When working in the ubiquitous 'base 10', the actual number 10 is used. However, when working in, say, base 5 (base five) the actual number cannot appear, as it's represented by '10', so is it just for convenience that we call it 'base 5'. Surely there must be a more scientific/mathematical way to portray the base we're in.
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This is why I try to write "base ten" rather than "base $10$" (though I sometimes forget). Of course, to make that work you have to insist that the number five (the number $1+1+1+1+1$) does not get renamed to ten just because you're working in base five and writing the number five using the string "$10$".
In any case, the scientific/mathematical way to deal with the problem of naming the base of a positional number system is to have a system for the name of such a number system that is independent from the particular base value of that system.
One possibility is that you avoid using any positional number system to describe the base of a positional number system. You might use the successor function $S$ to name base $S(S(S(S(S(S(S(S(S(S(0)))))))))),$ which is the base we conventionally call "base ten." But that's not very convenient--imagine having to write all those $S$s (or even count them!) every time you mention a base ten number in a context where base ten isn't assumed.
On the other hand, "independent" does not have to mean "never coincident with." It is very convenient to have a compact representation of any particular integer $b$ that you can substitute into the phrase "base $b$", and base ten (mainly because it is so familiar to so many people) is a good candidate for the most convenient compact representation for the names of bases, for example "base $5$" for the quinary number system or "base $16$" for hexadecimal.
So it's quite common to see numbers in various place-value systems written with the digits of the number itself in the desired place-value system but annotated with the name of the base in decimal notation, for example, $$ 10_{16} = 16_{10} = 31_5 = 10000_2. $$
You could also (and some people, including myself, sometimes do) write the same set of equations using words to identify the bases, for example $$ 10_{\mathrm{sixteen}} = 16_{\mathrm{ten}} = 31_{\mathrm{five}} = 10000_{\mathrm{two}} $$ or $$ 10_{\mathrm{hexadecimal}} = 16_{\mathrm{decimal}} = 31_{\mathrm{quinary}} = 10000_{\mathrm{binary}}, $$ but this can get very cumbersome very quickly if you have to write a lot of numbers or if you use large numbers for some of the bases.
I do not see anything particularly unscientific or unmathematical about fixing base ten as the conventional notation in which to name the value $b$ in the expression "base $b$". It is admittedly rather anthropocentric, and even worse, "centric" on only a proper subset of "anthropo" when you really get down to it. But it does at least have the advantage that it's a very easy way to communicate with virtually every person with whom a human living on Earth today might want to do mathematics; certainly it's a trivial obstacle to universal understanding compared to other obstacles we put up with, such as the fact that we use English on math.stackexchange.
David K
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I'd forgotten this had been posted! Only one answer in a yr+! Maybe because it's trivial, or too obvious/difficult to explain? I've taught base maths to 8 and 9yr olds, and the biggest stumbling block, even when they click, is the fact that 10 looks like 10 (decimal). – Tim Apr 01 '18 at 07:27
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Even educated adults who are very familiar with place-value systems other than base ten sometimes cannot agree on how to pronounce the numeral 10 in a non-decimal system; if the base is five, do you pronounce 10 as "five" or "ten"? (I say "five," but I don't always find other people agree.) So I don't fault young children for having trouble with this. – David K Apr 01 '18 at 16:53
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I introduce this as 'Martian Maths', because Martians have only one hand, with five digits. Thus, '10' equals one handful. One handful and no fingers. Kids love it, and it's good fun, but we grown ups really need a more grown up nomenclature! – Tim Apr 01 '18 at 17:10
the actual number 10 is used? – goodvibration Feb 20 '17 at 17:00