Homework Help: irrational numbers?!

Posted by Sanchia on Monday, June 15, 2009 at 11:10pm.

what are rational and irrational numbers??
i know that irrational numbers are numbers that cannot be expressed as a fraction, but i'm still confused.

there is this question:

which of the following are irrational numbers: √2, √8, 22/7, pi, 2√3

i know that 22/7 is rational, pi is irrational, but what about those square roots?

irrational numbers?! - drwls, Monday, June 15, 2009 at 11:49pm
Square roots of numbers that are not the squares of integers are all irrational. There is a way to prove that but I forgot the details. You have to assume that a fraction works and then prove that the assumption leads to a contradiction

Any number that does not meet the definition of being rational is irrational.
irrational numbers?! - Reiny, Tuesday, June 16, 2009 at 12:55am
- √2 is either rational or irrational.

- Assume that √ is rational so that
√2 = a/b, with a/b in lowest terms
- Square both sides to get
2 = a^2/b^2
then a^2 = 2b^2
- the right side of this eqation is clearly an even number, since anything multiplied by 2 is even
- so a^2 must be even. We also know that if we square an odd number the result is odd, and if we square an even number the result is even
so a must be even
so a could be written as 2k
- rewriting our equation as
2b^2= (2k)^2
2b^2 = 4k^2
b^2 = 2k^2

by the same argument as above 2k^2 is even , so b has to be even

which means a and b are both even, therefore a/b is not in lowest terms

BUT that contradicts my assumption that a/b was a fraction in lowest terms

so √2 = a/b is a false statement
therefore √2 cannot be rational, and
must then be irrational

the same argument could be used for √3 and all other square roots
irrational numbers?! - brandy, Tuesday, September 1, 2009 at 4:55pm
true or false does this number represent a rational number 0.20200200020000200000
irrational numbers?! - Amber, Thursday, April 21, 2011 at 5:15pm
Is √4 Irrational?

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