Decimals

Now that you have a solid understanding of decimals, it is time to "pull the rug out" from under you. As it turns out, every decimal expression "goes on forever". In fact, every whole number goes on forever!

Let's start with the whole number one. We can represent this quantity in decimal notation as one, or as 1.0, or 1.00, or 1.000. An infinite string of zeros continues to the right of the decimal point; we have just agreed to rarely, if ever, write them! That is:

$1 = 1.\overline{0} = 1.000000000000000\cdots$

It is also true that the whole number one has an infinite string of zeros that continue on the left before the 1 itself! Again, we have merely agreed to rarely (if ever) write these zeros.

This means that we have:

$\cdots 0000000000000001.000000000000000\cdots = 1$

Of course, this also applies to other decimal expressions, such as these:

$\begin{align*} \frac{1}{3} &= \overline{0}.\overline{3} = \cdots 000000000000000.333333333333333\cdots \\ \\ \frac{1}{4} &= \overline{0}.25\overline{0} = ...000000000000000.250000000000000\cdots \end{align*}$

Since including all of these zeros is cumbersome and distracting, we will not use this notation. We encourage you to similarly avoid it, but it is worth mentioning for two reasons. First, it helps demystify ongoing decimal expressions – they have been with us all along! These observations also help explain what decimal notation, and our usual notation for whole numbers, actually means.