I had discussed this before, but I thought it would be good to do a quick survey on this. Please note that this is not a trick question. Also please give what you seriously believe to be the correct answer; and please vote even if you think the answer is obvious.
kdackson says:
I vote even. If the number “2″ is even, and even numbers alternate, then counting backwards, “0″ is the next even number.
September 29, 2009, 12:30 pmyankee says:
As of right now, over 50% of respondents have given the wrong answer (i.e., anything other than “even”). Fortunately, the ability to call zero “even” doesn’t have much impact on the sort of math the vast majority of people have to do.
September 29, 2009, 12:31 pmMark N. says:
It seems this is a much-discussed subject.
September 29, 2009, 12:32 pmJohn Thacker says:
The Wikipedia article on this is very extensive and well-written.
It is possible to define prime numbers such that 1 is either prime or not. The reason it is now not considered prime is that far more theorems, like the Fundamental Theorem of Arithmetic, would require exceptions if 1 were treated as prime.
By contrast, while of course one could create a definition under which 0 was not even, it would require exceptions to essentially every theorem and postulate that deals with evenness and even numbers.
September 29, 2009, 12:32 pmalkali says:
Remarkably, there is a Wikipedia article about this very subject. (I won’t say whether the title is “Oddness of zero,” “Evenness of zero,” “Numbers that are neither odd nor even,” or “Simultaneous oddness and evenness of zero”.)
September 29, 2009, 12:35 pmJohn Thacker says:
It seems that volokh.com readers understand mathematics significantly less than do British 6th graders.
September 29, 2009, 12:36 pmJames J. says:
Only a Whole Number may be defined as odd or even. If I remember correctly, 0 is not considered a Whole Number and therefor cannot be odd or even.
I think a better poll would be if people think 0 is a Whole Number or not.
September 29, 2009, 12:39 pmbyomtov says:
It seems that volokh.com readers understand mathematics significantly less than do British 6th graders.
Illustrating the value of a law school education. :-)
September 29, 2009, 12:43 pmJohn Thacker says:
No, this would be an absolutely ridiculous definition. If you used this, then you would still need a word for “a number that can be expressed as 2n, where n is any integer” and “a number that can be expressed as 2n+1, where n is any integer.” They crop up a lot in mathematical theorems and facts. “Even” and “odd” are pretty good terms for those things.
September 29, 2009, 12:45 pmASlyJD says:
I understand the logic of defining “even” as “any number that that is divisible by two,” a definition under which zero certainly qualifies. However, the question of the oddness/evenness of zero seems far more akin to “how much dirt is in a hole 1 ft x 1 ft x 1 ft” than a serious mathematical inquiry.
Knowing odd & even rules are helpful as a simple check of one’s arithmetic.
(E+-E=E, E+-O=O, O+-O=E, E*/E=E, E*/O=E, O*/O=O)
Beyond that limited scope, it’s a trivial designation. In each of these cases, performing the operation with 0 doesn’t require use of the rule to check to insure the accuracy.
I will concede that math higher than Calc III may have some other use for the odd/even designation and am willing to be corrected if so.
September 29, 2009, 12:52 pmjab says:
James J.
You are mistaken. Zero is indeed a whole number.
September 29, 2009, 12:53 pmI think you are confusing whole numbers with natural numbers.
Natural numbers are the set of positive integers.
Henry says:
A typical mathematical definition has an intuitive core and then some technical cases around the edges when it comes to make the notion precise. We decide those edge cases based on what’s most convenient when the definition is used.
In the case of being even, the intuitive core is the positive even numbers, and the ambiguous edge is things like 0, negative numbers, and perhaps complex numbers. However the outcome is clear: it’s more natural and more useful to say that 0 is an even number, and that the negation of an even number is even. There is no benefit to restricting the property to only positive numbers, therefore we don’t.
September 29, 2009, 12:53 pmMike S. says:
0 = 0*2 so it is even (i.e. an integral multiple of 2).
The fact that there seem to be a number of posters who think a number can be both odd and even is disturbing.
September 29, 2009, 12:53 pmEinhverfr says:
The results here are really funny.
Less than 50% got the right answer?
September 29, 2009, 12:56 pmmath person says:
Actually, he is right, but both of you seem confused about what a whole number is. It means “not a fraction”, and is exactly the same as integer, be it positive, negative, or 0. What you are thinking of is natural numbers, which is positive integers, and possibly 0, depending essentially on personal choice.
So yes, 0 is whole, and so it’s even, but whether one chooses to call it natural or not is irrelevant.
September 29, 2009, 12:59 pmkdackson says:
For the next trick, let’s determine how many different infinities there are.
Hint: more than one.
September 29, 2009, 1:02 pmsteve says:
forget the math nerd rationale. zero is nothing. numbers that are odd and numbers that are even are something… and something can’t be nothing…. so zero is neither odd nor even.
September 29, 2009, 1:02 pmyankee says:
I kind of understand the people who believe that zero is neither odd nor even. They’re wrong but I can understand why a person without much familiarity with mathematics would believe the concept of oddness/evenness is not applicable to zero.
But what’s with the people who think zero is odd?
September 29, 2009, 1:03 pmkdackson says:
Yankee: I’d wager they think that because it’s divisible by 1, it’s odd.
September 29, 2009, 1:05 pmJohn Thacker says:
No, it affects whether the set of even numbers forms a group. It affects whether the even numbers form an ideal in the ring of integers with standard addition and multiplication.
Modern cryptography is based around the idea of mod 2(^n) arithmetic, and would need some term to describe the numbers that are 0 mod n. Those “simple checks” are fundamental to your everyday modern life, you just don’t think about it.
Calc III? Did you not learn about the concept of even and odd functions in calculus? f(x) = x^0 = 1 is fits the definition of an even function, not an odd function. The concept makes more sense if even includes 0.
The definition of the Moebius function is easier to state if 0 is even, and thus the derivation of the Moebius Inversion Formula.
Useful statements and things to do with the fundamental theorem of algebra (unique factorization theorem) work better if zero is even.
Yes, you could exclude 0 from even numbers. But you’d still use the set of numbers divisible by two all the time in advanced mathematics, and “even” is a good name for it.
The only useful time to exclude zero from “even” is if you want a nice term for a certain set of numbers that is a common bid in roulette. For everything else, adding zero makes things no worse and generally better.
More interesting similar concepts are discussed in the Wikipedia article on empty products.
September 29, 2009, 1:06 pmJohn Thacker says:
I’m not confused. He obviously meant a natural number, from his confusion about what “whole number” meant when he tried to define it. There’s a entirely separate discussion about what “whole number” means. Some people do use it to mean the natural numbers plus 0, but not the negative numbers. (I.e., the nonnegative integers.)
See for example the here. It’s silly to have “whole number” be a synonym for “integers.” Of course, the other definitions are silly too.
