Odd and even numbers

[user789653241]

Ds(9, yr5) asked me if decimal numbers can be defined as even or odd. I said no, but haven't got a clue why.

Can anyone give us some explanations please?

9toenails I'm out. I never really got set/number theory at university. Which is why my 3rd year options were definitely based on nothing at all from the second year.

It's great simple question from 9 years old created such an interesting thread. Though most of debate went over my head since I am not so good at mathy thinking.

noble, I would like to know how your discussion with class went, I had talk about "is 1 a prime ?" and " is a cylinder a prism?" with my ds in the past as well.

I wasn't allowed to take the number theory module at uni

irvine 1 is officially not a prime because if it's a prime it breaks the fundamental theorem of arithmetic.
The fundamental theorem of arithmetic says that every integer greater than 1 is either a prime, or can be written as primes multiplied together in exactly one way. For example, 35 = 5 x 7, 100 = 2 x 2 x 5 x 5 and you can't find any other way to multiply primes to get 35 or 100 (order doesn't matter). If 1 were a prime, then you could have 35 = 5 x 7 or 1 x 5 x 7 or 1 x 1 x 5 x 7 or.... so basically to solve this, 1 was binned from the list of primes, but you can still find old books where it's listed as one.

Officially, a cylinder isn't a prism, because technically the shape at the end has to be a polygon (no curved edges). However, it's useful to treat it just like a prism when finding the volume (area of cross-section x length of prism) and you'll find a lot of textbooks actually list it as a prism.

Thank you, noble.

I reviewed this interesting thread over the week-end and although I have nothing to add to the even-odd debate, the last couple of posts did trouble me a little.
I must be ancient because i was taught that "1" was indeed a prime. However, just the other day on Khan I discovered that "1" isn't a prime any longer (a bit like Pluto is not a planet any more...). I have read here and there about it, but I have not found a compelling reason as to why it shouldn't be a prime. OK, there is the "fundamental theorem of arithmetic", but that can be simply reformulated adding a "...multiplied with primes greater than 1...".
Here you can find other reasons:
primes.utm.edu/notes/faq/one.html
but I am not overblown by them. Perhaps there is some aspect of mathematics that is actually simplified by having "1" as neither a prime or a composite number, but to me now it is simply a matter of definition. A bit like the reason why a cylinder is/not a prism. Is there a deeper reason why it shouldn't be, besides "a circle is not a polygon"?

I would say that 2-1 is a calculation, but that the answer is an integer.

Noble, I'm still thinking about how best to answer that, but it makes me very uneasy as a mathematical statement.

I think a "digit" might be a good comparison to consider. You're thinking of integers as being an extension of the set of digits {0,1,2,3,4,5,6,7,8,9}. I'm thinking of them as being a set of numbers with associated operations and identities (+,-,*,0,1) with certain properties. For example + is a binary operation ZxZ -> Z which associates to every pair of elements a,b in Z an element a+b in Z. etc.

So if you consider Z purely as a set, fair enough to say a+b is not a member of it. But normally we consider Z with additional structure, at the very least as an additive group. And if you are even writing down the calculation 1+1 you are acknowledging the addition operation is there. What is it if it isn't a mapping ZxZ->Z ? What set is 1+1 in if it isn't in the integers? Where does it exist as a mathematical object?

I'm still thinking though, may be able to come up with a better way to express what bothers me about this.

Arkadia, I think the thing is that 1 is the identity element for multiplication. 1 is what you should get when you multiply together the empty set, just as 0 is what you get when you add up the empty set. It's the version of "nothing" for multiplication, whereas prime numbers are the smallest units of "something" for multiplication.

A prime is defined as having only two factors (itself and 1). One only has one factor (itself) so it fails the test

Change, still only a matter of wording. Hardly profound. If someone asked why, the answer is still "because it is".

I don't get you? One doesn't have two factors...so is not prime. Any number that does not have two factors is not prime.

Well, I could argue that it does have 2 factors, only that the two factors (1 and itself) coincide. How many solutions does (x-2)^2=0 have?
Besides, I am still not clear what "1" then is... It is not prime, it is not composite, but it is "other", which makes it difficult for me. I can't really follow the "unit" argument and besides it just comes down to a definition. Would need to ask the right kind of mathematician, but I don't think I am in touch with any.

Arkadia, you said you have discovered "1" isn't a prime anymore on Khan. Which video was it? Because I had a look, and Sal never says that 1 used to be prime but not anymore. Sal clearly says on tutorial video that prime is defined by having only 2 factors, 1 and itself, as Change states. I am also ancient, but I never learned 1 as a prime.

As for needing to ask mathematicians, there are plenty on this thread.

It didn't say "anymore", but he gave it for granted. It is a video in grade 4 where he introduces composite numbers.
When he started saying that there are primes, composites and "others" I thought he meant "zero", instead no...
The concept that 1 is not prime is totally new to me. I did engineering at university, but despite I had to sit LOTS of higher maths exams at no stage did anyone warn me that 1 is not a prime (or perhaps I was napping at the time...)

Perhaps you were napping, or perhaps it never came up in engineering. I certainly remember it being mention in my maths degree (although I couldn't claim to actually be a mathematician - more someone with an interest in maths).

Whether it came up in university or not, I know that year 6 know that one is not prime by that definition as their teacher quizzed them during lunch last week.

That sounds a bit mocking. What I meant was...thats how kids are learning it