etotheipi]

I believe there is nothing profound (nor very interesting ,...sorry!)

No you're completely justified, this whole thing is inherently an unbelievably dry subject matter! In hindsight I don't know why I voluntarily got myself into this mess... sorry everyone!

If I write a number
2C1_A
there is no doubt that this is equivalent to what normal humans would write as
321 (did I do that right?200+120+1 yes),
but wouldn't it be better to limit ourselves for uniqueness' sake...

Makes sense, I see what you mean .

 

[hutchphd]

Don't stop asking! But occasionally you may face a little "flak"...it is what friends are for after all...

 

[sysprog]

dec 25 = oct 31 wherefore christmas = halloween (joke not originated by me)

 

[etotheipi]

I think it seems like a good idea to label the number bases in text form, since this is unambiguous.

Even if we clicked onto a website which read 'As a convention, all number bases are expressed in base 10', somewhere or another they'd need to say that this is base-ten, or base-||||||||||, or base A, or base S(S(S(S(S(S(S(S(S(S(S(0))))))))))). I know special agreed symbols also exist for the common bases (d, h, b etc.), as well as the computer science conventions (like 0x, etc.).

Or at least something like “any unsubscripted number is implied to be base-TEN”.

And also, the notion of $\text{ten}^{2}$ is independent of the number base, since it just means we count ten, and then count ten of these, and put all of the sticks in a huge pile. The basis numbers (i.e. $10_{\text{ten}}, 10_{\text{ten}}^{2}$) can't be decomposed any further since they're just representations of the abstract concept of "ten-ness", which we use as the basis of the number system.

 

[jbriggs444]

I think it seems like a good idea to label the number bases in text form, since this is unambiguous.

I disagree with this. The KISS principle applies. ("Keep it simple, stupid"). We have an agreed upon form for expressing numbers in radix notation. It should be used.

If one takes pains to go outside the standard notation in the name of avoiding ambiguity, the result is to distract the reader from the essence of the communication. An important property of mathematical writing is that it is terse. One can express a concept simply and briefly.

Rather than writing $12_{\text{ten}}$, write $12_{10}$. Or better yet, just write $12$.

The more baggage that is tacked onto the writing, the more of the intended meaning is hidden. The reader is left thinking "why is the writer doing this -- is there some important meaning hidden in these baroque syntactical choices?"

Edit: In practice, there is little to argue about. The number of times I have needed to write down a numeric literal in standard place value notation using a radix other than 2, 3, 8, 10 or 16 can be counted on the fingers of one hand with five fingers left over.

 

[Dale]

The thing is, how can one write down ‘B’ without implicitly using another base?

To write a number in any base N you need N symbols. Those N symbols represent the integers from 0 to N-1 and N is represented by the combination of symbols with the symbol for zero in the first digit and the symbol for one in the second digit. By convention we use the symbols 0, 1, ..., 9, A, B, ..., but there is no need for that, it is purely conventional. Also by convention for any base other than base ten we use a subscript to denote the base in base 10 numbers. Again, it is convention and is not necessary if understood in context. To make the point, I will use base 16 numbers denoted by a (), b (*), c (**), d (***), e (****), f (**** *), g (**** **), h (**** ***), i (**** ****), j (**** **** *), k (**** **** **), l (**** **** ***), m (**** **** ****), n (**** **** **** *), o (**** **** **** **), p (**** **** **** ***), ba (**** **** **** ****)

So your $B3_{16}$ is my $ld$ which can be written purely in my notation as $ld = l \times ba^b + d \times ba^a$. So there is never any need to use base 10 digits, and any such use is purely a matter of convenience and convention.

