I’ve started reading a lot of sci-fi and the humans always attempt to communicate with aliens using prime numbers, but if they use a counting system that _isn’t_ base10, would the prime numbers still make sense?
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There is something bigger going on here that should be brought out: nearly *nothing in math* depends on base 10 because almost no important mathematical concepts are defined in terms of base expansions: polygons, curves, functions, square roots, pi, vector spaces, *primes*, metrics, groups, fields, limits, series, … absolutely none of this stuff involves a choice of base.
Perhaps, depending on your background, you think real numbers are defined in terms of bases. This isn’t really the case: a choice of base gives you a way to do numerical calculations (like estimating the square root of 2 very accurately, assuming you don’t know about continued fractions) but you don’t need base expansions to give a definition of real numbers, although maybe you have to take a higher-level course in real analysis to see that. Admittedly most people only know about real numbers as infinite decimal expansions (with some funny business like .999… = 1 sometimes), but that is because in school we are never taught any other way to conceptualize real numbers.
You may have heard about divisibility tests in base 10 for numbers other than 10, like a number is divisible by 3 (resp. 9) exactly when the sum of its digits is divisible by 3 (resp., 9). These things are base-dependent, e.g., in base 5 that rule for divisibility by 3 doesn’t work anymore: 147 in base 10 is divisible by 3 since 1+4+7 = 12 is divisible by 3, but in base 5 we have 147 = 1042*_5_* and the sum of the base 5 digits 1+0+4+2 is not divisible by 3. You can create divisibility tests in any base you want, but the test for a specific number will change from one base to the next. As an example, in base 10 the tests for divisibility by 7 look complicated, but in base 8 the divisibility test for 7 is just checking that the sum of the base 8 digits is divisible by 7: in base 10, 2989 is not obviously a multiple of 7, but in base 8 this number is 5655*_8_*, and the sum of those digits 5+6+5+5 is divisible by 7, so 5655*_8_* is divisible by 7 too.
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