Complexity Math

question marks are infinity symbol

It strange to think that the infinity of Reals or Rationals is larger then the infinity of integers, or how some infinities are greater then others. Can you have 2*infinity ?

think you meant irrationals?

On September 06 2016 07:47 FMLuser wrote: It strange to think that the infinity of Reals or Rationals is larger then the infinity of integers, or how some infinities are greater then others. Can you have 2*infinity ?

Infinity is still infinity, no matter how many times you divide or multiply it.

Infinity of Rationals and Integers is the same "size" (they can be put/mapped one-to-one). They are called "countable" (as is any set that can be mapped one-to-one to Integers)

Reals are "uncountable" - no matter how you try to map them to Integers, there will always be some (infinity) outside of the mapping.

See:
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

On September 06 2016 09:36 PoorUser wrote: think you meant irrationals?

yeah sorry used to using Z R Q N

IMO the most beautiful set of theorems in mathematics:
Between any two rational numbers there is an irrational number.
Between any two irrational numbers there is a rational number.

terror

there's a video on youtube about the different sizes of infinity.

I'm at work but I think it's this one

On September 06 2016 19:03 Daut wrote: IMO the most beautiful set of theorems in mathematics: Between any two rational numbers there is an irrational number. Between any two irrational numbers there is a rational number.

"Between any two rational numbers there is an irrational number."

I belive (but im not sure) that between any two rational numbers there is infinite amount of irrational numbers:

(-oo, +oo) = (0,1) = | 2*N | = | R |

(0.5, 0.6) =| R |

I belive

let A, B be real numbers. A > B, and B != 0.

then, there exists a real number C where A - B = C, where C > 0.

there exists a natural number N such that N > 1/C

so:

N > 1/C
<=> CN > 1
<=> AN - BN > 1

=> there is an integer X where:

AN > X > BN

<=> A > X/N > B

where X/N is a rational number

since A and B are real numbers, they can be both rational and irrational, and since X/N is rational, there are an infinite amount of rationals between A and B

i think that's right (i'm rusty)

euler's identity is probably the most beautiful thing in math imo


e^i(pi)+1=0

imo the fact that between 0 and 1 there are equally many (in the bijection sense) numbers as there are any numbers is pretty cool. although now that i think the "in the bijection sense" kind of ruins it and becomes somewhat trivial and boring, but I remember being impressed when a professor stated this without proof on like first lecture at the university.

Yeah i vote eulers identity. Also i like the proof that closed interval from 0 to 1 is uncountable. It shows that the reals is just faaaaaaaar bigger than integers.