1365.11 – Decimals On Screen

We start with

$$1/8: .1 2 5 0 0 0 0 0.$$

Think of this as: $.1 2 5\ \_\ \_\ \_\ \_\ \_$ , where there are 10 possible digits for each blank, thus $10^5$ possible 8-place decimal numbers greater than $\dfrac{1}{8}$ and beginning with $.125$. There is a similar result for $.126\ \_\ \_\ \_\ \_\ \_$, and for $.127\ \_\ \_\ \_\ \_\ \_$, and so on up to $.129\ \_\ \_\ \_\ \_\ \_$. This makes $5 \cdot 10^5$ numbers so far.

Next, for $.13\ \_\ \_\ \_\ \_\ \_\ \_$, $.14\ \_\ \_\ \_\ \_\ \_\ \_$, and so on up to $.19\ \_\ \_\ \_\ \_\ \_\ \_$, we get $7 \cdot 10^6$ more. And for $.2\ \_\ \_\ \_\ \_\ \_\ \_\ \_$, $.3\ \_\ \_\ \_\ \_\ \_\ \_\ \_$, and $.4\ \_\ \_\ \_\ \_\ \_\ \_\ \_$, we get $3 \cdot 10^7$ more.

Finally, there is .5 itself, i.e., $\dfrac{1}{2}$. One possibility.

Now add:

$$(5 \cdot 10^5) + (7 \cdot 10^6) + (3 \cdot 10^7) + 1 = 37,500,001.$$

That's a lot!