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Now, I know intuitively why it's called base 10: because there's 10 numbers.

But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at 9. Why isn't it called base 9?

If it's because 10 is the number we round up to,that logic doesn't seem to transcend to other bases because every base would literally be called base 10 if we were to extend that logic.

So, in using a system with 9 unique characters, and calling it base 10, this leads to inconsistencies with every other single base. Take binary, base 2, it's called base 2 though uses 1 character, hexadecimal, called base 16, uses 15 char (1-9 [not 10 though] and A-E).

Now in binary, octal, and hexadecimal, it's not really an issue, but what about an arbitrary base n? What do we call it then?

We could assign a name to it like we have with 'base 2' i.e binary, but if we want to work within a new base, then to communicate you're working in base n, it becomes tedious.

The most logical thing to do, in my opinion, would be to call the base by the last unique number you have to work with. This is easier to remember, and resolves the issue of nomenclature.

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    $\begingroup$ It's because you're counting in powers of $10$ $\endgroup$ – ClassicStyle May 24 '16 at 4:28
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    $\begingroup$ I think the OP is trying to talk about the fact that the number $n$ written in base $n$ is "$10$" which we usually call "ten." We just need to separate the notation "10" from the actual number it represents when we change bases. $\endgroup$ – angryavian May 24 '16 at 4:39
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    $\begingroup$ This question is why I generally try to write the name of our common numbering system as "base ten" or "the decimal system" instead of "base $10$". $\endgroup$ – David K May 24 '16 at 4:44
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    $\begingroup$ Just count zero as a digit (because you actually use it!) and you'll get ten different digits, $0..9$, not nine $1..9$, hence the base is ten, not nine. $\endgroup$ – CiaPan May 24 '16 at 7:19
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    $\begingroup$ When I tell you in English "This book is in French" I use the English word for French. It doesn't matter that in French "French" is "Francais" and I didn't make a mistake when I said "French" instead of "Francais".... because I'm speaking in English! Not French! So if I'm counting in a number system based on powers of 7 and in this number system the basis of the number system is called 10 I say "I'm counting in base 7" instead of the base 7 "I'm counting in base 10". Why? Base 10 is our native system. That's all. $\endgroup$ – fleablood May 24 '16 at 7:27
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The name refers to the way we choose to group our items. The way we've evolved, we found that nine counting symbols and a symbol for nothing suffice for our preferred base. We're able to recycle the glyphs $1$ and $0$ to denote our grouping, which is given by the combined symbol $10$.


To illustrate why it's all about the grouping, and not about the final unique symbol, let's roll with this example. Pick an amount of symbols that can use to enumerate your fingers. You may prefer $$1,2,3,4,5,6,7,8,9,X$$ but another could have just as easily chosen $\odot,\triangle,\square,\cup,\star,\vee,\uplus,\times,+,\LaTeX$. These are just glyphs at this point, markers for each of the fingers. For convenience, let's use my suggested set.

We have an issue, I want to add more symbols to keep track of toes. Okay, how about:

$$1,2,3,4,5,6,7,8,9,X$$ $$1',2',3',4',5',6',7',8',9',X'$$

Man, that's a lot of symbols. I like the number of my fingers, $X$, and in fact, you could say I have $X$ toes. Wow, grouping things by $X$ is convenient, both physically and visually. Can I keep track of these groups by repeating my symbols to enumerate larger quantities in groups of $X$?

Well, it would be nice if I started counting from nothing, so let's add a symbol for that. $$0$$ $$1,2,3,4,5,6,7,8,9,X$$

Now, how should this work? Well, I can have $X$ objects. Or I can have $X$ groups of $X$ objects each. Or, wow, even $X$ groups of $X$ groups of $X$ objects! It seems like reaching $X$ marks a sort of restart point. I could denote this regrouping by saying I have $1$ $X$ and $0$ other stuff. Okay, I feel inspired. Let's combine those symbols to replace $X$:

$$0$$ $$1,2,3,4,5,6,7,8,9,10$$

And on and on it goes... This was more or less a colorful way of expressing BigBearzzz's answer: each digit's location denotes a higher grouping. We don't group every nine objects, though. We group ten objects.


What about other bases? Well, say we choose to group every three objects. We would still want a symbol for nothing, $0$, another for one $1$, for two $2$, and once again we can recycle our glyphs to call one group of three $10$. That's where we regroup. Note that these are just glyphs, recycled from the base ten representations. The meaning is different here than in base ten due to where I choose to regroup. Why is this called base three? Well, I'd like to refer to this new system, and I choose to borrow the name from a convenient source: my favorite base of all, ten.

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  • $\begingroup$ So... it's a convention. $\endgroup$ – Asaf Karagila May 24 '16 at 16:13
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You're mixing the actual number with its representations. Yes, in base $n$, the string $10$ represents exactly $n$.

But now forego of the decimal representation of $10$, and think about it as "how many digits a healthy human being has on both their hands". This is your $n$, now. Let's for the sake of simplicity call this number "ten".

Now we are counting in base ten, and that's that. We are taught to innately think in terms of this base, which is why we call it base $10$.

