Phil Bowermaster is wondering if there’s something dodgy about the math here.

No, this is in fact a standard technique for determining the sum of an infinite series, which is in fact what 0.999… is (it could be expressed as the sum, from n=0 to infinity, of the expression 9 times 10 to the minus n). Perhaps, as one commenter notes, it’s the word “precisely” that’s hanging people up, but certainly that number is equal to one, whatever modifier you want to put on it or leave off.

[Update in the afternoon]

I’m not sure I follow the commenter’s objection. He claims that no matter what you start out with as “a” you get a=1. I don’t see that.

Try it with two, as suggested.

a = 2
10a = 20
10a – a = 20 – 2
9a = 18

Ergo, a = 2.

In fact, do it with 1.999…

a=1.999…
10a = 19.999…
9a = 18
a = 2

As I said, it’s a standard technique for expressing repitends as whole numbers or fractions.