# How many times can you round a number?

For a typical rounding algorithm, I'm wondering how long the rounding chain goes for when you round up a number. For example, if you have a decimal like 0.4445, you round the last 5 up, which would round it to 0.4450, which would round it to 0.4500, which would round it to 0.5000, which would round to 1.0000? Or does the rounding stop after the initial rounding?

So we are given a number $0.4445$. Now if we were to round this to three decimal places we would indeed get $0.445$. Rounding the resulting number to two places gives $0.45$. However this does not represent the rounding of the original number, $0.4445$ to two places, for that would be $0.44$.
So rounding successively has no real meaning. An even more obvious example than the one above is that successive rounding of $0.4445$ yields $1$, whereas rounding the number to the ones place initially gives $0$.