Irrational Numbers

We have seen that Rational Numbers can be written as a fraction of two integers.

p/q

But can every number be written in this form?

Consider the number:

√2

√2 is a perfectly valid number.

But can we write it as a fraction of two integers?

√2 = p/q ?

Surprisingly, no.

There is no pair of integers p and q, with q ≠ 0, whose fraction is exactly √2.

So √2 cannot be written in the form:

p/q

KEY IDEA

Numbers that cannot be written as p/q are called Irrational Numbers.

√2, √3, π, ...

Their decimal representations do not terminate or repeat in a fixed pattern.

NOW WE HAVE TWO GROUPS

Some numbers can be written as a fraction of integers.

Some cannot.

NEXT IDEA

What happens when we put these two groups together?