Significant Figures

Use of significant figures ensures that values are reported accurately based on how precise the measured values are.

Accuracy vs. Precision

Accuracy Describes how close the measured value is to the true value Precision Refers to the degree of exactness with which a measurement is made and stated Is the shape 4cm long or 4.2cm or 4.25cm? When using an instrument such as a meter stick, you must estimate one additional value beyond what the instrument shows you A typical ruler shows measurements to the nearest tenth (mm), thus you must estimate the hundredths place (usually as 0 or 5)

Significant Figures

• A calculation cannot be more precise than the least precise measurement used to find the answer
  • Let’s say you counted the leaves on three trees.
  • One tree had 4,115 leaves.
  • The second tree had 4,986 leaves.
  • When you counted the third tree, you got bored and estimated 3,000 leaves.
  • If you calculate the average, it isn’t going to be very precise because one value (3,000) was not precise.
  • You wouldn’t report the average as 4033.6666667. It wouldn’t make sense to be so precise when one measurement wasn’t precise.
  • You would have to report the average as 4,000 leaves.
• Significant figures help us to keep our answers appropriately precise for all measurements and calculations

Significant Figure Rules

• All digits 1 through 9 are significant
  • 54.238 → has five significant figures
  • Measured values including the one estimated value (such as the hundredths place on a meter stick) are significant
  • Zeroes are significant if they are not just place holders
    • Zeroes in between any two significant numbers ARE significant
      • 904 → has three significant figures
    • Zeroes in front of non-zero digits are NOT significant
      • 0.0854 → has only three significant figures
    • Zeroes that are at the END of a number AND AFTER a decimal ARE significant
      • 52.00 → has four sig. figs. 234.50 → has five sig. figs.
    • Zeroes that are at the END of a number AND BEFORE a decimal are ONLY significant if they are the first estimated digit (i.e.: from a meter stick) or if they have been measured.
      • From a meter stick (thus, measured): 70. cm → has two sig. figs.
      • 3,000 → has only one sig. fig.

Operations With Significant Figures

Addition and subtraction have different rules than multiplication and division

Addition and Subtraction

• Keep the same number of decimal places as the least precise value being added
• 23.6 + 3.59 → 27.19
• “23.6” has ONE decimal place, while “3.59” has two decimal places
• 27.19 must therefore be rounded to 27.2 which also has only ONE decimal place

Multiplication and Division

• The final answer must have the same number of significant figures as the least precise value being factored in
• 12.78 * 0.042 = 0.53676
• “12.78” has 4 sig figs, but “0.042” only has 2 sig figs
• 0.53676 must be rounded to 2 sig figs → 0.54