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

• B
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## Main Question or Discussion Point

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
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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.

• etotheipi
Merlin3189
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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
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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
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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]

• etotheipi
Mark44
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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).

• sysprog, Merlin3189 and etotheipi
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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
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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.

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• etotheipi
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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
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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.

• etotheipi and Klystron
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 . . .

• • Dale, etotheipi and FactChecker
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 and sysprog
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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.

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jbriggs444
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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.

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• Klystron, hutchphd and etotheipi
Mark44
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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 10B 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.

• Klystron, etotheipi and jbriggs444
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In any numbering system with base B, there are B digits: 0, 1, 2, ..., B-1. A number 10B 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.

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jbriggs444
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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.

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

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• sysprog and etotheipi
Mark44
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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, $64_{10} = 10_{64}$.

etotheipi said:
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.
etotheipi said:
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.
etotheipi said:
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.

• sysprog and etotheipi
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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
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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
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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.

• • Klystron, sysprog, Merlin3189 and 2 others
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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.

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"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.

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jbriggs444
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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"]

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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 **********.