Real numbers
| Philosophy of Mathematics |
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Hints for difficult problems
Irrational numbers Exercise 1.3.1 1.Write four possible irrational numbers between 4 and 5. Solution: Consider the squares of 4 and 5
42 = 16 and 52 = 25 We can also wright 4 and 5 as =4 and =5
Between and there exists
,,,,,,,
2.Write four rational numbers between and Solution: root 2 ~1.414 and root 3 ~ 1.732 Rational numbers of root2 and root 3 is in between 1.4 to 1.7 like 1.45,1.5,1.55, 1.6 ........
3.Prove that + is an irrational number Solution: Let us assume on contrary that is a rational number. Then there exists co-prime positive integers p and q such that
=
-=
(-)2 = ()2 +2- 2.= 5
=
p and q are integers is rational
But this contradicts the fact that is irrational. So, our assumption is wrong. Hence, is irrational Write an irrational
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