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 1/8 and beginning with .125. There is a similar result for .1 2 6 _ _ _ _ _, and for .1 2 7 _ _ _ _ _, and so on up to .1 2 9 _ _ _ _ _. This makes $5 \cdot 10^5$ numbers so far.

Next, for .1 3 _ _ _ _ _ _, .1 4 _ _ _ _ _ _, and so on up to .1 9 _ _ _ _ _ _, 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., $1/2$. One possibility.

Now add:

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

That's a lot!