September 29, 2009, 1:08 pmAnnoyingly Anonymous says:
This wasn’t a math problem. This was a reading comprehension problem.
“please vote even(,) if you think the answer is obvious”
September 29, 2009, 1:09 pmA.S. says:
This minor point is now worthy of multiple posts? Really?
I am beginning to think someone (Glenn Beck?) has hacked into Eugene’s blog and is posting under his name. What is going on here?
I’m just raising the possibility that Eugene might be getting a bit too worked up about what is a pretty minor point. One would think EV, who is after all a public servant, would have something better to do with his time.
/parody
September 29, 2009, 1:10 pmJohn Jenkins says:
Isn’t any number n even if n mod 2 = 0? If that is the test, then zero is definitely an even number.
September 29, 2009, 1:11 pmJohn Thacker says:
The term whole number is ambiguous and has had different definitions. Some people do use it for the non-negative integers.
Since it’s a redundant and unclear term, it doesn’t matter whether you use it as a synonym for the integers, the nonnegative integers, or the positive integers. When you need precision, you use one of those terms.
September 29, 2009, 1:12 pmJohn Thacker says:
OTOH, the set of numbers 0 mod 2 is useful and doesn’t have another good shorthand other than “even,” so there the definition is more important.
September 29, 2009, 1:13 pmA.S. says:
“Yankee: I’d wager they think that because it’s divisible by 1, it’s odd.”
Is this a joke? Because I’d hate to spoil it by pointing out the obvious.
September 29, 2009, 1:13 pmTomB says:
I managed to vote for two categories. I think that 0 is even, but I seem to recall being told that 0 is defined as neither odd nor even. Perhaps I am confusing that 0 is neither positive nor negative with its evenness. It certainly fits the 2n=E formula and not 2n+1=O. However, zero’s evenness is largely irrelevant. If you simply had a rule “0 is neither even nor odd,” 0 mod 2 could still be equal to 0. Evenness would just a definition.
September 29, 2009, 1:15 pmkdackson says:
Yes, it’s a joke. A very bad one.
But in all jokes there is a kernel of truth. How else to explain that 12 people voted “odd”?
September 29, 2009, 1:15 pmLester Hunt says:
Since it comes just before 1 it can’t be odd. But it can’t be even either, because it isn’t divisible by 2 without remainder. So it’s neither. But I’m not a mathematician and this is off the top of my head, so take it with a grain of salt.
September 29, 2009, 1:15 pmJohn Thacker says:
No, it isn’t. Trust me, as a mathematician. If you consider it irrelevant, then please, let us use the definition that makes everything easier.
The only people that have cause to be upset by the 0-inclusive definition are people who bet on roulette, as far as I can tell. (Well the house would be upset, really.)
September 29, 2009, 1:16 pmCDU says:
Really? What’s the remainder of 0/2 then?
September 29, 2009, 1:17 pmkdackson says:
Lester:
Pray tell, what is the remainder when you divide 0 by 2?
September 29, 2009, 1:17 pmJohn Jenkins says:
It most certainly is divisible by 2 without a remainder. 0/2 = 0.
September 29, 2009, 1:18 pmNorthern Dave says:
The problem is one of language and philosophy. In terms of simple mathematical definitions it is considered even for practical purposes.
Philosophically, however, the concept of zero is rather different from a standard even number. To illustrate, one can divide by any even number – except zero – and get a specific answer.
Zero is also a null concept rather than one with content – again different from *any* even or odd number.
So to conclude, zero *is* an even number, as long as one doesn’t look too closely :-)
( For real fun one could explore Non-Standard analysis and the areas around zero and the different infinities :-))
September 29, 2009, 1:18 pmJohn Armstrong says:
Yale math Ph.D. here: even. No “practical purposes” bull**** about it. Northern Dave (and others) are trying to snow you. Ask any mathematician and they’ll tell you the same thing.
Just to be clear on my bona fides
September 29, 2009, 1:22 pmJohn Thacker says:
My thesis adviser was Dr. Greg Lawler. His thesis advisor was Ed Nelson, who wrote an entire book on non-standard analysis and applying it to stochastic processes. Trust me when I say that non-standard analysis provides no support for the idea that 0 is not even. The hyperreals, infinitesimals, and all that don’t affect it; the hyperreals contain the reals (and thus the integers) as a subgroup. The definition of even including 0 is still more useful.
September 29, 2009, 1:24 pmCrackmonkeyjr says:
The question is basically a definitional one rather than a conceptual one. That is to say, “even” is just a term that has either been or not been defined to encompass “zero.”
If I recall correctly though, “even” is generally considered to exclude zero, but it is a debated subject, although I think there is a general consensus that it is not odd.
September 29, 2009, 1:26 pmTomB says:
Ok, editing a comment is probably useless when comments are happening frequently. Let me refine my statement.
Zero’s evenness is largely irrelevant. If you simply had a rule “0 is neither even nor odd,” 0 mod 2 could still be equal to 0. 0 could still be part of the set of numbers represented by 2n, you just wouldn’t call that set the set of even numbers. Evenness would just be a definition.
September 29, 2009, 1:27 pmMathDude says:
I’m amused at the exceedingly high number of comments that are trying to tackle the metaphysics of the number zero. Math is not philosophical, people. (It can be when we ask whether or not the axiom of choice is valid or not, but this question isn’t one of those).
Evenness of zero is a simple question with a simple answer. Zero is an integer and, hence, the integer congruence (mod) operation is well-defined for it. 0 := 0 mod 2 and therefore 0 is even because *this is the definition of an even number*.
I like it more when the discussions are on law.
September 29, 2009, 1:27 pmNorthern Dave says:
Easy to say Dr. Armstrong, but if you have a Group element with different characteristics from other Group elements give me a reason to restrict it’s classification. Please answer the example that you can divide by any even number except 0 and get a specific answer.
September 29, 2009, 1:28 pmJohn Thacker says:
No, you don’t. Even is generally considered to include zero. Yes, it’s “possible” to define “even” otherwise, but it’s utterly useless and counterproductive. The set of numbers 0 mod 2 is a set that crops up all the time in mathematics. The same set excluding 0 does not. (The set of all integers excluding 0 does come up, sure.)
Roulette wheels are still the only exception I can think of.
September 29, 2009, 1:28 pmJohn Thacker says:
0 is a member of the group of even integers. Without 0, the even integers are not a group. Thus, it is useless to exclude 0.
To be a group, it must be closed; adding two elements together must yield an element still in the group. 2 + (-2) = 0. If 2 and -2 are both even, then the even numbers are not a group unless 0 is even.
Stop it, you mathematical crank. ARGH! I should stop arguing with you.
September 29, 2009, 1:30 pmTomB says:
And to be clear, I think 0 is even. But now that I have claimed its evenness is irrelevant, I will so argue.
September 29, 2009, 1:30 pmJohn Thacker says:
The set of even integers is an ideal in the ring of integers only if 0 is even.
