Significant Figures

The numbers of digits in a measurement about which we are reasonably sure are called significant figures. In another way, the digits (figures) in a given number which can be realized are called significant figures (s.f.). The concept of significant figures is important in scientific fields where accurate representation and data communication are crucial.

Some important rules for significant figures

Below are some important rules for indicating significant figures:

1. All non-zero digits are significant figures.
   For example, The numbers 123 and 6789 have significant figures of 3 and 4 respectively as all digits are non-zero.

Note: Significance of ZERO depending on their types

2. All zeros between two non-zero digits are significant figures.
   For example, 205, 88005, 170904 have significant figures of 3, 5, and 6 respectively. It is because the zeros are in between the non-zero numbers and are significant.
   
3. Zeros that come before non-zero integers are never significant.
   For example, The zeros in 098, 0.3, and 0.000000000389 are not significant because they are all in front of non-zero integers.
   
4. If the zeros come after non-zero integers and are not followed by a decimal point, the zeros are not significant.
   For example, The zeros in 1000 , 330500 are not significant because they are not followed by a decimal point.
   
5. If the zeros come after non-zero integers and are followed by a decimal point, the zeros are significant.
   For example, The zeros in 1000. , 9277.00, 8782.0900 are significant because they are after non-zero integers and are followed by a decimal point.
   
6. Power of 10 are not significant. For example: 2.30 × 10⁹, 5.624 × 10⁻⁷ have significant figures of 3 and 4 respectively.
   
7. Universal constants have infinite significant figures. For example: π = 22.7, c = 3 x 10^⁸ m/s have an infinite number of significant figures.

Addition and Subtraction in Significant figures

When adding or subtracting the numbers in significant figures case, the answer should have the same number of decimal places as the limiting term. The limiting term is the number with the least decimal places.

For example,

\begin{align*} 6&.22\\
53&.6\\
14&.311\\
+ 45&.09091\\
\hline
119&.22191
\end{align*}

In this case, the limiting term, 53.6, has 1 decimal place. So, the answer must be in 1 decimal place. After rounding the above sum value to 1 decimal place, we get 119.2 as our answer.

Let’s see another example of subtraction:

\begin{align*} 5365&.982\\
-210&.53109 \\
\hline
5155&.45091
\end{align*}

In this case, the limiting term, 5365.982, has 3 decimal places. So, the answer must be in 3 decimal places. When we round the above results to 3 decimal places, we will get 5155.451 as our final answer.

Note: When rounding the digits, if the digits next to the limiting decimal place is greater than or equal to 5, we have to add 1 to that number. For example, in the above answer, the number next to 0 is 9 which is greater than 5. So, we add 1 to the 0 and the answer is 5155.451, but not 5155.450.

Multiplication and division of significant figures

When multiplying and dividing, the answer should have the same number of significant figures as the limiting term. The limiting term is the number with the least number of significant figures.

503.29 \times 6.177 = 3108.82233

Here the term with the least number of significant figures is 6.177 which is 4 significant figures. So, the answer must be in 4 significant figures. On rounding the above answer to 4 significant figures, we will get 3109 as our answer.

Again remember here that, the digit with 4 significant figures in the above answer is 3108 and the number next to it is 8 which is greater than 5. So, when rounding up the number, we add 1 to the 8 and the answer therefore is 3109.

Let’s look at another example:

1000.1/243 = 4.11563786\overset{round}{\rightarrow}4.12

In this example, 243 is the term with the least number of significant figures which is 3. So, the final answer is in 3 significant figures.

Significant figures in conversions

When converting a number, the answer should have the same number of significant figures as the number started with.

For example, converting 52.4 inches to ft. (Note, the question has only 3 significant figures, so our answer should be in 3 S.F.)

52.4 \times \frac{1}{12} ft = 4.366666667 ft\xrightarrow{round}4.37ft