# Significant Figure Calculations

The more rigorous approach for determining the precision of a calculated result is to propagate the uncertainty in all your measured quantities through the calculation. The topic of error propagation through calculations, however, is outside the scope of this course. Thus, we will present the simpler rules below for determining the approximate number of significant figures in a calculated result.

## Addition and Subtraction

**Rule:**- With Addition and Subtraction keep only the number of decimals in the result that occur in the least precise number.

Note that only the final answer is rounded. There is no rounding numbers in the intermediate steps of the calculation.

## Multiplication and Division

**Rule:**- With Multiplication and Division, the final result should only have as many significant figures as the term with the least number of significant figures.

## Mixing Operations

It can be tricky keeping track of the number of significant figures in a calculation that combines addition, subtraction, multiplication, and division. For example,

My calculator gives 0.013099698, but how many significant figures should be in the final answer? All numbers in the calculation have 3 significant figures, but if you break up the calculation into steps

#### Exact numbers

Exact numbers are known with infinite precision. For example if there are 10 students in a classroom, that number is an exact number. We treat exact numbers as numbers known with infinite precision

**What is the average of 10.2, 11.4, and 10.9?**

#### Powers of 10 and Logarithms

When taking 10 to the power of a number the final answer will have the same number of significant figures as the fractional part of the number.

**Calculate 10 ^{-9.2067} with the correct number of significant figures.**

We split the calculation into the integer and fractional part:

10^{-9.2067} = (10^{-9})(10^{-0.2067}) = (10^{-9})(0.621__2__9806352) → 0.621__3__ x 10^{-9} = 6.21__3__ x 10^{-10}

When taking the logarithm of a number the final answer will have the same number of significant figures as the logarithm argument.

**Express the logarithm of 6.213 to the correct number of significant figures.**

log(6.21__3__) = 0.793__3__013536 → 0.793__3__

When taking the logarithm of numbers expressed in scientific notation, remember that the exponent of 10 is an exact number and therefore has infinite precision.

**Express the logarithm of 6.213 x 10 ^{-10} to the correct number of significant figures.**

Recalling that the log(AB) = log(A) + log(B) we can write:

log(6.21__3__ x 10^{-10}) = log(6.21__3__)+log(10^{-10}) = 0.793__3__013536 + (-10) = -9.206__6__986464 → -9.206__7__

You should only round off numbers when reporting your final result. Do not round off numbers in the middle of a calculation.