I have this memory from highschool math classes that while most positive integers are even or odd, the numbers 0-2 were classified as neither, but were instead called “unique” numbers. (I’m more certain I learned this for 0 and 1; less so for 2.)
Sadly, my Google-fu turns up nothing to support my recollection, and this wikipedia link seems to actively dispute it.
Does the notion of these numbers being unique, rather than even or odd, have any validity? Is this something that used to be taught but is no longer? Or am I merely suffering from a case of faulty memory?
The only thing unique is that 0 is the identity value for addition, 1 is the identity value for multiplication, and 2 is the only even prime number. Maybe it’s little factoids like that you’re recalling?
IANAM(athemetician) , but I can’t see how 1 and 2 are not considered respectively odd and even, given that in my mind, at least, even means divisible by 2.
I don’t remember ever being taught that. 1 was odd; 2 was even. 0 is more problematical, as it has many unique properties, but not having it be even would create problems of its own.
The one exception I can think of is the elementary definition of an odd number - a number that leaves a remainder of 1 when divided by 2 - which obviously would leave 1 out. However, that is not a good definition of odd number. Try these:
And if you use this definition, even the divide by 2 rule is accommodated:
There is no question at all that 2 must be even, though. And check the definition of even number:
I’m forced to believe that your memory is just plain wrong.
More generally, we speak of two numbers a and b being congruent mod m if a - b is divisible by m. This has some really nice properties (and generalizes further in a nice way, but that’s getting pretty far afield). We define the even numbers to be those that are congruent to 0 mod 2 and the odd to be those that are congruent to 1 mod 2.
I have no clue where you might have gotten that factoid. But AFAIK there’s no validity in it.
What might be the case is a faulty memory of the singularity: One is not prime, by arbitrary rule (and because one being prime would muck up some important premises in number theory), despite the fact that it meets the casual everyday definition of prime (“Evenly divisible only by itself and 1.”) However, the unique character of primes is that they and only they have exactly two and only two discrete natural factors: themselves, individually, and 1. One is completely unique in being the only number with only one discrete natural factor. Not having precisely two discrete natural factors, one is not prime.
Rather obviously, two is the only even prime, since “even” means “divisible by 2 within the natural numbers” and hence any other even number would exhibit more than two possible natural factors: 1, 2, and at minimum itself (the case with 4), more often itself and itself-divided-by-two, e.g. 46 (factors of 1, 2, 23, 46).
We’ve had that 1/infinity discussion before and the related 1/0=infinity discussion.
The problem is that infinity isn’t a number. If 1/infinity=0, what does 2/infinity equal? Zero again, right? So if we treat infinity as just another number then 1=2. So then we have to say that infinity can’t be used in regular old algebra, and that means that expressions like 1/infinity aren’t well defined.
Because he was waiting for you to reach basic Calculus, probably, when limits and such are introduced. As the denominator approaches infinity, the overall fraction approaches zero, but since the first condition is never actually fulfilled, neither is the second. For practical purposes, though, it’s zero.
When I said “elementary” definition I was referring to the definition as taught in elementary school - at least back in my time - before classes got to the complexity of getting zero times for an answer. I also said it was wrong. It was a WAG as to why the OP might be remembering 1 as not odd, not a formal expression of math.
what you are thinking of is the fact that 0 and 1 are “neither prime nor composite,” and that 2 is the only even prime.