 

[etotheipi]

Awesome, thanks a bunch for everyone's help. I think I have a better understanding now. Namely,

• The base of a number is specified in decimal by convention (note I'm careful not to write "10" here , I'd rather not have to deal with so-called regresum ad infinitum...). If the base is base-10, then we do not write the subscript.
• There exists a distinction between numbers and numerals (their representations). $10$, $14_6$, $A_{16}$ all refer to the number ten.
• $\text{abc}_{b}$ can be expressed as a power series, $\sum \alpha_{i}b^{i}$. This is usually done in base-10, but it does not have to be so. We could have $2B_{16} = 3 \times {{21}_{6}}^{1} + 4 \times {{21}_{6}}^{0}$; of course, this is needless complexity, but goes to show that the base $b$ - like all algebraic variables - exist independently of bases!
• The base $b$ could theoretically be negative, non-integer, complex etc. (i.e. we might have base $\phi$...)

It's just a peculiar thing since everyone gets an intuitive feel for numbers and arithmetic through school, but this intuition is unhelpful to understand bases, since it's easy to be inclined to resort to "well, this is obviously such and such" reasoning. Do let me know if I've gone mad.

 

[sysprog]

LaPlace tended to say things along the lines of 'from here, it is easily seen that . . .' when he didn't want to do the proof . . .

 

[jbriggs444]

The base of a number is specified in decimal by convention

Numbers do not have bases. Numerals (i.e. representations) sometimes do.

A course in computer languages, parsing and syntax nails some of these things down in painful detail and adds some useful terminology.

 

[etotheipi]

Numbers do not have bases. Numerals (i.e. representations) sometimes do.

A course in computer languages, parsing and syntax nails some of these things down in painful detail and adds some useful terminology.

Whoops, please do excuse my sloppiness...

 

[etotheipi]

Though it does beg the question of what a number actually is, rigorously. If "ten", "10", "$*7@1" are all numerals. I think like some have mentioned, it's for the best that I stop speculating and just get a hold of some actual study resources, I found some of Tao's notes which look like they cover some fun stuff:

https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

Anyway thanks for putting up with me for this long (!)

 

[jbriggs444]

Though it does beg the question of what a number actually is, rigorously.

At a simple level, a number is the abstract concept that all of those representations are trying to get at. It is, for instance, the abstract notion of four-ness shared by four rocks in your hand, four sheep in your field or four sides on a square.

Typically we have an operational notion that is good enough to count sheep and verify that the buyer has paid us for all of them. We typically leave it at that.

One way of approaching the foundations is to define numbers in terms of their properties. For instance, the Peano axioms. From memory (and there are a lot of flavors of these).

1. There is an undefined entity called "zero" that we say is a "number".
2. For every number A there is a next number that we refer to as the "successor" of A. We refer to it as S(A).
3. For every number A that is different from zero, there is exactly one number B such that A=S(B).
4. There is no number A such that 0 = S(A).
5. [Induction] -- If you have a set of numbers that contains zero and also contains the successor of every number in the set then that set contains all of the "numbers".

The key takeaway from this is that we've not defined what these "numbers" are. But we've described how they behave. That turns out to be good enough to do a massive amount of arithmetic and a ton of proofs.

You can spend a couple of semesters going from here to a description of the real numbers.

 

[etotheipi]

The key takeaway from this is that we've not defined what these "numbers" are. But we've described how they behave. That turns out to be good enough to do a massive amount of arithmetic and a ton of proofs.

Freaky... the more I learn the less I know...

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

 

[PeroK]

Though it does beg the question of what a number actually is, rigorously.

The intuitive idea of whole numbers is as follows. You have a flock of sheep say. Every day you have to go out and bring all your sheep into a field for the night. How do you ensure you have all your sheep?

One approach is: as each sheep exits the field in the morning you add a small stone to a pile. Then, as each sheep enters the field at the end of the day you take a stone out of the pile. If there are stones left in your pile, then you know there are still sheep to be tracked down.

Note that this is a basic form of counting your sheep.

You do the same for everything: cows, chickens, pigs, whatever. For everything you need to keep track of you have a set of "counters" of some sort. But, you need a different set of counters for everything you need to "count"

You can improve on this by having a general set of things called numbers. You memorise the pattern of the sequence of symbols or words and each time a sheep leaves the field you move to the next number. Instead of having a set of stones, you write down (or remember) the number: 35 or XXXV or 100011 or thirty-five.

