is 0 even or odd?
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Define the argument to win the argument.
Orange could be "even" if you define it as an integer; assuming orange divided by two is still orange.
People get confused when they equate zero with "nothing", but they are two different concepts. While "nothing" is an ill defined concept describing absence. Zero is a number rigidly defined by the properties assigned to it. To mix up the two is like comparing apples and oranges.
Orange could be "even" if you define it as an integer; assuming orange divided by two is still orange.
People get confused when they equate zero with "nothing", but they are two different concepts. While "nothing" is an ill defined concept describing absence. Zero is a number rigidly defined by the properties assigned to it. To mix up the two is like comparing apples and oranges.
feather240 said:
Ah. I was viewing it as a fraction (12 over 3 x (3+3)). I'm used to seeing the division sign when it's written in this context. Of course, since keyboards typically don't have those, this makes sense.
An element of the integers is even if it is divisible by 2. In other words, is 0/2 an integer. Even more precisely, is there an integer n such that 0 = 2*n. There is such an n - let n = 0.
Hence, 0 is even.
Alternatively, we can define the even integers to be all integers in the set {2k | k is an integer}, but that feels like cheating a little. (Equivalently, define the evens to be all elements of Z/2Z, to bring in some group theory).
And again, we could also define n to be even if n is congruent to 0, modulo 2. Hence, 0 is even.
Oh, and assuming you're willing to agree that all integers are numbers, then 0 is a number - if the integers must include 0, or Z would fail to be an additive group.
And all this assumes you're talking about 0 the mathematical object, not as a philosophical concept of nothingness (although if you're talking about parity, then it seems reasonable to assume you're talking about 0 the mathematical object).
Hence, 0 is even.
Alternatively, we can define the even integers to be all integers in the set {2k | k is an integer}, but that feels like cheating a little. (Equivalently, define the evens to be all elements of Z/2Z, to bring in some group theory).
And again, we could also define n to be even if n is congruent to 0, modulo 2. Hence, 0 is even.
Oh, and assuming you're willing to agree that all integers are numbers, then 0 is a number - if the integers must include 0, or Z would fail to be an additive group.
And all this assumes you're talking about 0 the mathematical object, not as a philosophical concept of nothingness (although if you're talking about parity, then it seems reasonable to assume you're talking about 0 the mathematical object).
Regarding syntax of a sequence of integers, it is an even number.
It is a "concept" just like any other numerical value. (Come to think of it, when was 'zero' conceptualized and utilized as a quantification? Makes me think how this was established..)
One is a single unit, a quantitative value.
Zero is also a quantitative value, and exists as no groups of a number of objects.
The grouping exists regardless. Zero is even, plain and simple.
EDIT: Ah, Babylonians [http://en.wikipedia.org/wiki/0_%28number%29#Early_history].
It is a "concept" just like any other numerical value. (Come to think of it, when was 'zero' conceptualized and utilized as a quantification? Makes me think how this was established..)
One is a single unit, a quantitative value.
Zero is also a quantitative value, and exists as no groups of a number of objects.
The grouping exists regardless. Zero is even, plain and simple.
EDIT: Ah, Babylonians [http://en.wikipedia.org/wiki/0_%28number%29#Early_history].
Numbers have a sequence: even odd even odd., as 1 and -1 are odd, zero, the number in between the is even, where were you during year 1 maths? Additionally... [http://lmgtfy.com/?q=0&l=1]
I believe you want 2Z, not Z/2Z. Z/2Z has a total of 2 elements. (By the way, it makes more sense to deal with rings here.)fuzzygenius said:
Zero is even. It makes sense if you think about it in context of its surrounding numbers. 1 is clearly odd and so is negative one. Numbers alternate being even or odd with every integer. For example, 3 and 5 are clearly odd therefore 4 must be even. By the same principle we can assume 0 must be even.
If that's a bit confusing, consider this:
An even number can be defined as a number that, after division by 2, is still a whole number. 4/2 = 2 while 3/2 = 1.5. If we consider 0 to be a whole number then we can see 0/2 = 0 which means that after division by 2, 0 is still a whole number. If we don't consider 0 to be a whole number then we can't qualify it as either so the discussion doesn't matter.
If that's a bit confusing, consider this:
An even number can be defined as a number that, after division by 2, is still a whole number. 4/2 = 2 while 3/2 = 1.5. If we consider 0 to be a whole number then we can see 0/2 = 0 which means that after division by 2, 0 is still a whole number. If we don't consider 0 to be a whole number then we can't qualify it as either so the discussion doesn't matter.
Either it's been edited, or you appear to be slightly dyslexic...Purple Shrimp said:
EDIT: It appears to have been edited, but it was before you posted?
Zero is an even number. In other words, its parity?the quality of an integer being even or odd?is even. Zero fits the definition of "even number": it is an integer multiple of 2. As a result, zero exhibits the properties shared by all even numbers: 0 is evenly divisible by 2, 0 is surrounded on both sides by odd integers, 0 is the sum of an integer with itself, and 0 objects can be split into two equal groups. Zero fits into the rules for sums and products of even numbers, such as even − even = even, so any alternate definition of "even number" would still need to include zero. Within the even numbers, zero plays a central role: it is the identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively generated. Every integer divides 0, including each power of 2; in this sense, 0 is the "most even" number of all.
http://en.wikipedia.org/wiki/Parity_of_zero
http://en.wikipedia.org/wiki/Parity_of_zero