The integers can be divided into two equivalence classes, odd and even, only if 0 is even.
Northern Dave, you’ve named some other random property that 0 doesn’t share. But it’s a stupid and useless thing to base “evenness” on.
You might as well claim that 1 is not odd because, unlike all other odd numbers, it is not divisible by a prime. Or that 2 is not a positive even integer, because unlike all others (so far), it doesn’t follow Goldbach’s conjecture and cannot be written as the sum of two odd primes.
September 29, 2009, 1:33 pmShelbyC says:
Well, in these here parts we have a little thing called a weggie that we administer to folks with a certain level of understanding of mathematics…
September 29, 2009, 1:33 pmTomB says:
I can’t argue with the group of even numbers argument. 0′s evenness is relevant. Though still probably irrelevant to lawyers.
September 29, 2009, 1:35 pmJohn Thacker says:
It’s evenness is relevant because the “set of numbers that can be written 2n for an integer n” crops up all the time in mathematics. If you excluded 0 from “even,” then you’d have to come up with another term for that set. There’s no good reason to exclude 0, and lots of good reasons to include it.
September 29, 2009, 1:35 pmptt says:
Let’s just call it uneven.
September 29, 2009, 1:36 pmNorthern Dave says:
Hence my point that the even-ness of zero is a functional definition rather than a necessary one. I can’t lay my hand on old Dr. Hermann’s textbook on it offhand – it isn’t where I thought it was, but it *IS* a great subject (Non-Standard Analysis) for considering some of the ramifications of what we think we mean when we consider subjects like infinities (and zero, which the division by creates: the closer one approaches zero the larger the Infinity).
September 29, 2009, 1:38 pmTomB says:
Suddenly, I feel like one of the little Godzillas in a commercial for The Ladders (the job website).
September 29, 2009, 1:41 pmNorthern Dave says:
I have no problem with being proved wrong, but think about your argument. You are completely correct that for the group to be closed operating on the two elements must yield an element in the group. However, adding zero to an Odd number yields an Odd number! By this argument zero must be an element of the Odd numbers!
(Note, personally I think you’ve just used the wrong argument.)
September 29, 2009, 1:41 pmNorthern Dave says:
In case it wasn’t clear, 1 + (-1) = 0
September 29, 2009, 1:43 pmJohn Thacker says:
Only in the sense that any definition anywhere is functional rather than necessary. We call a Manx as a cat, despite it lacking a tail. You could come up with a definition of “cat” that excludes tail-less Manxes, but you would still want a common term for Felis catus.
September 29, 2009, 1:43 pmNorthern Dave says:
Bowing out with thanks to Prof. Volokh for the thought-provoking question :-)
September 29, 2009, 1:44 pmJohn Thacker says:
OK, you don’t understand what a group is or what it means to be closed.
The odd numbers do not form a group. Add two odd numbers together and you get an even number. 1 + 1 = 2, which is not odd, in case that isn’t clear.
In fact, add any even number to an odd number and you get an odd number. This is evidence in favor of 0 being even.
September 29, 2009, 1:45 pmSara says:
Don’t blame us if your precious 0 is misidentified, according to you. Your the ones who decided to name it “zero,” which means ‘nothing, empty, or void’ – so it is logically neither this nor that.
September 29, 2009, 1:48 pmJohn Thacker says:
Northern Dave, you also don’t understand the difference between necessary and sufficient conditions, or if and only if, besides the fact that odd integers are not a group in the algebraic sense.
The odd integers can be viewed as an equivalence class under the equivalence relation defined by mod 2, but that’s not a group. It is a coset, though.
September 29, 2009, 1:49 pmkarrde says:
Much more interesting than the discussion of “X times less than” verbiage.
And yes, 0 is even.
When I was in elementary school, I learned that any number whose lowest-place-value digit is from the set {0,2,4,6,8} is even. The better explanations about 0′s membership in the Group of Even Numbers under Addition are good, but the original definition is useful and simple.
September 29, 2009, 1:52 pmNorthern Dave says:
Actually *those* arguments sunk me! It’s been 20 years since I last played with Group and Ring Theory and you are right and I need to go back and review from first principles.
September 29, 2009, 1:54 pmJohn Thacker says:
You have a confused definition of the word “logic.” We’re also the ones who decided to name “even” “even,” and that means “divisible by 2,” which 0 is. See how these silly prescriptivist arguments work?
The definition including zero is much, much more useful than a definition without zero. Please, give me an example where you would possibly want to exclude zero, outside of roulette.
September 29, 2009, 1:55 pmJohn Jenkins says:
The most important lesson of this thread is never subscribe to math threads at VC. Just don’t.
September 29, 2009, 2:01 pmnevinscrna says:
True MathDude.
But where people get hung up is on the cultural significances of odd vs even. They have names and usage in common language because even means things can be pared off with no one (odd) thing left without paring.
Odd and even are lay terms for these cultural references; they parallel some terms used in mathematics, such as parity that also describe the same.
It is no accident that there is no set of three words in common use for 0 mod 3, 1 mod 3, and 2 mod 3.
So in common lay language, is zero odd or even? Can’t be paired off, so could, by a non-mathematic lay definition be therefore not even.
In some respects it depends on whether you are asking a folks for a common lay usage of even (pairs), or a mathematic definition.
John Thacker: why is it that the British 6th graders seem to have regressed in their understanding from the 5th grade?
September 29, 2009, 2:03 pmEinhverfr says:
Up to 18 votes that 0 is an “odd” number.
Probably because most of us start counting at 1, and hence 0 is a bit strange. ;-)
September 29, 2009, 2:04 pmKenB says:
I voted and then I looked it up. I got it wrong, though so did a plurality of others.
September 29, 2009, 2:07 pmBruce Boyden says:
I think what has to be remembered here is that “even” might have one meaning among specialists, and another meaning among the general public. In such a case, which answer is “right” depends on context. Eugene didn’t give us any context, therefore the definitive declarations above about the “right answer” strike me as overly confident.
In general usage I’m not sure there is a clear answer, but I voted “even,” just because the evens and odds seem to be defined more by their alternating pattern than anything else, and zero is between two odds, or next to an odd, depending on what numbers we’re talking about. But I could see someone saying “neither even nor odd” was correct.
September 29, 2009, 2:07 pmTuck says:
EV either won or lost a bet on this one, or is leaning back in his chair hitting refresh and laughing hysterically.
September 29, 2009, 2:07 pmEinhverfr says:
So, what does this say about the virtues of democracy?
September 29, 2009, 2:12 pmJMA says:
Of course it’s even. You can split it between two children and they hate you instead of one another because the two are “even.” (Doesn’t work with 1, 3, 5, 7, etc…) This is the definition for “even” my teacher gave me in first grade.
