How to add these in accordance with the rules of significant figures?

Question

$2300$ has 2 significant figures (2,3). Thus $2300+123 = 2423$. However, we can also write this as $100\cdot(23+1.23) = 100\cdot( 24.23 )$ since 23 has no decimals, 24.23 is rounded to 24. The answer becomes 2400. Which is correct?

Answer 1

When adding numbers using significant-figure rules, the rule is that the position of the least significant figure of the result should not be in the same position, or to the right of, any insignificant digits in the addends. In your case, we have

$$\begin{array}{rr}&2300\\+&123\\\hline&2423\end{array}$$

The first addend is only significant up to the hundreds place, while the second one is significant up to the ones place. So we must round the result off to the less-precise of these numbers, namely 2400.

It is occasionally taught that you're supposed to "round the answer to the LEAST number of places in the decimal portion of any number in the problem." This is inconsistent, as this case illustrates. Really, you're supposed to look at the positions of the lowest significant figures of each addend, and then round off at the highest one of these "lowest significant positions".

Answer 2

Since the uncertainty in "2300", which we'll state is +/- 50 (Half the last digit), you can see that the uncertainty is far greater than the uncertainty in the second number. As such, you could write
" 2423 +/- sqrt(50^2 + 0.5^2) "
But since that error term is practically equal to 50, then you can see why the final answer, without explicit statement of error, is "2400."

HOWEVER
This is why we use "scientific notation" to avoid confusion. Then we start with

2.3E3 + 1.23E2
And it's much easier to propagate the uncertainty correctly.