# Exploring the Irrationals

Math > Math Concepts  > Numbers  > Irrationals

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## Irrationals

 Irrational numbers are numbers that cannot be expressed as the ratio of two integers.   See rationals.   One of the first irrational numbers discovered was √2.  The Greeks discovered this number, and really didn't know what to do with it.   They constructed a right triangle whose legs were equal to 1. The hypotenuse of this triangle has the immeasurable length of  √2.   And this triangle wasn't the only one they discovered with irrational lengths.     Here's how the story goes to show √2 cannot be expressed as the ratio of two integers.  (This is how the Greeks proved this fact.)       Let's say, instead,  that  √2 can be expressed as the ratio of two integers in reduced form,  say  m/n.  So we have                        √2  =  m/n.       (reduced form means m and n have no                                                 common factors.) Now let's square both sides and get    2  =  m2 /n2 So, now we have  2n2 = m2        this means that m2 is even. Now and odd number multiplied by itself results in an odd number, so m  must be even, which means m equals 2 times some number say x. m = 2x.      So, m2 = 4x2.    And using the equation 3 lines back we have 2n2 = 4x2.    Dividing by 2 we have n2 = 2x2.  This equation tells us n2  has to be even, so n equals 2 times some number say y. n = 2y.      We have a problem here, m = 2x and n = 2y means m and n have a common factor, 2, which goes against our original claim that √2 can be expressed as the ratio of two integers in reduced form.   Other irrational numbers include √3, √5, ∏, -∏, etc.   This is another FREE Algebra PRINTABLE presented to you from the Algebra section of K12math.com