…everything I need to know, I learned in elementary school. Damn. :)
September 29, 2009, 2:15 pmbc1234 says:
Why can’t we define an even number as any number x, for which it is is true that (1/x ) * x^2 = 2n (where n is any integer). If I’m not mistaken, it would have the exact same set as all even numbers normally defined, except for zero. I mean, it is sort of philosophical, in that in the end it just boils down to convention and semantics.
September 29, 2009, 2:16 pmKarl Lembke says:
Odd numbers are in the set of {1, 3, 5, 7, 9, …}.
In general, odd number N+1 is two more than odd number N, and odd number N-1 is two less than odd number N.
The set of odd numbers can be extended to {-1, -3, -5, -7, …}
Even numbers are in the set of {2, 4, 6, 8, …}
In general, even number N+1 is two more than even number N, and even number N-1 is two less than even number N.
The set of even numbers can be extended by subtracting 2 from the smallest number in the above set: {0, -2, -4, -6, …}
There is no way to get to 0 by adding 2 to or subtracting 2 from any odd number. Getting to 0 by adding 2 to or subtracting 2 from some even number is trivially easy.
Even numbers are defined as numbers that can be expressed as N = 2 * M, where M is an integer.
Odd numbers are defined as numbers that can be expressed as N = (2 * M) + 1, where M is an integer.
There is no integer M where doubling it and adding one yields 0.
Zero is even.
Oddly enough!
September 29, 2009, 2:20 pmAdrian says:
EV either won or lost a bet on this one, or is leaning back in his chair hitting refresh and laughing hysterically.
Well, I don’t know about him, but I sure am (laughing)! (Btw, I didn’t corrupt the results with the correct answer. I wonder if you took the mathematicians out of the results how they would look.)
September 29, 2009, 2:21 pmJohn Jenkins says:
You mean other than because your equation reduces to x = 2n and zero is an integer?
September 29, 2009, 2:21 pmKarl Lembke says:
Is it just my browser, or are all the comments numbered 1?
September 29, 2009, 2:21 pmCato The Elder says:
Why don’t people buy the argument that 0 is even because 0 = 2 * 0? Is it because “0″ is on both sides of the equals sign? If so, that’s silly.
September 29, 2009, 2:26 pmJohn Jenkins says:
@ Karl: Does that make all of the comments odd?
September 29, 2009, 2:30 pmSara says:
Oh John, nothing divided in two yeilds nothing. It’s still nothing.
September 29, 2009, 2:31 pmbc1234 says:
I suppose you’re right– I don’t know much about math. Still, it seems an abuse of the word “reduce,” when the “unreduced” equation gives a different value for the same input.
September 29, 2009, 2:32 pmKarl Lembke says:
@John:
It would appear to.
September 29, 2009, 2:37 pmBut are they prime?
blargh says:
bc1234, are you sure? (1/x) * (x*x) = x…
September 29, 2009, 2:47 pmtamerlane says:
I hope Professor Volokh will eventually explain the reason for this poll. I’m guessing it’s the result of a bet about how mathematically ignorant Volokh readers are (maybe after a few beers?). If it was a bet, I’m also curious who won. I’ve been teaching recently at the university level but I’m still shocked at the widespread ignorance among this country’s educated elite.
September 29, 2009, 2:48 pmCato The Elder says:
Now they are. (#79)
September 29, 2009, 2:48 pmLaura(southernxyl) says:
Well, I’ll bite.
As a mere lass I thought that zero was even because it just intuitively made sense to me that it was, until I got to one of my upper-level math courses and was told that it’s not, it’s neither odd nor even, and people who thought it was even were ignorant.
I don’t remember the reasoning – I graduated from college in 1982 and all of my math use since then has been in pursuit of more or less real stuff in which it didn’t freakin matter whether zero was even or not, thank God – but I do remember this b/c of my surprise that zero was not even.
So this is why I voted neither even nor odd. And I suspect that the same is true of others.
Beats me why people want to use obscure, arbitrary stuff like this – not to ask the initial question, b/c it’s kind of interesting to think about – but to beat other people over the head with and call them ignorant. Especially when the right answer differs according to who’s beating you over the head. It’s just another game of “gotcha” which is neither mature or attractive.
In my opinion.
September 29, 2009, 3:05 pmDonBoy says:
The advanced class is going to attack “what’s 0 to the 0 power?”, which is legitimately troublesome. We’re told that 0 to any power is 0, and that anything to the 0 power is 1. So…somebody’s getting discontinuous tonight, boys.
September 29, 2009, 3:07 pmKarl Lembke says:
That’s best handled with a limit process. Calculus, anyone?
September 29, 2009, 3:09 pmTHESMOPHORON says:
0 modulo 2 = 0. QED.
September 29, 2009, 3:09 pmChrisTS says:
I believe the Pythagoreans were correct in just keeping the whole zero-thing secret.
September 29, 2009, 3:19 pmKarl Lembke says:
@ChrisTS:
September 29, 2009, 3:22 pmSo the Pythagoreans taught nothing was sacred?
Arkady says:
Well, if 0 is an even number, it is unique in that division by it is null (in the sense of having no meaning) as opposed to division by every other even number. But if mathematicians wish to say it is even because otherwise whole regions of mathematics would be rendered nonsensical, why shouldn’t that be enough?
Socrates (in New Jersey): What is Mathematics?
Theatetus: Why, that’s what mathematicians do.
September 29, 2009, 3:30 pmASlyJD says:
My point was exactly the one Laura made — as an engineering major, all I cared about in mathematics were those trivial cases where numbers are real and positive.
September 29, 2009, 3:36 pmjab says:
Karl,
Limit is undefined because it converges to two different values.
For example:
September 29, 2009, 3:37 pmTry 0^0.000000000000000001 = 0
but 0.0000000000000001^0 = 1
For the function z = x^y, you get different limits depending on how you approach (x,y) = (0,0).
So 0^0 REALLY is undefined.
bc1234 says:
Right… for every x except zero, which ends up not equaling zero, but equaling “undefined”. It’s kind of a cheap trick, I guess– I’m just interested that reduction can actually change the output in these types of situations.
September 29, 2009, 3:37 pmMarcW says:
Well, we can define an even number as you say. We can also define an even number as any integer greater than 10. But, of course, we don’t. We define an even number as any integer which is equal to 2 × n where n is an integer.
September 29, 2009, 3:44 pmjab says:
(1/x)*x^2 is undefined at x=0, but it does indeed have a very well defined limit as x approaches zero.
Same for sin(x)/x. It is undefined at x=0, though it can be made continuous by defining the value to be equal to 1 at x=0. You are not allowed to divide by zero. Period. But often, if the limit is defined as you approach zero, you can redefine your function to take on the value equal to the limit.
It’s the difference between f(x) = sin(x)/x
versus
f(x) = sin(x)/x for x != 0
and f(x) = 1 for x==0.
The first case, the function is undefined at x=0 (but has a well-defined limit),
and in the second case, you redefine the function to take the value of the limit.
I’m a physicist, and often we get sloppy and almost always mean case 2, even when we write it as case 1.
