Chemistry and Measurements: Measured Numbers and Significant Figures

Chemistry and Measurements

Introduction to Measured Numbers

In chemistry, measurements are fundamental for quantifying physical properties such as length, mass, and temperature. Understanding how to record and interpret measured numbers is essential for accuracy and reliability in scientific work.

• Measured numbers are obtained when you measure a quantity, such as height, weight, or temperature.

• To write a measured number:

  • Observe the numerical values of the marked lines on the measuring instrument.

  • Estimate the value between the marks; this estimated digit is the final digit in your measurement and reflects uncertainty.

• Example: Measuring the length of an object using a ruler, you might record 4.5 cm, where '5' is the estimated digit.

Measured Numbers for Length

When measuring length, the value recorded includes all certain digits plus one estimated digit.

• Example measurements:

  • 4.5 cm

  • 2.55 cm

  • 3.0 cm

• Each measurement includes an estimated digit, which is the last digit recorded.

Significant Figures

Definition and Importance

Significant figures (SFs) in a measured number are all the digits known with certainty plus one estimated digit. They are used to represent the amount of error associated with a measurement.

• All nonzero digits are significant.

• Zeros between nonzero digits are significant.

• Zeros at the end of a decimal number are significant.

• Zeros that act as placeholders before digits are not significant.

Rules for Identifying Significant Figures

The following rules help determine which digits in a measured number are significant:

Rule	Measured Number	Number of Significant Figures
Nonzero digit	4.5 g	2
Zero between nonzero digits	205 °C	3
Zero at the end of a decimal number	50. L	2
Zero at the end of a decimal number	16.00 mL	4
Coefficient of a number written in scientific notation	4.8 × 104 m	2
Coefficient of a number written in scientific notation	5.70 × 103 g	3

Zeros That Are Not Significant

Some zeros in a measured number are not significant, depending on their position:

Rule	Measured Number	Number of Significant Figures
Zero at the beginning of a decimal number	0.0004 s	1
Zero at the beginning of a decimal number	0.075 cm	2
Zero as a placeholder in a large number without a decimal point	850 000 m	2
Zero as a placeholder in a large number without a decimal point	1 250 000 g	3

Significant Zeros and Scientific Notation

Scientific notation is used to clearly indicate which zeros are significant in large or small numbers.

• In scientific notation, only the digits in the coefficient are significant.

• Example: 3.0 × 102 m has two significant figures.

• Example: 4 × 106 g has one significant figure.

• Example: 4.6 × 10-3 g has two significant figures.

Exact Numbers

Definition and Examples

Exact numbers are numbers that are not measured and do not have a limited number of significant figures. They are typically obtained by counting or by definition.

• Numbers obtained by counting (e.g., 8 cookies, 6 eggs).

• Numbers from definitions in the same measuring system (e.g., 1 min = 60 s, 1 kg = 1000 g).

• Numbers from definitions that compare two units (e.g., 1 qt = 4 cups).

Exact numbers are not used to determine the number of significant figures in a calculated answer.

Summary Table: Measured vs. Exact Numbers

Type	Examples	Significant Figures
Measured Number	7.902 cm (diameter of a circle)	4
Exact Number	8 coins, 1 min = 60 s	Unlimited

Practice Problems

Identifying Significant Figures and Scientific Notation

• Identify the significant and nonsignificant zeros in each of the following numbers, and write them in correct scientific notation:

  • 0.002650 m

  • 3.026 g

  • 44,000 L

Classifying Numbers as Measured or Exact

• Identify the numbers below as measured or exact, and state the number of significant figures in each measured number:

  • 8 coins

  • Diameter of a circle is 7.902 cm

  • 1 min = 1 h

Key Equations

• Scientific Notation: $a \times 10^n$ where $a$ is the coefficient and $n$ is the exponent.

Additional info: Scientific notation is especially useful for expressing very large or very small numbers in chemistry, making it easier to identify significant figures and perform calculations.