Number Theorems and Number Bases

[Euan]

I am probably not phrasing this question precisely enough but ...

... are Number Theorems true regardless of the Number Base?

In particular, is any given Number Theorem that is true in Base 10 equally true in Base 2?

Thank you.

Euan

 

[d_leet]

What do you consider a number theorem to be?

 

[CRGreathouse]

Some things are true regardless of base. (a + b)(a - b) = a^2 - b^2 in base 2, base 10, or any other base.

Some things are true only in particular bases. Mersenne primes are all 1s in base 2 but not base 10; a number is divisible by three iff the sum of its digits is divisible by three in base 10 (and base 7, base 4, base 13, ...) but not in base 2.

 

[HallsofIvy]

Assuming it really is a number theorem and NOT about "numerals", then, yes, every theorem is true independent of number base.

Number base and "numerals" are how we represent numbers, not the numbers themselves.

By the way, the philospher, Friedrick Engels, co-author, with Karl Marx, of the "Communist Manifesto", was, toward the end of his life, working on applying "dialectic realism" to science and mathematics. Just how much he actually understood of mathematics, at least, can be judged by his saying that many mathematics ideas only applied to some bases. For example, the number "15" is odd in base 10, but would be written as "30", and so be even, in base 5!

In order to write that you have to have a complete misunderstanding of what "even" and "odd" mean.

 

[CRGreathouse]

By the way, the philospher, Friedrick Engels, co-author, with Karl Marx, of the "Communist Manifesto", was, toward the end of his life, working on applying "dialectic realism" to science and mathematics. Just how much he actually understood of mathematics, at least, can be judged by his saying that many mathematics ideas only applied to some bases. For example, the number "15" is odd in base 10, but would be written as "30", and so be even, in base 5!

In order to write that you have to have a complete misunderstanding of what "even" and "odd" mean.

Of course, Engels *was* a kook...