September 29, 2009, 3:45 pmbc1234 says:
Right which was sort of my original point. It;s all convention and semantics.
September 29, 2009, 3:52 pmhedberg says:
What sort of engineering can you do without resort to complex numbers?
September 29, 2009, 3:57 pmKarl Lembke says:
@bc1234:
In this case, “convention and semantics” are pretty darn important. If we set them aside too easily, we wind up with the accountant in the joke who, asked what 2+2 equals, replies “What number did you have in mind?”
September 29, 2009, 4:00 pmjab says:
A mathematician, an experimental physicist, and an engineer were asked to prove that all odd numbers are prime…
The mathematician starts, 3 is prime, 5, is prime, 7 is prime… by induction, all odd numbers are prime.
The physicist goes, 3 is prime, 5 is prime, 7 is prime… experimental data justifies that odd numbers are prime.
The engineer goes, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, 13 is prime, 15 is prime.
Badump-bump… well… they all come out looking bad, but we used that to make fun of engineers who believed anything you told them… for the record, I’m a physicist who has much respect for engineers ;)
September 29, 2009, 4:09 pmAllan Leedy says:
It would be odd if it were not even.
September 29, 2009, 4:10 pmMarcW says:
bc,
OK, fine it’s all convention and semantics. But if that makes it unimportant (and if that’s not your bigger point than forgive me for reading too much into your statement), then you can say the same about pretty much any question: Is 1 positive? Depends on the meaning of positive. Is pi rational? Depends on the meaning of rational. Does the power set of the natural numbers contain uncountable antichains? Depends on the definition of uncountable. But positive, rational and uncountable have definitions in mathematics, so the answers are yes, no and yes. No ifs ands or buts.
September 29, 2009, 4:12 pmBama 1L says:
What do mathematically literate people use even/odd for? Please tell me there is some value to the categories besides sharing apples.
September 29, 2009, 4:14 pmJohn Jenkins says:
Then you didn’t make any point at all. That’s all any definition is, a convention. But in this case, the convention is that zero is an even number and that convention makes sense because it satisfies ALL of these criteria while n=0 & k is an integer.
n mod 2 = 0
n = 2k where k is an integer (0)
n +/- any odd number is odd
n +/- any even number is even
n * any integer is even (iff zero is even)
n+1 = odd
n-1 = odd
If you define zero as not even, then all of those rules by which we identify even numbers would have to have an exception for zero, just because you want to multiply an equation by x/x and do the simplification on one side and not the other.
So, yes, defining zero as an even number is a convention, but it’s a convention that makes sense, as opposed to defining it as non-even (it can’t be odd) and not making sense in the context of the parity of other integers.
September 29, 2009, 4:16 pmJeff R. says:
But we’re approaching (x,x) = (0,0). And looking at 0.000000001^0.000000001= 0.999999979 certainly seems to make it obvious which limit is dominating when both terms go at the same time.
September 29, 2009, 4:20 pmjab says:
Jeff,
September 29, 2009, 4:28 pmNo, we are analyzing a function of 2 variables: z=f(x,y)=x^y.
If you’re going to vary both the base and the exponent, you must consider this a function of 2 variables.
I see what you’re getting at, but
the point is, for the limit to be defined, it has to be converge to the same answer from ALL directions, not just most directions.
Neal says:
This thread is frightening…
September 29, 2009, 4:32 pmMike G in Corvallis says:
The attorney stood up, closed the blinds, locked the door, sat down next to the client, and whispered,
“Which one do you want it to be?”
September 29, 2009, 4:35 pmMike G in Corvallis says:
jab wrote:
The version we used in physics class had the physicist saying, “Three is prime, five is prime, seven is prime … Hmm. Nine doesn’t appear to be prime, but that could be due to experimental error … Thirteen is prime, fif— … I think we have enough data now!”
September 29, 2009, 4:48 pmJust Passing By says:
I am not a student of the law, but this kind of disagreement about what I consider to be a simple property of 0 makes me worried about the quality of legal disagreements.
September 29, 2009, 4:50 pmEinhverfr says:
BC1234
But the limit as X approaches 0 is still defined. So we could conclude that it does lead to the same approach.
September 29, 2009, 4:55 pmShelbyC says:
There’s only one reason I can think of why someone would post this question on a law blog.
September 29, 2009, 4:56 pmjab says:
Mike G… you’re right! I forgot about that…
and other really bad math jokes from college:
What do you get when you cross an elephant with a grape?
Ans: Elephant grape sin(theta).
What do you get when you cross an elephant with a mountain climber?
Ans: Silly, you can’t cross anything with a scalar…
Uuuuuuuugh… um yeah, I should get back to grading my student’s physics homework.
September 29, 2009, 4:58 pmepignosis says:
Even numbers are those whole numbers (counting numbers & 0) that are evenly divisible by 2. Zero is evenly divisible by 2 since the result is zero. Therefore, zero is an even number.
September 29, 2009, 5:15 pmJeff R. says:
You may be analyzing such a function, but I’m analyzing the function of one variable y=f(x)=x^x. Which appears to have a single limit.
As does, say, (x^2+x)/(x). Rewriting it as (x^2+x)/(y) doesn’t make that stop having a single limit at x=0
September 29, 2009, 5:16 pmMike G in Corvallis says:
What do you get when you cross an elephant with a mountain climber?
Ouch! I hadn’t heard that one. Thanks … I think.
This one might be relevant to Eugene’s question:
What’s the difference between and engineer, a physicist, and a mathematician?
An engineer believes equations approximate the real world.
A physicist believes the real world approximates equations.
A mathematician sees no connection between the two.
September 29, 2009, 5:18 pmjab says:
Jeff,
Ok, if you are exploring f(x)=x^x as x->0, you still have to consider the limit from the negative side.
And that is still undefined.
The way you would write this is:
As x–>0+ lim x^x = 1;
As x–>0- lim x^x = undefined;
As x–>0 lim x^x = undefined.
So i will grant you that the *one-sided* limit of the function f(x)=x^x is 1.
The fact that the limits from different directions converge on different values means the function is discontinuous.
September 29, 2009, 5:24 pmEMG says:
Many people who didn’t vote for even probably didn’t really believe you when you said it wasn’t a trick question. Or, not quite a trick question, but along the lines of “well EV probably knows a lot more math than I do, so the answer can’t be the one that seems intuitively obvious – else why would he bother asking?” Something like that was my first thought – when I find a smart person treating a question I find easy as though it were interestingly difficult, my first instinct is to doubt whether I really knew what I thought I did.
September 29, 2009, 5:32 pmJeff R. says:
Well, sure. Maybe. I wish I had a calculator ap that did complex numbers…maybe Wolfram Alpha? Ah, yes. According to it, (-0.000001)^(-0.00001) is well on it’s way towards approaching 1+0i. So I’d say that “1+0i” works perfectly fine as a two-sided limit, no?
