Why is base-10 decimal?

Question

A way to write a number in various bases is: 101₂, 3Fa₁₆, 519₁₀.

The thing here, is that we apparently specify the base in decimal by default. This makes sense in everyday life, since we're not really doing base conversion when grocery shopping. But 10 is ambiguous when working with different bases. So why is 101₁₀ not binary or ternary or octal when working specifically with base conversion?

How about 101₁₀₂? Is 111₁₁₁₃ = 183₁₀ = 183?

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Related:

• Why do they call it base 10?
• Name for "decimals" in other bases?
• Base ten is called "decimal"; what's the name of numbers in base 15?

Answer 1

Yes, you could indicate the base of the base too, but eventually you need a base-indicator that is specified in some "default" base. We normally use base ten so if nothing else is stated that is what the base that is assumed.

Answer 2

As far as I know, the base is always specified in decimal, but it is up to you to change that convention, provided you state it clearly.

The following shorthands are universally adopted: $b$ for two, $o$ for eight, $d$ for ten, $h$ for sixteen.

If you want a system that can denote any base without extra conventions, you can resort to unary:

$$10_{||||||||||||||||}=16_{||||||||||}=51_{|||}.$$

This is why th Babylonians dropped their base $60_{||||||||||}$ numeration ($_{||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||}$).