Exploring the 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