The significant figures are digits that are significant or meaningful in a numerical value. For example, you know the distance between two cities to be \(144\text{km}\) within the uncertainty of \(\pm 1\text{km}\) then all the digits in \(144\) are meaningful or significant because you are sure up to the final digit within an uncertainty of \(\pm 1\) and there are three significant figures.
Consider \(45892\pm 1\). The final digit has uncertainty of \(\pm 1\). It means all the digits are meaningful and there are 5 significant figures. In \(0.0034\) there are two significant figures because the zeros in the left hand side are only the place holders for the decimal point and they are not significant and this number obviously has the uncertainty of \(\pm0.0001\).
Now what happens when you add or subtract the numbers having uncertainties? When you add two or more numbers having uncertainties the result will have the same uncertainty as in the number with fewest digits to the right of the decimal point.
For example \(3.54+3.3=6.8\) not \(6.84\). The number \(3.54\) has the uncertainty of \(\pm 0.01\) and that of \(3.3\) is \(\pm 0.1\). Now the result has the same uncertainty as there is in \(3.3\) due to only one digit to the right of the decimal point. And the same rule applies for the subtraction as well, for example \(4.43-2.6=1.8\) not \(1.83\).
When you add or subtract numbers having uncertainties the computed number will also have the uncertainty equal to the uncertainty with the number having fewest digits to the right of the decimal point.
The general rule of the multiplication and division of the numbers with uncertainties generally in the lack of complete analysis of uncertainties is that the result always has the same number of significant figures as there is in the number with least number of significant figures.
For example, multiply \(7.86\) and \(3. 112\), the result of multiplication has the same number of significant figures as there is in the number with fewest number of significant figures. So the value of multiplication of \(7.86\) and \(3. 112\) is 24.5 not 24.53892.
When you multiply and divide the numbers having uncertainties the result has the same number of significant figures as in the number with fewest significant figures.
For example let's find the total number of significant figures in \(\frac{(2.59)(3.2)}{2.69}\).
As we have discussed before the total number of significant figures in the result of multiplication and division is equal to the number of significant figures in the number with least number of significant figures in the calculation. So the answer is 3.1 not 3.081040892 as your calculator gives. And note that do not forget to round your final answer.
And what about the total number of significant figures in \(5.3-3.21\)?
As we have already known that for the addition and subtraction the location of decimal point matters not the number of significant figures, that is the result will have the same uncertainty as there is in the number with fewest digits to the right of decimal point. The answer is \(2.1\) not \(2.09\). You should not forget to round your answer.