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    $\begingroup$ @user, usually we contort ourselves and come up with an appropriate Latin or Greek prefix to prepend to "-ary"... :) $\endgroup$ – J. M. is a poor mathematician May 24 '16 at 4:47
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    $\begingroup$ @user2901512 Would calling them base two, base eight and base sixteen not work well? The upshot is that ten has an unambigous meaning without reference to any base but the meaning of $10$ varies from base to base. $\endgroup$ – Jyrki Lahtonen May 24 '16 at 4:47
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    $\begingroup$ @Cole: While it is tempting to argue that because we have ten fingers, we count in base ten, it is probably false. Many ancient cultures counted in base 60, and some bushmen tribes in Africa do not have names for specific numbers, even those we can represent on a single hand (and look at little children having difficulty counting above three (also "this many!" with some fingers raised)). But in the modern society we are being taught to innately count in base ten. $\endgroup$ – Asaf Karagila May 24 '16 at 7:01
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    $\begingroup$ @user2901512: I'm sorry, but tough luck for you, then. Everything begins with conventions. Why do we chose the inference rules as we did? We did we call $0$ "zero" and not "flugelhorn"? Why is Hausdorff space "every two points have disjoint open sets separating them" and not "every cover has a finite subcover"? These are all conventions, used to communicate the ideas between one mathematician to another in an unambiguous. $\endgroup$ – Asaf Karagila May 24 '16 at 7:34
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    $\begingroup$ And we are consistent when we refer to base n with the decimal expression of n. There is nothing inconsistent or ambiguous about it. $\endgroup$ – fleablood May 24 '16 at 7:41
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This is more like a comment but it's too long, so I'm putting it as an answer instead, please accept my apology.

In base $10$, the "symbol" $78152_{10}$ represents the number

$78152_{10}=7\cdot 10^4 + 8\cdot 10^3 + 1\cdot 10^2 + 5\cdot 10^1 + 2\cdot 10^0$.

In base $n$, the "symbol" $78152_n$ represents the number

$78152_n=7\cdot n^4 + 8\cdot n^3 + 1\cdot n^2 + 5\cdot n^1 + 2\cdot n^0$.

You can see that we prefer to call a base by the number that is raised to the power of its position from the last digit. That is why we call it the base.

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  • $\begingroup$ But wouldn't you still call it "base 10" if "10" referred to * * instead of to * * * * * * * * * * ? $\endgroup$ – Simpson17866 May 24 '16 at 4:37
  • $\begingroup$ But, then, isn't it always base 10? In binary, we stop counting at 1, the last digit, and then move to 10. $\endgroup$ – user2901512 May 24 '16 at 4:38
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    $\begingroup$ You should ask, "isn't it always base ten?" No, it could be base two, base three, etc. $\endgroup$ – littleO May 24 '16 at 4:39
  • $\begingroup$ Yeah but in base 2, you don't have the number two to express two. To understand what I mean, it might help to work the other way. For e.g, in higher bases, say base 16, we express say 15 as F, but we not in base 10 $\endgroup$ – user2901512 May 24 '16 at 4:41
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    $\begingroup$ @littleO I like this new convention: always spelling out the number (two, three, ...) instead of using the base ten representation of the new base. We should make it more popular :) $\endgroup$ – Simpson17866 May 24 '16 at 4:42
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When I work with numbers in base 16 or base 2, the string "10" is not called "ten", but "one zero". We generally don't have names for specific integers that reflect some other base, so although a full 16 bit value is ffff and then 1 0000 is significant to me, I don't have a spoken name for it that corresponds to thousand. It's only known as "sixty four K", which is nothing like how it's written.

The spoken "sixteen" refers to the specific natural number, no matter how else it is noted. The English name comes from a base-10 herritage, but we could have had a special word for it that stands alone, like dozen, gross, or mole. Maybe someone will introduce one and it will catch on.

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  • $\begingroup$ Ah but you see, this isn't an issue with what we call the numbers in other bases, as Asaf says, we use base 10 as the standard. The matter is what we call the bases, if base 10 is base 10 because we round to 10, then every base is base 10, apart from some exceptions like binary, octal, etc. Now, we do have names for the first n bases, but no names for anything beyond base n (whatever n means). So for e.g, what is base 15, 17, 23, 101? We can't just keep coming up with new names like binary, it becomes impractical. $\endgroup$ – user2901512 May 24 '16 at 6:31
  • $\begingroup$ Note also the logical inconsistency between how we call binary base 2, and decimal base 10. In the latter, we do so because it's the next number in base 10, but in base 10, we do so because it's what we round up to. So not only is it logically inconsistent, but introduces complexities further down the line for other bases. $\endgroup$ – user2901512 May 24 '16 at 6:35
  • $\begingroup$ Round up to? I don't understand. Inconsistent, no: regardless of the name or current notation in use, the integer that serves as the base of the polynomial is the successor of the largest one used for a single digit. If "ten" bothers you, use the chinese symbol 十 instead. $\endgroup$ – JDługosz May 24 '16 at 6:43
  • $\begingroup$ ''the integer that serves as the base of the polynomial is the successor of the largest one used for a single digit''. If that were the case though, then base 10 would be called base 9, and Binary would be referred to as base 1. $\endgroup$ – user2901512 May 24 '16 at 6:57
  • $\begingroup$ Are you not paying attention? 9 is the largest digit, the successor of that is ten, not nine again. No number is its own successor. $\endgroup$ – JDługosz May 24 '16 at 17:44

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