September 29, 2009, 5:43 pmBruce Hayden says:
Who would have expected so many people who actually know something about mathematics to show up here. Intuitively, I immediately thought even, and then back filled and did the x mod 2 = 0 thing.
But then I read the comments, and kept going, oh, yeh, I remember that. And that. I do seem to remember that zero was still even when we first took groups, because I remember the distinction was made between the even being a group, and the odds not. That would have been around 40 years ago, which is why my memory is quite hazy. I loved the upper division classes, because you could go and do a lot of “let’s pretend” in class (I like science fiction maybe for the same reasons), while the rest of the students were reading long books and writing papers. But, then, I graduated, only took a couple more real math classes after that, got into programming, did that for awhile, and then have been doing the lawyerly thing the last 20 years. And then, after this thread of comments, I look back and remember how much fun I had back then.
September 29, 2009, 5:48 pmAnother Math Prof says:
Here is just one cute example of how defining 0 to be even is useful. It’s about a game called Nim. The game begins with several piles of stones. On your turn, you may remove as many stones as you want from any one pile. (You must remove at least one stone.) The winner is the person to remove the last stone. The question is how do you know what positions will guarantee that you will win if you move to them and don’t make any mistakes in the future. The game, and the winning strategy are described on the Wikipedia page for it.
Briefly, the winning strategy is to write the number of stones in each pile in binary and align these binary numbers in rows. The winning positions to move to are exactly those where the collection of binary numbers you write down has an even number of ones in each column.
We could say “The winning positions to move to are exactly those where the collection of binary numbers you write down has an even number of (or zero) ones in each column,” but that would just be annoying.
September 29, 2009, 5:50 pmCornellian says:
Let me just state for the record that I find it totally awesome that this post has generated more comments than virtually any other post not dealing with guns or gay people. Clearly VC is the nerdiest legal blog in existence. It’s why I keep coming back.
September 29, 2009, 5:55 pmjab says:
Jeff,
Ah, very nice… that is true.
Well… I’ll concede that then.
as x–>0 lim x^x = 1.
Still, I think the reason why 0^0 is still considered to be undefined is
September 29, 2009, 6:01 pmbecause when you explore z=x^y, that gives different limits.
I can think of no other reason why 0^0 would be considered undefined.
Off Kilter says:
As I write this, over 50% of voters think it is possible for something to be either BOTH odd AND even, or NEITHER odd NOR even. Wow.
Eagerly awaiting EV’s quiz on 4 sided triangles…
September 29, 2009, 6:18 pmNoah Snyder says:
While there are lots of times where slightly different definitions make sense for different circumstances, even just isn’t one of those. I can’t think of any situation where you wouldn’t want to think of 0 as even.
September 29, 2009, 6:22 pmMike McDougal says:
If 0 is not even, then the group of evens (2, 4, 6, 8 ) is smaller than the group of odds (1, 3, 5, 7, 9).
AND THAT’S NOT FAIR.
September 29, 2009, 6:31 pmChrisTS says:
Karl Lembke:
Ohh, MUCH is sacred.
But that not-really-a-number thing that is purported to be less than the arche of ‘number’- i.e., the Monad – …. well, somethings are just profane.
September 29, 2009, 6:58 pmChrisTS says:
Mike G:
What’s the difference between and engineer, a
I thought the last line was supposed to be: A mathematician believes the real world is an equation. (They are all Platonists, yes?)
September 29, 2009, 7:01 pmDeezrightwingnutz says:
I like how this is supposed to be “obvious” and how “British 6th graders know it” better than readers of this site. But the 6th graders don’t comprehend it, I bet… the reasons why it is useful to include zero in the set of even numbers. They just memorized it, and a lot of 35 year old lawyers forgot it.
If I recall, any number divided by zero is “undefined,” except 0/0, which is “indeterminate.” I don’t know what the hell the difference is, so I’m not going to make fun of someone who forgot what he memorized for a test back in 11th grade.
September 29, 2009, 7:09 pmCornellian says:
I took a lot of math courses in college (despite majoring in the humanities). I can’t recall much about number theory now, but I remember thinking it was the first course I had taken that really seemed to be telling me something fundamental about the nature of mathematics itself, rather than just using mathematics as a tool to solve real-world problems. But like Bruce and like EV I ended up being a lawyer working with words instead of numbers, and now I just read about math for fun.
September 29, 2009, 7:11 pmSara says:
119.Another Math Prof says:
Here is just one cute example of how defining 0 to be even is useful. It’s about a game . . .
Cute and Useful to you, maybe.
September 29, 2009, 7:15 pmDeezrightwingnutz says:
What’s 1.5… odd or even? What’s your angle, Off Kilter? Anyway, I find your response obtuse (demonstrated a lack of acuity). And a response has to be either obtuse or acute, right?
September 29, 2009, 7:28 pmBellisaurius says:
Lots of amusement here. People who don’t have a dog in the fight about the evenness of zero (ie nonmath types) seem pretty bothered by the concept of zero’s evenness.
Pragmatics probably support the the evenness convention (except on the roulette table). If we had to divide a group into to on the basis of a final digit, even/odd is what we’d probably pick, and zero would have to be even in this case (1,3,5,7,9 vs 0,2,4,6,8). In fact, the numbers 10, 20 etc.. should make us a bit more comfortable here since they represent zeros in the unit group. So, on the surface, useful level, zero is basically even on a sort of aesthetic grounds.
John Thacker at 0124 brings up a good point that if you look very hard, zero is kind of neither on a very deep level. In this sense, zero is is like y, and functions in an unusual way. However, most people who took the neither route probably weren’t thinking like john, so right answer/wrong reason could apply.
September 29, 2009, 7:39 pmA. Zarkov says:
A horse has an infinite number of legs.
Proof:
A horse has his fore legs in front, and two legs in back. That makes six legs. But six legs is an odd number of legs for horse to have, so the number legs a horse has is both odd and even. But only infinity is both odd and even (not zero). Thus a house has an infinite number of legs.
QED.
Told to the class in a graduate course in analysis. The guy next to me took copious notes. I coundn’t convice him to sit back and enjoy the fun.
September 29, 2009, 7:56 pmEric E. Coe says:
To a programmer, the test for odd/even in integers is easy:
if(num & 1) {
printf("Number %d is odd.\n", num);
}
else {
printf("Number %d is even.\n", num);
}
... the combination of the bit-and operation (
September 29, 2009, 7:59 pm&) and a value of 1 selects the lowest bit of the integer stored innum. If the result is 1 (non-zero) then the number is odd; a zero result means the number is even. This works for both positive and negative integers because of the twos-complement integer arithmetic used is most (almost all) modern computers. It also means that 0 is even, because0 & 1 == 0David Gaw says:
Zero is evenly divisible by two, and is therefore even.
