Mystery of 2.3: Is it Even or Odd?

In summary, the terms "even number" and "odd number" are only used for whole numbers and are not defined for non-integer numbers. Some authors consider "whole numbers" to include zero, while others only include positive integers. The terms "natural numbers" and "whole numbers" can also be used interchangeably, depending on the context. In order to extend the concept of even and odd to rational numbers, one can break down numbers into prime factorizations, but this is not a commonly used method. The uniqueness of factorization still holds for rational numbers, and this can be proven using Bezout's Identity. However, these distinctions may fail for infinite sets.
  • #1
The Rev
81
0
Sorry if this is the most elementary question ever, but hey, I gots ta know man!

Is the number 2.3 even or odd?

:confused:

The Rev
 
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  • #3
Neither......
 
  • #4
The terms “even number” and “odd number” are only used for whole numbers.
 
  • #5
Evo said:
The terms “even number” and “odd number” are only used for whole numbers.
Actually, they are used for all integers.

[tex]for~n~\epsilon ~ \mathbb{Z},~~n = 2k,~~k~ \epsilon ~ \mathbb {Z} => n~even,~~else ~ n ~odd[/tex]
 
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  • #6
Gokul43201 said:
Actually, they are used for all integers.

[tex]for~n~\epsilon ~ \mathbb{Z},~~n = 2k,~~k~ \epsilon ~ \mathbb {Z} => n~even,~~else ~ n ~odd[/tex]
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
 
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  • #7
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"..
 
  • #8
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.
 
  • #9
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
 
  • #10
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!
 
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  • #11
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...
 
  • #12
One way to extend odd and even to the rationals would be to break down numbers into prime factorizations [tex]N=2^{a_0}3^{a_1}5^{a_2}\cdots[/tex], calling N even if [tex]a_0>0[/tex] and odd otherwise.

[tex]2.3=2^{-1}5^{-1}23[/tex] would be odd under that definition, while [tex]\frac49=2^23^{-2}[/tex] would be even.
 
  • #13
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 ?
 
  • #14
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!
 
  • #15
But congruence modulo 2 is base invariant... :confused:
 
  • #16
HallsofIvy said:
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!)
 
  • #17
Gokul43201 said:
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 [tex]\mathbb{Q}[/tex]. They're defined as numbers that can be represented as [tex]a/b,a\in\mathbb{Z},b\in\mathbb{Z},b\neq0[/tex]. 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 [tex]3.2=\frac{32}{10}=\frac{16}{5}=2^45^{-1}[/tex], for example.
 
  • #18
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.
 
  • #19
If 0 is even, then can we say that there's exactly 1 more even number than odd?
 
  • #20
Gokul43201 said:
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...
 
  • #21
CRGreathouse said:
I was just showing that factoring them in the way I described makes the factorization unique...
Of course it does. :redface:

I guess I was just too hung up on Bezout ! :grumpy:
 
  • #22
Icebreaker said:
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?
 
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  • #23
Icebreaker said:
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.
 

1. Is the number 2.3 even or odd?

The number 2.3 is neither even nor odd. It is a decimal number and does not fit into the categories of even or odd numbers.

2. What is the significance of the number 2.3?

The number 2.3 is significant because it is a part of the decimal number system and represents a fraction between 2 and 3. It is also an irrational number, meaning it cannot be expressed as a ratio of two integers.

3. Can the number 2.3 be simplified?

No, the number 2.3 cannot be simplified as it is already in its simplest form as a decimal number.

4. How is the number 2.3 represented in different number systems?

In the binary number system, 2.3 is represented as 10.0100110011... and in the hexadecimal number system, it is represented as 2.4C.

5. What is the significance of the decimal point in the number 2.3?

The decimal point in 2.3 separates the whole number (2) from the decimal part (0.3). This allows us to represent fractions and values between integers in the decimal number system.

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