Rounding with significant digits I'm dealing with significant digits right now, and recently I've been having a nagging question in my mind. When we have digits past the last significant digit in a quantity, do we round the last SD by the first non-SD? For example, say we have a random number, $3.456722$. However, only 3 SDs are allowed because in the multiplication, the quantity with the least amount of SDs had 3. Do we round $3.45$ to $3.46$? I'm iffy about this because I think rounding using that non-SD makes the value pseudo-contain that non-SD, and thus is more precise than it should be. Should we round, or should we not?
P.S. Put the extra 4 digits at the end to prevent complaints about 3 digit and 4 digit numbers not multiplying to produce 5 digit numbers.
 A: When you round to 3 significant digits your possible results are
$$ 3.45 \pm 0.005 \qquad \text{ or } \qquad 3.46 \pm 0.005 $$
The second of these intervals is the one that contain your true result, so that is what you should round to.
Normally we don't write the $\pm 0.005$, but it is implicit when we write a rounded number with two digits after the decimal point.
We choose to let the approximation $3.45$ stand for "a true result somewhere in the interval $[3.445,3.455]$" rather than the interval $[3.45,3.46)$ because it makes for smaller errors to use an approximation roughly in the middle of the intervals. Since we're deciding to use approximations with three digits written down (in order to save ink), this is most easily done by choosing the intervals we can represent to be such that our three-digit numbers lie in the middle of them.

P.S. There is actually no 3-by-4-digits multiplication that produces $3.456722$ exactly. The closes we get is $0.720\times 4.801=3.456720$.
It is also important to remember that "number of significant digits" is not an exact procedure, but a rule of thumb that allows you to estimate the size of the round-off errors roughly without too much work.
If your actual numbers to multiply are $0.720\pm0.0005$ and $4.801\pm0.0005$, the true product is somewhere between $3.45395975$ and $3.45948075$. When we say, "oh, three significant digits" and round to $3.46$ on that basis, we're in effect rewriting the interval $[3.45395975,3.45948075]$ into $[3.455,3.465]$.
This means the the true result could possibly be outside of the interval we're giving as a result. That is the price we pay for the convenience of not writing the precision down explicitly but instead letting it be implicit in the number of digits we write down.