September 29, 2009, 8:00 pmMarcW says:
Bellisaurius:
No. No. No. The definition is very clear. Zero is even. There is no “level,” “deep” or otherwise in which zero is not even. You want to say it’s different than the positive even numbers because it’s not positive? Fine it’s different in that it’s not positive. You want to say that it’s different from all the other even numbers because it’s the only one that’s zero? Fine. But these differences have no bearing on it being even. Philosophical folderol does not enter the picture.
A. Zarkov:
Please don’t. I know it’s just a joke, but if you say that pretty soon someone is going to quote you in an attempt to justify the claim that infinity is both even and odd.
September 29, 2009, 8:16 pmSara says:
Hey! McGraw-Hill’s Catholic High School Entrance Exams, 2ed:
Zero is an interger that is neither even nor odd.
September 29, 2009, 8:25 pmJust Visiting says:
Is “0″ even or odd?
Depends upon which answer paid me a retainer….
September 29, 2009, 8:43 pmJohn Jenkins says:
Even numbers are numbers divisible by 2.
The integer 0 is even and is not odd.
There is no rational reason to exclude zero from the definition of even numbers since it meets ALL of the tests.
September 29, 2009, 9:01 pmJohn Jenkins says:
Even numbers are numbers divisible by 2.
The integer 0 is even and is not odd.
There is no rational reason to exclude zero from the definition of even numbers since it meets ALL of the tests.
September 29, 2009, 9:01 pmfishbane says:
AS: I’m just raising the possibility that Eugene might be getting a bit too worked up about what is a pretty minor point. One would think EV, who is after all a public servant, would have something better to do with his time.
I like to think that Eugene is simply trying to expand lawyer’s knowledge of math beyond calculating billable hours.
Given the poll results, this is an extremely useful public service.
I mean, really, this is basic, basic stuff. And fine, not remembering a (simple) definition from school outside your core professional pursuit isn’t anything anyone will hold against you. But fighting about simple definitions in math that would mean the world would be a very different place if they were incorrect, even as a lawyer, isn’t something I’d bet the vacation home or the nanny on.
September 29, 2009, 9:07 pmC.P. Snow says:
My goodness, I didn’t know the half of it.
September 29, 2009, 9:08 pmfishbane says:
But it can’t be even either, because it isn’t divisible by 2 without remainder.
What is the remainder of 0/2? Feel free to use a calculator.
September 29, 2009, 9:10 pmbuck says:
Finding less than 50% of VCers failing to identify 0 as an even number is troubling enough. But someone actually dragging out this dinosaur of an argument combined with an anti-intellectual snark is just plain offensive.
There is no debate possible here–a definition is simple and precise. Any measure by which you may approach the definition, 0 will end up being even. Not both, not neither–only even.
It is important to note that Aristotle did not have a concept of 0 at all. For that matter, Aristotle did not think that 1 was a number either–1 was a unity, a whole and thus could not be counted. So, if Volokh readers were contemporaries of Aristotle, they would get a pass. But they are not, therefore, they are illiterate.
Thanks for the laugh, Eugene. Next, ask how many of your readers believe that the Earth is flat.
September 29, 2009, 9:15 pmTyrone Slothrop says:
I voted “even” and I voted even though I am merely an engineer. I voted “even” because of an engineering tool, Occam’s razor. Don’t overthink. Accept the definition that provides the most utility in the real world
September 29, 2009, 9:15 pmEinhverfr says:
Not quite. The angle would either have to be acute or obtuse if it is not right (or correct). Therefore “acute” and “obtuse” display the different quantities of wrong (not “right”) points of view (i.e. “angles of approach”).
September 29, 2009, 9:17 pmEinhverfr says:
Re “undefined” vs “indetermine” the answer becomes apparent when starting to do basic calculus and it has to do with questions of limits.
Suppose I take the limit of 1/x as x goes to 0. As x goes to 0, 1/x goes to +/- infinity. Same with 1/2x and 2/x. This isn’t strictly defined because the limit approaches different values from different sides of the graph (from the negative side, it approaches negative infinity, and from the positive side infinity or possibly vice-versa depending on the equation).
Now, consider the following equations: 3x/4x^2, x^2/4x, and 8x/x and take the limits there as x goes to 0. In the first case, the limit if 3x/4x^2 goes to +/- infinity, x^2/4x goes to 0, and 8x/x goes to 8. Hence while the limit of any number divided by 0 can be defined as approaching infinity or negative infinity in any case, many equations which could yield 0/0 could result in ANY limit. These could result in a defined limit, but of the infinite number of equations that could create a limit, any possible answer is possible.
Hope this helps, but it has been 15 years since I took a math class.
September 29, 2009, 9:26 pmjab says:
Sara… but that is for Catholic School entrance exam… ;) (I’m a recovering Catholic.)
September 29, 2009, 9:29 pmEinhverfr says:
I wouldn’t hold your breath. On the other hand, spherical triangles are pretty interesting. I did, however, fail at teaching myself how to solve spherical trig problems though outside of triangles with at least two right angles (which are easy).
September 29, 2009, 9:34 pmKarl Lembke says:
@buck: If you define “flat” as following a single gravitational equipotential surface, you could say the Earth is pretty close to flat.
September 29, 2009, 9:41 pmBruce Boyden says:
Lighten up, mathematicians. This thread is not the downfall of society. The fact is that “even,” in addition to being a mathematical term of art, is also an ordinary English word, subject to the normal variation in definition of any ordinary English word. Mathematics cannot give you the answer as to ordinary usage. You have no bottles of milk. Do you have an even number of bottles? 0 mod 2 = 0 does not give you the answer to that question, unless you hang out with a really geeky crowd.
September 29, 2009, 10:35 pmBellisaurius says:
Marc, if the identity for an even number is just the remainder test or the 2n test, I’m completely in agreement (my answer was zero is even as well) about zero’s evenness. PLus, as some have pointed out, there’s a usefulness to zero evenness.
However, there are worlds beyond the simple definition where zero’s status is a bit fuzzier, like with goldbach’s conjecture. Are they trivial or unusual enough to disregard? Perhaps, but they’re still there. There used to be some debate about whether one was prime, as it violates one of the uniqueness part of a prime’s identity. A little ambiguities not a bad thing here. It allows for a greater appreciation of the subject.
September 29, 2009, 10:37 pmThe Volokh Conspiracy » Blog Archive » Yow! says:
[...] was shocked at the number of people who took the view that 0 was neither even nor odd. (I was even more shocked by those who thought 0 [...]
September 29, 2009, 11:50 pmJames in MA says:
Surprised no one has told this joke yet …
2 is an odd number.
Proof: 2 is the only even number that is prime. Isn’t that odd?
September 30, 2009, 5:31 ambuck says:
Actually, it can. If you add one more bottle, your number becomes odd. So, yes, you do have an even number of bottles.
