maikochan
User #69
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- Nov 12, 2008
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Just had a thought while in the shower that I wanted to write down, even though it's pretty trivial. I'll just use this spot for any thoughts I get that I'd like to share, though likely not important ones.
Anyway, on to the inaugural musing.
Is Zero an Even Number or an Odd Number?
Yes, I know this is a pretty simple argument in number theory, and a pretty trivial one in general, but hey, I wanted to practice some logic.
So, this question has two parts, first, if the number Zero is Even, and second, if it's Odd. Let's tackle the first part, well, first.
First we need to establish what an even number is. I've come up with three criteria as follows: Using Integers for simplicity and because I'm not entirely sure of how fractional or irrational numbers are defined as even or odd.
First: If Zero is divided by two, is the result an integer?
Yes. Zero is an integer (by convention). Zero divided by any number results in Zero. Formally, 0/X = 0. As Zero is an integer, criteria #1 is met.
Second: If Zero is multiplied by any number, will the resultant number pass criteria #1?
Yes. Zero multiplied by any number results in Zero. Formally, 0*X = 0. As Zero is shown to pass the criteria #1, Zero also passes criteria #2.
Third: Can the number two be reached by steps of two from Zero?
Yes. Simply adding two to Zero results in two. Formally 0+2=2. Thus criteria #3 is met.
From these criteria we can conclude that Zero is an even number.
To determine if Zero is an Odd number, we can assume that no number can be both even and odd, and since we have proven that Zero is an even number, Zero cannot be an Odd number.
So that's that. Any number theorist or a university student who is even tangentially involved with mathematics could probably point to any number of problems with my proofs. But it's been a couple of years since I did any proofs like this, and it just felt good to go through the motions, as it were. And again, the result is trivial, but it's the journey, they say, and not the destination, that matters.
I welcome any input on this or any other musing I post. For now, Maiko out.
Anyway, on to the inaugural musing.
Is Zero an Even Number or an Odd Number?
Yes, I know this is a pretty simple argument in number theory, and a pretty trivial one in general, but hey, I wanted to practice some logic.
So, this question has two parts, first, if the number Zero is Even, and second, if it's Odd. Let's tackle the first part, well, first.
First we need to establish what an even number is. I've come up with three criteria as follows: Using Integers for simplicity and because I'm not entirely sure of how fractional or irrational numbers are defined as even or odd.
- If a number divided by two produces an integer answer, then the number is even. Formally: X is an integer. If X/2 is an integer, then X is even.
- If a number is multiplied by any other number, and the result creates an integer when divided by two, then the first number is even. Formally: X is an Integer. Y is an Integer. If (X*Y)/2 is an integer, then X OR Y is even. (Logical OR) This can also be read as: If (X*Y) passes criteria #1, then X OR Y is even.
- Integers alternate between even and odd as one progresses along the number line. If a number can reach the number two by adding or subtracting in multiples of two, then the number is even. Formally: Not going to try this as it would require induction, which I'm not so great at.
First: If Zero is divided by two, is the result an integer?
Yes. Zero is an integer (by convention). Zero divided by any number results in Zero. Formally, 0/X = 0. As Zero is an integer, criteria #1 is met.
Second: If Zero is multiplied by any number, will the resultant number pass criteria #1?
Yes. Zero multiplied by any number results in Zero. Formally, 0*X = 0. As Zero is shown to pass the criteria #1, Zero also passes criteria #2.
Third: Can the number two be reached by steps of two from Zero?
Yes. Simply adding two to Zero results in two. Formally 0+2=2. Thus criteria #3 is met.
From these criteria we can conclude that Zero is an even number.
To determine if Zero is an Odd number, we can assume that no number can be both even and odd, and since we have proven that Zero is an even number, Zero cannot be an Odd number.
So that's that. Any number theorist or a university student who is even tangentially involved with mathematics could probably point to any number of problems with my proofs. But it's been a couple of years since I did any proofs like this, and it just felt good to go through the motions, as it were. And again, the result is trivial, but it's the journey, they say, and not the destination, that matters.
I welcome any input on this or any other musing I post. For now, Maiko out.