Overview
This lecture explains how calculator results can be misleading when handling measurements and emphasizes the correct use of significant figure rules in calculations.
Calculator Limitations & Measurement Precision
- Calculators often display more decimal places than your measurements justify, leading to false precision.
- Measurement tools limit the number of significant figures (sig figs) you can claim in your results.
Significant Figures in Calculations
- For multiplication/division, the result must have the same number of significant figures as the least precise measurement used.
- Example: 2.4 cm × 2.4 cm = 5.76 cm² on the calculator, but only two sig figs are allowed: 5.8 cm².
- Always round your results to match the correct number of significant figures.
- Propagation of error occurs when rounded values are reused in further calculations.
Notation Techniques & Scientific Notation
- Insignificant digits carried for future calculations can be written as a subscript (e.g., 5.7₆).
- Use scientific notation to clarify significant digits, especially when zeros could be ambiguous.
- Example: 7212 × 1.4 = 1.0 × 10⁴ (2 sig figs).
Examples & Practice
- When measurements have ambiguous zeros without a decimal, count them as insignificant for multiplication/division.
- Use scientific notation for clarity: 409000 → 4.09 × 10⁵ (3 sig figs).
- Calculators can produce more digits than needed; always reduce to the correct sig figs.
- Multiplying or dividing numbers in scientific notation does not alter sig fig rules.
- Example division: 1.879e4 ÷ 1.82e8 = 1.0 × 10⁻⁴ (2 sig figs).
Algebraic Reminder for Exponents
- 10ˣ × 10ʸ = 10ˣ⁺ʸ.
- 10ˣ ÷ 10ʸ = 10ˣ⁻ʸ.
Key Terms & Definitions
- Significant Figures (sig figs) — Digits in a measurement that are known with certainty plus one estimated digit.
- Scientific Notation — A method of writing numbers as a product of a number (with significant digits only) and a power of ten.
- Propagation of Error — The effect of carrying rounded or estimated figures through subsequent calculations.
Action Items / Next Steps
- Review and apply significant figure rules to all calculator-based results.
- Watch videos on entering scientific notation in your calculator if unsure.
- Practice rounding calculator results to correct significant figures on homework problems.