[Master]

#The number of evens

There are as many even numbers as natural numbers.

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• The natural numbers are the "counting" numbers, like 1,2, 3, 4, et cetera.
  
• There are an infinite number of natural numbers.
  
• There are also an infinite number of even numbers.
  
• You'd imagine that there are more natural numbers than even numbers because the natural numbers
  are comprised of both the evens and the odds.
  
• You would also be wrong.
  

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We can set up a one-to-one correspondence between the natural numbers and the even numbers that shows that for every natural number there is also an even number.

• Think about it this way. Every natural number has a number that is twice as large as it, and every even
  number has a natural number that is half its size:

1<——>2
2<——>4
3<——>6
4<——>8
5<——>10
6<——>12
7<——>14
8<——>16
...<——>...

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What does this mean?

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• For every natural number, there is also an even number.
  
• This means that both infinite sets are the same size, what we call "countable infinite."
  
• This distinguishes it from sets that are “uncountable infinite," like the real numbers or the complex numbers.
  
• For example, you can't set up a one-to-one correspondence between the natural numbers and the real numbers.
  
• Other countable infinite sets include the rational numbers and the odd numbers
  

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Sources

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The Most Controversial Math Problems
http://uk.businessinsider.com/the-most-c...013-3?op=1

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[roger_smith]

at first I was like "I call shenanigans" but after reading this again I realized what you're saying. This is actually quite interesting. It's quite intriguing that odd number x2 seems to always equal an even number.

[Eclipse]

It's nice to see someone else post under the [Mathematics] tag.

OT: It's an interesting topic, and one that defies logic. Nice work.

EDIT: Wait, I think there's something wrong with that. Let me do some research and I'll get back to you.
EDIT2: No, never mind. OP is correct. +1

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at first I was like "I call shenanigans" but after reading this again I realized what you're saying. This is actually quite interesting. It's quite intriguing that odd number x2 seems to always equal an even number.

In algebra, to represent an odd number, we take advantage of this and do: 2n + 1, where n is an integer.