Difference between revisions of "Real numbers"
| Line 145: | Line 145: | ||
= Hints for difficult problems = | = 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 | ||
= Project Ideas = | = Project Ideas = | ||
Revision as of 06:19, 23 February 2014
| 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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