Random really interesting proof I discovered today, it's really awesome (but a very famous proof)
Basically, if I have a set of an infinite amount of real numbers (ex: 1, 5.75, -4, Pi, 1038579458.46857593, etc) then does that set contain EVERY real number? Intuitively you'd think, well, of course! IT has to in a way, it's a set that never ends which means eventually every real number must be read and if you can come up with a real number then it should be in this set.... right?
Wrong!
In fact, there will always be a real number that will never be in your infinite set, even though your set is infinite. The proof goes as follows, say we convert real numbers into infinitely long binary numbers, they mean the same thing it's just one is in base 10 and the other is in base 2. Let us look at our set of infinite amount of infinitely long binary numbers
S:
101001001....
100100101....
001010011....
100100011....
100101101....
....etc
That's just an example of the set. Lets say we wanted to create a new binary number, called B. Now suppose we definethe nth digit of B to be the nth digit of the nth binary number in our set. That basically means B is a binary number that makes up the diagonal of the set. Let us look at our set again
[1]01001001....
1[0]0100101....
00[1]010011....
100[1]00011....
1001[0]1101....
....etc
The digits with brackets around them are our digits of B, so we would B in this case would be equal to 10110..... etc.
Now suppose we invert every digit in B. So anytime we see a "1" we convert it to a "0" and every time we see a "0" we convert it to a "1". This means our B will now equal 01001.... etc
This new number we created, B, is now different to EVERY SINGLE number in our set because there is always going to be ONE value in B that is different than some binary number in our set, because we took the diagonal and inverted it. Thus, even though there are an INFINITE amount of real numbers in our set, we have created a number that can never belong in our set!
I think that's absolutely awesome! Even infinity has its limit so to speak
Topic: The Random Thread (Read 2053083 times)