




1 < 7 < 8 =>
1,9^{3} = 6.859 < 7 1,91^{3} = 6.967871 < 7 < 1,92^{3} = 7.077888 => 1,91 < < 1,92 1,911^{3} = 6.978821031 < 7, 1,912^{3} = 6.989782528 < 7, 1,913^{3} = 7.000755497 > 7 => 1,912 < < 1,913 
=> 1,912 < < 1,913 

proof
(a) is irrational (b) is irrational, if p is a prime number 
(a) assume:
is rational =>
stands for natural numbers p, q teilerfremd: p, q can not be
divided by
the same


(b) assume:
is rational =>
a, b teilerfremd: a, b can not be divided by the same natural number ist durch p teilbar: can be divided by p im Widersrich dazu: this contradicts dass a und b teilerfremd sind: that a and b can not be divided by the same natural number 
Setting x = y it follows, if x^{2} = x * x can be divided by
p then x can be divided by p.