In fact, by combining zero and even in the same question, one essentially removes the ambiguity of “ordinary usage”. Under ordinary usage we can talk about a surface being “even”, but that clearly does not apply here. We can talk about an “even” amount, which basically means “just right”–again, that makes little sense here. We can talk about dividing something evenly, but the context does not specify division. If we are going to introduce division into the context, however, there is no reason not to rely on a minor mathematization of the question–can 0 bottles be divided evenly between two people? And, of course, the answer is “Yes”–both get the same number of bottles, none!
So, you can stretch it any way you like, but the fact remains that 0 is an even number.
September 30, 2009, 7:05 amSome dude says:
What I want to know is: Is 0.999… an integer?
September 30, 2009, 7:55 amCan't find a good name says:
I’m disappointed that after all this time, we still haven’t gotten a majority of votes in favor of 0 being even.
When most people first heard of odd and even numbers, some of them probably learned that numbers that end with 1, 3, 5, 7, or 9 are odd, and numbers that end with 2, 4, 6, 8, or 0 are even. The advantage of this rule is that if one can memorize this rule, one can figure out whether any number is odd or even just by looking at it, and one need not know anything about multiplication or division to do that determination. And if you do know a definition of odd and even numbers based on multiplication or division, you will find that the rule is still true.
In fact, the only significant objection that mathematicians would have to the original rule is that they would say that the rule applies to integers as opposed to all numbers.
September 30, 2009, 8:01 ambyomtov says:
This thread is truly astonishing. As Fishbane says, to forget even a simple thing you don’t ordinarily deal with is no crime, but to argue about it is ridiculous.
I wonder how many of the commenters arguing this point would ridicule someone who sees the world wholly in subjective terms.
September 30, 2009, 8:11 amE Garland says:
I voted ‘Even’, even though I’m aware that as a signifier for ‘nothing’, Zero is neither odd nor even…
The issue is, in our numbering system we use it as an even number in counting up or down…+2, +1, 0, -1, -2…
September 30, 2009, 8:16 amagesilaus says:
Eh? 2 is divisible by 1, do you think it is odd?
September 30, 2009, 9:34 amzuch says:
Of course 0 is even.
Better question is whether 0 != -0.
Back before sanity took hold and mod 2 arithmetic slew all others, some computers used the more ‘intuitive’ sign-magnitude arithmetic .. which had the property of allowing 2 different numbers for 0 (“+0″ and “-0″). One beast I worked on would “jump on zero” for both “+0″ and “-0″. But if you did the “skip if unequal” instruction, it would skip when comparing “+1″ and “-1″. What was worse is that which “zero” you got depended on how you got there (adding to a negative number, or subtracting from a positive number).
Needless to say, those days have since passed and computers nowadays use the 2′s complement arithmetic, which has less quirkiness (the main one being that the negative of the largest negative number is itself).
Cheers,
September 30, 2009, 10:01 amSigivald says:
I voted “yes” to all four just to point out that the vote widget is deeply flawed.
September 30, 2009, 10:17 amDuffy Pratt says:
Pi is neither odd nor even. So, yes its perfectly possible for something to be neither.
Infinity may be both odd and even. Think of a switch. You will be switching it on and off for one second. The first switch will occur at 1/2 a second. The next at 3/4, then 7/8, then 15/16, and so on… When exactly one second has expired, is the switch on or off? Both? Neither?
Sorry bout that.
September 30, 2009, 1:32 pmKarl Lembke says:
At the end of one second, the switch will be broken from over-use. :-)
September 30, 2009, 2:08 pmKarl Lembke says:
OK, I’m changing my vote. Option 5: A vowel.
September 30, 2009, 2:12 pmDavid III says:
Theorem: All positive integers are interesting.
Proof: We will prove this by induction. 1 is interesting since it is the multiplicative identity and is neither prime nor non-prime. 2 is interesting because it is the only even prime. 3 is interesting because it is the smallest odd prime. Assume that a positive integer k is the last interesting positive integer. That means that the integer k+1 is the first uninteresting integer. Isn’t that interesting?
September 30, 2009, 2:36 pmharley says:
My math teacher used to tell me that:
x^x=even
for all even x
what then for 0?
September 30, 2009, 4:56 pmother math person says:
Strictly speaking “not a fraction” is NOT the same as “integer”. For example, PI (or any irrational number) is “not a fraction”, but is also not an integer.
September 30, 2009, 5:19 pmWakefield Tolbert says:
Zero is EVEN, you mollusks.
It IS a number in the sense that it is a placeholder.
It’s position in connection with numbers that are also divisible by “2″ should have made this painfully obvious.
Ig not, count backwards, starting at ten if you like, and then you’ll arrive at the number before “1″, which is the smallest odd number.
The confustion comes in here for ZERO’s place in the notion of “null” or “nothingness.”
Take it up with the mathematicians and before them, the mysticism of the Hindus who created this notion of “something that is, yet is not” simultaneously.
It is one of those nifty elements the Western world uses on a daily basis in ratio-centric thinking and smug snarkyness about its superior reasoning skills to the mystic muck of the old world, but nontheless is used fairly much as is.
“That which is, and yet is not”
Half-dead kitty-cats have met their match here.
singed,
(a real estate appraiser in South Carolina)
September 30, 2009, 5:23 pmbuck says:
Why are people trying to outdo each other in defending their own ignorance?
September 30, 2009, 10:39 pmAndrew Myers says:
I was initially aghast at reading the results of the poll and the thread. Actually, I still am. But I begin to understand why people think that zero isn’t an even number. Fundamentally, they think zero *isn’t a number* in the sense that is meant. That’s why they don’t think you can split 0 things evenly into 0 and 0. In a sense, they have a point: if by “number” we mean 1,2,3,…, then indeed 0 is not an even number.
Unfortunately that doesn’t explain how you decide it’s an odd number. Maybe the logical process is, “everything must be even or odd” (actually, a false premise if “everything” includes reals, etc.), “0 is not even”, hence “0 must be odd”.
October 1, 2009, 4:26 amLinda F says:
If you subtract an even number from an even number, the result will be even. Hence, zero qualifies.
October 4, 2009, 5:03 amBill Cheswick says:
Eric Coe (above) is halfway to the correct, or perhaps most generally useful, answer, if uses there be. He gave a piece of computer code that is certainly the most likely computation to be used these days.
In MS Excel, the iseven function truncates the argument, so 2.5 is even in that world. VB gives our C example, above.
Google gives about 48000 hits for the term “iseven”, a likely name for the function.
The best legal answer I can see is Coe’s answer, because code is law (see Larry Lessig.)
October 4, 2009, 12:15 pmRehan butt says:
zero is an even no.
July 4, 2010, 4:58 amProof
we know that
sum of two odd nos= even no……….(1)
1+3=4
7+9=16
if 0 is odd then
1+0=1
7+0=7
which is not true from 1
we also know that
sum of two even nos= even no……….(2)
2+8=10
8+16=24
if 0 is even then
2+0=2
8+0=8
which is true from 2
so zero is even no