Every measurement or calculation we do with instruments in chemistry has some level of uncertainty called experimental error. Significant figures are necessary to communicate a level of accuracy and precision while dealing with this uncertainty.
Rules for Number of Signficant Figures
Rule 1: If your number has a decimal point move from left to right. Start counting once you get to your first non-zero number and keep counting until you get to the end.
Trailing zeros are a sequence of zeros after the decimal point once you’ve passed all non-zero integers. They would be significant.
Rule 2: If your number has NO decimal point move from right to left. Start counting once you get to your first non-zero number and keep counting until you get to the end.
If a value has no decimal point then the final zero would mean it has an infinite number of significant figures.
Exact Numbers
Exact numbers contain an infinite number of significant figures because they represent the amount of a particular item and not a measurement.
For example, when we say that a decade is equal to 10 years, there are 150 students in your chemistry class, or NH_{3} contains 1 atom of nitrogen and 3 atoms of hydrogen.
Addition & Subtraction
When dealing with addition or subtraction the least number of decimal places will determine the final answer.
Adding the two values gives the initial value of 5.375 x 10^{6}, but the final answer should have only 1 decimal place and so the answer is rounded.
Multiplication & Division
When dealing with multiplication or division the least number of significant figures will determine the final answer.
Multiplying the two values gives an initial value of 103.6, but you must round your final answer to significant figures.
Furthermore keep in mind when dealing with mixed operations we must follow the same arithmetic guidelines as we always do.
Logarithm & Natural Logarithm
When taking the log or ln of a value, the number of significant figures it has determines the number of digits after the decimal point for the answer.
Using your calculator, the log of 1.53 x 10^{-3} gives an initial value of - 2.01531. The value of 1.53 x 10^{-3} has 3 significant figures and so the final answer should have 3 digits after the decimal point.