## Monday, January 20, 2014

### 1: Is zero Odd or Even?

I'd say even ...
• to fit the pattern. 6, 4, 2, 0
• to contradict the formula that an odd number can be written as 2n + 1, n ∈ integer.

Any others?
Any reasons to call it odd?

1. Even for me:
- Add two odd numbers, you will get an even number. 7+0 is not even, so 0 is not odd.
- Add an odd number with an even one, you get an odd. 7+0 is odd, so 0 can maybe be even.
- Even numbers match numbers whose modulo 2 is 0. As 0 modulo 2 is 0…

2. My 8 year old says even because you can share an even number of candies equally with a friend. For 0 candies you both get 0 candies which is equal.

3. We usually define "even number" as "a number that can be written as a multiple of 2." So 10 is even because 10 = 2*5. More precisely, n is even if there exists an integer k such that n = 2k.

So then -- does zero fit this description? Surely it does, since 0 = 2*0. Since 0 is an integer, we have written zero as an integer multiple of 2, so zero is even!

4. Even. If 1 is odd.
Also check an old BBC article on a surprising application:
http://m.bbc.co.uk/news/magazine-20559052

5. Division makes it all messy.

6. I'm surprised by the persistence of this question. I have had three different AP students ask me this this year alone. Is it because we have trained kids to think of 0 as some mysterious different type of number?

7. I think the question, "is zero an even number?" illustrates very well the importance of precision in mathematics. The question is not really mathematical in nature until we agree on what is meant by the word "even". Once we have that agreement then it is possible to meaningfully discuss the problem and to come to a conclusion. I can certainly understand how a child might be confused by the question, in same sort of way that they might confused by the question, "Is zero a multiple of 17?" The answer to this second question depends on what, precisely, you mean by the phrase "multiples of 17". In some settings it would be most natural to say a number is a multiple of 17 if it can be written as 17*n, where n is an integer, and in other settings it would be most natural to say a number is a multiple of 17 if it can be written as 17*n, where n is a positive integer. The answer to the question depends on the definition.

8. When my students argued this, about half said neither odd nor even. It sparked a huge debate wondering if 0 has a value of 0, or if it has no value at all because it is nothing. (I hope you see the difference there, it's a pretty poor explanation)