Significant figures

Significant Figures in Chemistry and Physics

In scientific measurements, precision and accuracy are crucial. Significant figures (also known as significant digits) represent all the known digits in a measurement, plus one estimated digit. They reflect the precision of a measuring instrument and help maintain consistency in calculations and reporting.

Definition of Significant Figures

Significant figures are the digits in a number that carry meaningful contributions to its precision. This includes all non-zero digits, zeros between significant digits, and trailing zeros in the decimal part.

Importance of Significant Figures

• They indicate the reliability of measurements.
• They help in avoiding overstatement of precision.
• They are important for rounding off calculated results in physics and chemistry.

Rules for Determining Significant Figures

1. All non-zero digits are significant.
   Example: 123.45 has 5 significant figures.
2. Any zeros between two significant digits are also significant.
   Example: 1003 has 4 significant figures.
3. Leading zeros are not significant.
   Example: 0.0056 has 2 significant figures.
4. Trailing zeros in a number with a decimal point are significant.
   Example: 50.00 has 4 significant figures.
5. Trailing zeros in a whole number without a decimal point are not significant (unless specified).
   Example: 1500 has 2 significant figures, but 1500. has 4.

Examples

\begin{aligned} 1234 &\Rightarrow \text{4 significant figures} \\ 0.00450 &\Rightarrow \text{3 significant figures} \\ 3.00 &\Rightarrow \text{3 significant figures} \\ 1200 &\Rightarrow \text{2 significant figures (if no decimal)} \\ 1200.0 &\Rightarrow \text{5 significant figures} \end{aligned}

Significant Figures in Calculations

1. Multiplication and Division:
The result should be reported with the same number of significant figures as the measurement with the fewest significant figures.

Example:
4.56 \times 1.4 = 6.384 \Rightarrow 6.4 \ (\text{2 significant figures})

2. Addition and Subtraction:
The result should have the same number of decimal places as the number with the least decimal places.

Example:
\[ 12.11 + 18.0 = 30.11 \Rightarrow 30.1 \ (\text{1 decimal place}) \]

Scientific Notation and Significant Figures

Scientific notation is often used to express very large or small numbers. It clearly shows the significant figures.

Example:
\[ 3.00 \times 10^8 \ \text{(3 significant figures)} \\ 1.2 \times 10^{-4} \ \text{(2 significant figures)} \]

Rounding Off Significant Figures

When rounding to the correct number of significant figures:

• If the next digit is less than 5, round down.
• If the next digit is 5 or more, round up.

Example:
\[ \begin{aligned} 4.367 \Rightarrow 4.37 \ (\text{to 3 sig. figs.}) \\ 7.845 \Rightarrow 7.85 \ (\text{to 3 sig. figs.}) \\ 2.449 \Rightarrow 2.45 \ (\text{to 3 sig. figs.}) \end{aligned} \]

Exact Numbers

Exact numbers have an infinite number of significant figures. They arise from counting (e.g., 20 students) or defined quantities (e.g., 1 inch = 2.54 cm exactly). These numbers do not limit the number of significant figures in a calculation.

Tips for Using Significant Figures

• Always consider the measuring device’s precision.
• Use scientific notation to avoid confusion in large or small numbers.
• Be consistent when performing multi-step calculations.

Practice Questions

1. How many significant figures are there in:

1. 0.00340
2. $6.022 \times 10^{23}$
3. 150

2. Round the following to 3 significant figures:

1. 0.045678
2. 123456
3. 9.995

Conclusion

Significant figures are fundamental in reporting scientific data with accuracy and honesty. They prevent overstating the precision of measurements and ensure that calculations stay within the bounds of the measuring tools’ limitations.