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It might be a silly question but I was just wondering if this was unprovable or false...
Thanks guys
Thanks guys
g_edgar said:There are some concepts of "density" of a set of natural numbers. The set of even numbers has density 1/2.
http://planetmath.org/encyclopedia/AsymptoticDensity.html"
http://mathworld.wolfram.com/SchnirelmannDensity.html"
CRGreathouse said:The asymptotic density of the even numbers is indeed 1/2. But the Schnirelmann density is 0.
g_edgar said:...and the Schnirelmann density of the odd numbers is 1/2
HallsofIvy said:So the Schnirelmann density of the even numbers is 0 and the Schnirelmann density of the odd numbers is 1/2? Peculiar!
Office_Shredder said:It's a poorly worded question. How many natural numbers are there, and how do you divide that by half?
As a start up to this kind of stuff, consider the function
f(n) = 2n
You now have a bijection, or a 1-1 and onto function, from the set of natural numbers to the set of even numbers. So for every even number you give me, I can give you a natural number, and for every natural number, I can give you an even number. Hence there must be the same number of even numbers as there are natural numbers.
This is more related to set theory than number theory (counting the size of different sets)
Wikipedia said:Consider a hypothetical hotel with infinitely many rooms, all of which are occupied – that is to say every room contains a guest. Suppose a new guest arrives and wishes to be accommodated in the hotel. If the hotel had only finitely many rooms, then it can be clearly seen that the request could not be fulfilled, but because the hotel has infinitely many rooms then if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, you can fit the newcomer into room 1. By extension it is possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room N to room 2*N, and all the odd-numbered rooms will be free for the new guests.
Natural numbers are also known as counting numbers. They are positive integers that start from 1 and continue indefinitely. In other words, they are the numbers we use for counting objects.
A number is considered even if it is divisible by 2, meaning it has no remainder when divided by 2. In other words, an even number is any number that ends in 0, 2, 4, 6, or 8.
No, it is not possible for half of all natural numbers to be even. Since natural numbers start from 1, half of them would be 0.5, which is not a natural number. Therefore, half of all natural numbers cannot be even.
There are an equal number of even and odd natural numbers. For every even number, there is a corresponding odd number that is one more than the even number.
Yes, all natural numbers can be classified as either even or odd. This is because every number has a remainder of either 0 or 1 when divided by 2, making it either even or odd.