Significant Figures and Their Application in Calculations

Significant Figures

Definition and Importance

Significant figures are the digits in a measurement that are known with certainty plus one digit that is uncertain. They are crucial in scientific measurements because they communicate the precision of the measured value and help maintain accuracy in calculations.

• Significant digits include all measured digits and the last uncertain digit.

• They are used to indicate the reliability of a measurement.

Rules for Identifying Significant Figures

• Trailing zeros (zeros at the end of a number):

  • Significant if they are to the right of a decimal point (e.g., 30.00 has four significant figures).

  • May be ambiguous if there is no decimal point (e.g., 2500 may have two, three, or four significant figures depending on context).

• Leading zeros (zeros before the first nonzero digit):

  • Never significant; they only indicate the position of the decimal point (e.g., 0.00150 has three significant figures).

• Captive (embedded) zeros (zeros between nonzero digits):

  • Always significant (e.g., 205 has three significant figures).

Examples: Counting Significant Figures

• 30.00: Four significant figures (trailing zeros after decimal are significant).

• 0.00150: Three significant figures (leading zeros are not significant; trailing zero after decimal is significant).

• 2500: Ambiguous; could be two, three, or four significant figures depending on notation (e.g., scientific notation clarifies: $2.5 \times 10^3$ has two significant figures).

Significant Figures in Calculations

Rules for Mathematical Operations

• Multiplication and Division:

  • The result should have the same number of significant figures as the measurement with the fewest significant figures.

• Addition and Subtraction:

  • The result should have the same number of decimal places as the measurement with the fewest decimal places.

Practice Exercises

Below are sample calculations applying significant figure rules:

• 15.55 / 0.00350

  • 15.55 has four significant figures; 0.00350 has three significant figures.

  • Result should have three significant figures.

  • $\frac{15.55}{0.00350} = 4442.8571 \approx 4.44 \times 10^3$

• 43.74 - 32.6

  • 43.74 has two decimal places; 32.6 has one decimal place.

  • Result should have one decimal place.

  • $43.74 - 32.6 = 11.14 \approx 11.1$

• 2.650 / 0.25

  • 2.650 has four significant figures; 0.25 has two significant figures.

  • Result should have two significant figures.

  • $\frac{2.650}{0.25} = 10.6 \approx 11$

• 8.3 × 104 / 452

  • 8.3 × 104 has two significant figures; 452 has three significant figures.

  • Result should have two significant figures.

  • $\frac{8.3 \times 10^4}{452} = 183.628 \approx 1.8 \times 10^2$

Additional Practice Problems

• (93.4 + 7.521) / 3.543 × 10-2

  • Sum: 93.4 + 7.521 = 100.921 (rounded to one decimal place: 100.9)

  • Division: $\frac{100.9}{3.543 \times 10^{-2}} = 2850.5 \approx 2.85 \times 10^3$ (three significant figures)

• (7.956 - 7.813) / 5.667 × 10-2

  • Difference: 7.956 - 7.813 = 0.143 (rounded to three decimal places: 0.143)

  • Division: $\frac{0.143}{5.667 \times 10^{-2}} = 2.524 \approx 2.52$ (three significant figures)

• (8.991 - 8.261) × 225

  • Difference: 8.991 - 8.261 = 0.730 (rounded to three decimal places: 0.730)

  • Multiplication: $0.730 \times 225 = 164.25 \approx 164$ (three significant figures)

• (25.2 × 7.46) + 85.34

  • Multiplication: 25.2 × 7.46 = 188.0 (three significant figures)

  • Addition: 188.0 + 85.34 = 273.34 (rounded to one decimal place: 273.3)

• (5.132 × 16.73) - 81.62

  • Multiplication: 5.132 × 16.73 = 85.85 (rounded to four significant figures)

  • Subtraction: 85.85 - 81.62 = 4.23 (rounded to two decimal places)

Summary Table: Significant Figure Rules

Type of Zero	Significant?	Example
Leading zeros	No	0.0025 (2 significant figures)
Captive zeros	Yes	205 (3 significant figures)
Trailing zeros (after decimal)	Yes	30.00 (4 significant figures)
Trailing zeros (no decimal)	Ambiguous	2500 (could be 2, 3, or 4 significant figures)

Example: Application in Chemistry

• When reporting the result of a chemical analysis, the number of significant figures reflects the precision of the measurement and the reliability of the result.

• For instance, a mass measured as 2.50 g (three significant figures) should be used in calculations so that the final answer is reported with three significant figures.

Additional info: Scientific notation is often used to clarify the number of significant figures in a measurement, especially when zeros are present.