Mystery of 2.3: Is it Even or Odd?

[The Rev]

Sorry if this is the most elementary question ever, but hey, I gots ta know man!

Is the number 2.3 even or odd?



The Rev

 

[hypermorphism]

AFAIK, the quality of being even or odd is defined only for integers. See http://mathworld.wolfram.com/EvenNumber.html .

 

[phoenixthoth]

Neither......

 

[Evo]

The terms “even number” and “odd number” are only used for whole numbers.

 

[Gokul43201]

The terms “even number” and “odd number” are only used for whole numbers.

Actually, they are used for all integers.

for~n~\epsilon ~ \mathbb{Z},~~n = 2k,~~k~ \epsilon ~ \mathbb {Z} => n~even,~~else ~ n ~odd

 

[Evo]

Actually, they are used for all integers.

for~n~\epsilon ~ \mathbb{Z},~~n = 2k,~~k~ \epsilon ~ \mathbb {Z} => n~even,~~else ~ n ~odd

Yes, but that's what I was referring to. I guess that's not as clear though.

"Some authors also interpret "whole number" to mean "a number having fractional part of zero," making the whole numbers equivalent to the integers.

http://mathworld.wolfram.com/WholeNumber.html

 

[arildno]

Well, I have always thought "whole numbers" was just a colloquialism for integers, so I was surprised that others, like Gokul, use it for the naturals.

In fact, the correct Norwegian word for "integers" is "heltall" which, literally translated, is "whole numbers"..

 

[Hurkyl]

Actually, the way I usually see them used, "whole number" and "natural number" are usually different by one number -- one includes zero, and the other doesn't.

 

[HallsofIvy]

I go with Hurkyl here:

"Natural numbers" (also called "counting numbers") are 1, 2, 3,...

Peano's axioms originally included 0- since most texts do now start with 1, we have the "whole numbers" which includes 0, 1, 2,...

The integers include all negatives of the natural numbers

 

[Data]

Actually I usually use it the other way! Whole numbers to mean without zero and naturals with zero... which doesn't actually make much sense. I think I'll stick with positive, nonnegative, nonpositive, and negative, from now on!

 

[Alkatran]

This is why you can't extend it:
Let's say we have a numer:
1.5

We want to say it's odd because it ends with a 5. But what happens if we convert to base 8?
1.5(base 10) = 1.4(base 8)

We want to say it's even in base 8.

So we want to say the number is both even and odd, which is impossible, so it must be neither.


I suppose that you could technically say that a non-repeating rational number was 'base 10 even,' but I don't know of any application of that...

 

[CRGreathouse]

One way to extend odd and even to the rationals would be to break down numbers into prime factorizations N=2^{a_0}3^{a_1}5^{a_2}\cdots, calling N even if a_0>0 and odd otherwise.

2.3=2^{-1}5^{-1}23 would be odd under that definition, while \frac49=2^23^{-2} would be even.

 

[Gokul43201]

Forgive my naivete' (I'm a math ignoramus), but I didn't know that the fundamental theorem held outside the naturals. Is the proof of this a trivial extension of the proof of uniqueness (of factorization) within the naturals ?

 

[HallsofIvy]

Friedrich Engels (co-author of the "Communist Manifesto" with Karl Marx!), toward the end of his life, was working on a book applying "material dialectic" to the philosophy of science and mathematics. How much he actually understood of science and mathematics himself may be indicated by this:

He argued that the concept of "even" and "odd" was not a proper mathematical concept because it depended on the base! The number "8" in base 10 is even, but in base 5 it is "13", which is odd!

 

[Gokul43201]

But congruence modulo 2 is base invariant...

 

[Alkatran]

Friedrich Engels (co-author of the "Communist Manifesto" with Karl Marx!), toward the end of his life, was working on a book applying "material dialectic" to the philosophy of science and mathematics. How much he actually understood of science and mathematics himself may be indicated by this:

He argued that the concept of "even" and "odd" was not a proper mathematical concept because it depended on the base! The number "8" in base 10 is even, but in base 5 it is "13", which is odd!

Now there's someone who doesn't understand bases. lol! (it's a 3, 3 means odd! NO THAT'S IN BASE 10!)

 

[CRGreathouse]

Forgive my naivete' (I'm a math ignoramus), but I didn't know that the fundamental theorem held outside the naturals. Is the proof of this a trivial extension of the proof of uniqueness (of factorization) within the naturals ?

I was talking about the rational numbers \mathbb{Q}. They're defined as numbers that can be represented as a/b,a\in\mathbb{Z},b\in\mathbb{Z},b\neq0. Put a and b in lowest terms (no common factors) and write out a facorization, putting all of the factors of a positive and b negative.

So 3.2=\frac{32}{10}=\frac{16}{5}=2^45^{-1}, for example.

 

[Gokul43201]

CRG,

I understand how to factorize a rational using integer exponents.

What I wanted to prove is that the factorization is unique. A little thinking (which I was lazy to do, the first time) has led me to believe that the proof of the fundamental theorem (through Bezout's Identity) can be extended to the rationals without too much trouble. So forget I asked.

 

[Icebreaker]

If 0 is even, then can we say that there's exactly 1 more even number than odd?

 

[CRGreathouse]

What I wanted to prove is that the factorization is unique. A little thinking (which I was lazy to do, the first time) has led me to believe that the proof of the fundamental theorem (through Bezout's Identity) can be extended to the rationals without too much trouble. So forget I asked.

I was just showing that factoring them in the way I described makes the factorization unique, which obviates the need for other methods of proof.

Not that I have anything against Bezout's Identity, of course...

 

[Gokul43201]

I was just showing that factoring them in the way I described makes the factorization unique...

Of course it does.

I guess I was just too hung up on Bezout !

 

[honestrosewater]

If 0 is even, then can we say that there's exactly 1 more even number than odd?

I hope someone answers, because I think such distinctions fail for infinite sets. Consider the set of positive integers N and the set of nonnegative integers M. Every member of N is a member of M and there is exactly one member of M (0) which isn't a member of N, so N is a proper subset of M. However, N and M are bijective (consider the function f(x) = x + 1 from M to N). So in what way can M have more members than N?

 

[hypermorphism]

If 0 is even, then can we say that there's exactly 1 more even number than odd?

The set of all even numbers has the same cardinality as the set of all odd numbers (This means that there exists a 1-1 function between even numbers and odd numbers). Given that the cardinality is countable, the addition of a countable set of numbers to either set will not change the cardinality.