Significant Figures in Calculations and Measurements
Understanding Significant Figures
Numbers and Measurements: Reflect our knowledge about a physical object or entity in the real world.
Uncertainty in Measurements: All measurements, whether taken by an instrument or manually, have a degree of uncertainty. This depends on the device used.
Estimated Digit: The last recorded digit is usually an estimated digit, marking the point where certainty ends and estimation begins.
Representing Measurements
Different instruments provide different levels of certainty.
Example: Two balances may provide different degrees of certainty for the same object.
Simple balance: certainty runs out in the tenths place.
Precise balance: certainty goes down to the thousandths place.
Significant figures give us an idea of the level of uncertainty in a measurement.
Rules for Counting Significant Figures
Non-Zero Digits: Always significant.
Example: 213.5 has 4 significant figures.
Leading Zeros: Not significant.
Example: 0.0023 has 2 significant figures (2 and 3).
Trailing Zeros: Significant if they are to the right of the last non-zero digit and to the right of the decimal place.
Example: 19.3400 has 6 significant figures.
Captured Zeros: Zeros between two significant figures are significant.
Example: 13.6009 has 6 significant figures.
Ambiguous Zeros: Can be confusing; use scientific notation to clarify.
Example: 24,000 can be written variously in scientific notation to show different significant digits.
Exact Numbers: Have an infinite number of significant figures (e.g., definitions and counting numbers like 12 inches = 1 foot, 14 sheets of paper).
Calculations Involving Significant Figures
Multiplication and Division: The number of significant figures in the result is limited by the measurement with the smallest number of significant figures.