Now you can dispense with all the different sets of counters and use your general set of numbers to count everything.

In this intuitive context, the whole numbers form a defined, ordered sequence of symbols or words. This ties into the Peano axioms as "ten" is then the successor of the successor ... of the first symbol or word.

 

[zoki85]

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

Makes me wonder how badly Russell banged his head against wall when he heard of Gödel's theorems

 

[pbuk]

Awesome, thanks a bunch for everyone's help. I think I have a better understanding now.

Yes I think you do, but one of the beautiful aspects of mathematics is that you never have to stop learning!

• The base of a number is specified in decimal by convention (note I'm careful not to write "10" here , I'd rather not have to deal with so-called regresum ad infinitum...). If the base is base-10, then we do not write the subscript.

A mathematician would write "numbers are expressed in radix 10 by convention" and know that every other mathematician would understand exactly what she meant.

• There exists a distinction between numbers and numerals (their representations). $10$, $14_6$, $A_{16}$ all refer to the number ten.

I would rather write

• There exists a distinction between numbers and and the symbols we use to represent them. $10$, $14_6$, $A_{16}$ and $\text{ten}$ all refer to the same number.

• $\text{abc}_{b}$ can be expressed as a power series, $\sum \alpha_{i}b^{i}$. This is usually done in base-10, but it does not have to be so. We could have $2B_{16} = 3 \times {{21}_{6}}^{1} + 4 \times {{21}_{6}}^{0}$; of course, this is needless complexity, but goes to show that the base $b$ - like all algebraic variables - exist independently of bases!

Not sure where you are going with this: $a_2 a_1 a_0_{b}$ implies the partial sum $\sum_{i=0}^{2} \alpha_i b^i$, not any other partial sum $\sum_{i=0}^{k} \beta_i c^i$ that happens to have the same value.

• The base $b$ could theoretically be negative, non-integer, complex etc. (i.e. we might have base $\phi$...)

Yes but with non-integer bases we are into a whole other game and we are no longer in a domain which includes Peano arithmetic. Although non-integer bases seem interesting and exotic at first, there are far more interesting things to discover in the "mainstream" analysis syllabus so I recommend you don't go down that track.

It's just a peculiar thing since everyone gets an intuitive feel for numbers and arithmetic through school, but this intuition is unhelpful to understand bases, since it's easy to be inclined to resort to "well, this is obviously such and such" reasoning. Do let me know if I've gone mad.

To learn mathematics you have to put your intuitive feel for everything you think you understand to one side and start again with an empty mind. Don't worry about this, once you have grasped the modern concept of an integer, and a real number, and a line, and a set, and an infinite set... they will be just as intuitive as those old notions and much more powerful and consistent.

I found some of Tao's notes which look like they cover some fun stuff:
https://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

That might work but it would be better to hear Terry Tao deliver that course in person - it may be on YouTube. Failing that, it would be better to work from his book ISBN 978-981-10-1789-6 covering the same material. But whatever you do - move on from this fascination with number bases, it is not useful.

 

[charminglystrange]

Freaky... the more I learn the less I know...

Makes you appreciate the fact that people like Mr Russell (https://en.wikipedia.org/wiki/Principia_Mathematica) had to spend hundreds of pages to prove what people learn at around 3 years old, since nothing is ever "obvious" if you look at it in enough detail!

Don't forget the classic Four Colour Theorem, which took 139 pages, and some patient time in front of a computer to prove!

 

[Mark44]

One approach is: as each sheep exits the field in the morning you add a small stone to a pile.

And, in Latin, these stones are calculi (singular calculus), which makes the connection between what a dentist cleans from your teeth and the mathematical study of derivatives, integrals, and so on.

 

[PeroK]

And, in Latin, these stones are calculi (singular calculus), which makes the connection between what a dentist cleans from your teeth and the mathematical study of derivatives, integrals, and so on.

Calculus, calculum, calculi, calculo, calculi, calculos, calculorum, calculis.

I think that's right.