Normally when we are taught how to add numbers with regards to significant figures, we are told to round the result to the rightmost place of the least precise digit. $3.55 + 4 = 7.55$, for example, would be rounded off to $8$. But for my argument I will be considering the addition of two numbers $9$ and $2$, both of which are significant to the ones digit.
Following the usual rule for addition, $9 + 2 = 11$ and now we have two significant figures. However, whenever I think about significant figures I would intuitively think of these two definitions:
1 - Digits to the right of the last significant digit, can be any number between 0 and 9.
2 - Any digit that has the possibility of being more than one numerical value, is not a significant digit.
Using the first definition, we can put $9$ and $2$ in these forms:
where $A$ and $B$ are arbitrary digits ranging from $0$ to $9$. Thus, when we add these two numbers, we would end up with:
$$9.A + 2.B$$
$$= 9 + 2 + 0.A + 0.B$$
$$= 11 + 0.A + 0.B$$
Finally when we assign some meaningful values to $A$ and $B$ such as $(A , B) = (0 , 0)$ or $(A , B) = (9 , 9)$, we can observe something very interesting:
- For $(A , B) = (0 , 0)$:
$$11 + 0.0 + 0.0 = 11$$
- For $(A , B) = (9 , 9)$:
$$11 + 0.9 + 0.9 = 12.8$$
In the set of all possible numbers that can result from adding $9.A$ and $2.B$, both of which are significant to the ones digit, the minimum happens to be $11$ and the maximum happens to be $12.8$ (the set becomes even greater if we add more uncertain digits to the right of $A$ and $B$, and in that case the maximum resulting value starts to approach $13$ as you add more and more 9's to both $9$ and $2$).
Lastly, we can see that tens digit is always $1$, and using my second definition of significant digits, we can claim that the tens digit is significant. On the other hand, because ones digit can either be $1$ or $2$, the ones digit is not significant. Thus, using my two definitions of significant figures, we can claim that the addition between $9$ and $2$ results in one significant figure (at the tens place), rather than two significant figures one would get from the usual rule of thumb.
So my question is, are my definitions of significant digits correct? If yes, then I can say the general rules of thumb for adding numbers with significant figures is wrong and is being blindly taught and learned in high schools and universities. If not, then can you show me why not and if you have any credible sources.