<s>[INST] What are the methods of dividing numbers to get weird values like $16\over 17$ without a calculator? I tried estimating it to somewhere near $16\over 20$, but it's a far stretch from getting the actual $16\over 17$. How can one do so? Conventionally, I think for numbers such as $50\over 17$, or for any large numbers, we have methods to do division and we can get $2+{16\over 17}$, but we're still left with $16\over 17$, which I have no idea how to find (or at least estimate till a good accuracy).
More interestingly, how would computers or calculators even find these values?
 [/INST] Please don't upvote me or downvote me as I have obviously copied and edited Mr. Hardy's answer. I agree with Mr Hardy that long division is the answer, but there is a small trick that makes subtraction a bit easier
Long division:
$$
\begin{array}{cccccccccc}
& & & 0 & . & 9 & 4 & 1 & 1 & 7 & 6 \\  \\
17 & ) & 1 & 5 & . & 9 & 9 & 9 & 9 & 9 & 9 \\
& & 1 & 5 & & 3 \\  \\
& & & & & 6 & 9 \\
& & & & & 6 & 8 \\  \\
& & & & & & 1 & 9 \\
& & & & & & 1 & 7 \\  \\
& & & & & & & 2 & 9 \\
& & & & & & & 1 & 7 \\  \\
& & & & & & & 1 & 2 & 9 \\
& & & & & & & 1 & 1 & 9 \\  \\
& & & & & & & & 1 & 0 & 9 \\
& & & & & & & & 1 & 0 & 2 \\  \\
& & & & & & & & \text{etc.}
\end{array}
$$
</s>