Number Theorems and Number Bases

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In summary, the conversation is discussing the validity of number theorems across different number bases. It is mentioned that some theorems are true regardless of base, while others are only true in particular bases. The idea of "numerals" and their representation of numbers is also brought up. The conversation ends with a comment about philosopher Friedrick Engels and his misunderstanding of mathematics.
  • #1
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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  • #2
What do you consider a number theorem to be?
 
  • #3
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.
 
  • #4
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.
 
  • #5
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...
 

Related to Number Theorems and Number Bases

1. What is a number theorem?

A number theorem is a mathematical statement or rule that explains the properties and relationships of numbers. These theorems are used to prove various mathematical concepts and can help us understand the behavior of numbers in different situations.

2. What is a number base?

A number base, also known as a radix, is the number of different digits or symbols used to represent numbers in a counting system. The most commonly used number base is the decimal system, which has 10 digits (0-9). Other number bases include binary (2 digits), octal (8 digits), and hexadecimal (16 digits).

3. What is the difference between a prime number and a composite number?

A prime number is a number that is divisible only by 1 and itself, while a composite number is a number that has more than two factors. In other words, a prime number can only be divided by 1 and itself, while a composite number can be divided by at least one other number.

4. How are number theorems and number bases related?

Number theorems and number bases are closely related because many number theorems are about the properties and relationships of numbers in different bases. For example, Fermat's Little Theorem states that for any prime number p, a^(p-1) is congruent to 1 (mod p), which is useful in understanding the behavior of numbers in different bases.

5. How are number theorems and number bases used in cryptography?

Number theorems and number bases are heavily used in cryptography, which is the science of writing and solving codes. In particular, modular arithmetic and prime numbers are essential in creating secure encryption methods. Number bases are also used in encoding and decoding messages, as different bases can represent characters and symbols in a code.

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