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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Inherently bases are just a way of writing numbers down. To make an analogy, I am typing this using the Roman alphabet with 26 letters. I could equally write the same words using the Greek alphabet with 24 letters, or the Cyrillic alphabet with 33 letters. Each letter has a sound meaning, so I could write English using one of those other alphabets, and it would communicate the same basic meaning. The meaning is not dependent on the writing system. The same applies for mathematics. I can write numbers using conventional Arabic numberals in base 10, but I could equally write them using Roman numerals, or use some other notation system. The actual properties of the numbers and how they work mathematically does not depend on how the numbers are written. Indeed the concept of algebra is based on the idea that the value of a number is not important, but the relationship between them is, so we write numbers using letters to indicate that the number in question could have any value.
This is *why* they use prime numbers to communicate.
I’m unclear on how it would work electronicaly, but in person, I’d show them the symbols for 1, 2, 3, 5, 7, 11, 13, 17 etc, and then expect them to show me *their* symbols for the same things.
We have a universal truth (more or less) that is easily expressed and understood *independent* of the language or symbols used. That lets us swap symbols.
It’s kind of like holding up an apple, and saying “apple” slowly, only the aliens probably don’t know what an apple is, nor do they know I’m not saying “red,” or “round” or “fruit” or any of several other options. But the know perfectly well what prime numbers are.
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.
You have 16 rocks. You can arange this in several grids. 4×4, 2×8. 1×16. With 7 rocks there is only one arrangement (ignoring difference between 1×7 and 7×1).
Now think of different bases as just naming numbers a different way. That’s really all it is. So although you’d call 7 rocks something different, it still retains that property of only being able to be arranged one way.
I feel like other comments do not address this properly.
`13` in `base15` is not a prime. Since it’s divisible by `2` in same base.
But `13` as in thirteen apples will always be a prime no matter if you write it as `13` or `D`
Are prime numbers still prime in French?
Others have said the answer that primes are primes are primes, no matter what. But I think they have all failed to mention why the base notation of the number does not matter. Base notation is a way of using shorthand to write large numbers and that is literally it. It does no more than that. Without a base numbering system you have tallies where one tick represents one thing. If you want to represent 1M things you would have 1M ticks. You want to expand to 2M things, that is another 1M ticks. But with base 10, if you want to represent something up to 1B, it requires 9 digits. Want to expand to 10B, your notation requires just one more digit. So again, the properties of the number and how it divides is no different. Base 10 is useful because things in counts that we typically run across are readable when put in base 10 notation.
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