Irrational numbers are numbers that cannot be
expressed as the ratio of
two integers. See rationals.
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
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.
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
Other irrational numbers include √3, √5, ∏, -∏, etc.
This is another FREE Algebra PRINTABLE presented to you from the
Algebra section of