Non-Repeating Decimals

Let's go on a brief optional tangent for those who are so inclined and see if you can identify a decimal situation we have failed to address. We have discussed terminating decimals (decimal expressions that "stop" at some finite location beyond the decimal point) and repeating decimals (decimal expressions that do not "stop" at some finite location beyond the decimal point. Rather, they begin repeating a fixed string of digits at some finite point beyond the decimal).

But what about a third possibility? What happens when you have an unending decimal expression that is not repeating?

For example, consider an unending decimal expression whose first several digits are:

$3.141592653589793238\cdots$.

Because there is no evident pattern of repetition, we cannot convert this expression into a fraction. There are also numbers, such as the Champernowne Constant, that have a clear and never-ending pattern, but we cannot express them as a fraction either:

Champernowne Constant $= 0.123456789101112131415161718192021\cdots$

We cannot even convert even a seemingly simplistic but non-ending decimal expansion such as this one into a fraction:

$0.1010010001000010000010000001\cdots$

What is going on? What types of numbers do they represent If we cannot convert these "strange decimal expressions" into fractions? These expressions represent (or approximate) a new kind of number we call an irrational number. We call them irrational precisely because we cannot express them as a ratio or fraction of whole numbers.

In other words, both decimal notation and fraction notation fail to help explain these quantities. But mathematicians have derived these numbers because they help explain other, deeper mathematical operations that are at play. For example, the first irrational expression above approximates the famous number $\pi$, which we use to measure the lengths of circles. We use the expression $\sqrt{2}$ to measure certain right triangles. But we cannot express it as a ratio of whole numbers either.