Overview
This lecture covers significant figures (sig figs): how to count them, rules for rounding, application in addition, subtraction, multiplication, division, and scientific notation, including complex and combined operations.
Counting Significant Figures
- All nonzero digits are significant; e.g., 126 has three sig figs.
- Zeros between nonzero digits are significant (e.g., 404 has three sig figs).
- Trailing zeros are only significant if there is a decimal point (e.g., 8000. has four sig figs).
- Leading zeros (zeros before the first nonzero digit) are not significant.
- In scientific notation, only the digits in the coefficient are counted as significant figures.
Rounding to Significant Figures
- To round, keep the required sig figs and replace the rest with zeros (or adjust the decimal point accordingly).
- Use the next digit to decide whether to round up (≥5) or down (<5).
- For very large or small numbers, use scientific notation to control the number of sig figs when needed.
Addition and Subtraction Rules
- Align decimal points for addition/subtraction.
- The answer should have the same decimal place as the least precise measurement (rightmost uncertain digit).
- Draw a line after the rightmost digit of the least precise number and round accordingly.
Multiplication and Division Rules
- The result should have as many sig figs as the measurement with the fewest sig figs.
- Only the number of sig figs matters, not the decimal place.
Combined Operations
- Calculate stepwise, carry enough digits through intermediate steps, and only round at the end.
- Apply addition/subtraction rules first, then multiplication/division rules to the final result.
Scientific Notation
- A positive exponent shifts decimal right (large numbers), negative exponent shifts left (small numbers).
- To convert to scientific notation, move the decimal to create a single nonzero digit to the left; exponent shows shifts.
- When adding/subtracting in scientific notation, exponents must match.
- When multiplying/dividing, multiply/divide coefficients and add/subtract exponents.
Special Cases
- Exact numbers (e.g., counting objects, defined constants) have infinite sig figs and do not affect sig fig limits in calculations.
Key Terms & Definitions
- Significant Figures (Sig Figs) — Digits that carry meaning in a measured value.
- Leading Zeros — Zeros before the first nonzero digit, not significant.
- Trailing Zeros — Zeros after the last nonzero digit; significant if there is a decimal point.
- Scientific Notation — Expression of numbers as a coefficient times a power of ten, e.g., 4.2 × 10³.
- Uncertain Digit — The last significant digit, subject to rounding.
Action Items / Next Steps
- Practice identifying and rounding significant figures in sample numbers.
- Complete additional exercises converting between standard and scientific notation.
- Review class examples on addition, subtraction, multiplication, and division with sig figs.