Number Theorems and Number Bases

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Discussion Overview

The discussion revolves around the validity of number theorems across different number bases, specifically questioning whether theorems true in Base 10 hold true in Base 2 and other bases. The scope includes theoretical considerations of number representation and properties of numbers in various bases.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the definition of a number theorem and its applicability across bases.
  • Another participant asserts that some mathematical statements, like (a + b)(a - b) = a^2 - b^2, are universally true regardless of the base.
  • It is noted that certain properties, such as Mersenne primes and divisibility rules, can vary between bases, indicating that not all theorems are universally applicable.
  • A participant argues that if a statement is truly a number theorem, it should hold independent of the base, emphasizing that number bases are merely representations of numbers.
  • There is a critique of Friedrich Engels' understanding of mathematics, particularly regarding the concepts of even and odd numbers in different bases, suggesting a misunderstanding of these properties.

Areas of Agreement / Disagreement

Participants express differing views on whether all number theorems are universally applicable across bases, with some asserting that certain theorems are base-dependent while others argue for a universal applicability of true theorems.

Contextual Notes

The discussion highlights the complexity of defining number theorems and the implications of base representation on mathematical properties, with no consensus reached on the universality of theorems across different bases.

Euan
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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
 
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What do you consider a number theorem to be?
 
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.
 
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.
 
HallsofIvy said:
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...
 

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