Overview
This lecture covers significant figures and scientific notation, focusing on definitions, rules for identifying significant digits, and how to apply these concepts in calculations.
Types of Numbers
- Exact numbers come from counting or definitions (e.g., 10 fingers, 100 cm in 1 m).
- Inexact numbers result from measurements and contain some uncertainty.
- In measurement, the last digit reported is an uncertain or estimated digit.
Significant Figures (Sig Figs)
- Significant figures represent all known digits plus one estimated digit in a measurement.
- All nonzero digits are always significant.
- Zeros between nonzero digits ("sandwiched zeros") are always significant.
- Zeros after a nonzero digit and after a decimal point are significant.
- Zeros at the beginning of a number or at the end without a decimal are not significant (they are placeholders).
Counting Significant Figures: Rules for Zeros
- "Sandwiched" zeros between nonzero digits count as significant.
- Trailing zeros with a decimal point are significant (e.g., 620. has three sig figs).
- Trailing zeros without a decimal are not significant (e.g., 620 has two sig figs).
- Leading zeros (in front of the first nonzero digit) are never significant.
Rounding Rules & Math Operations
- For multiplication/division: Final answer should have as many sig figs as the measurement with the fewest sig figs.
- For addition/subtraction: Final answer should match the least precise decimal place (not number of sig figs).
- When rounding, if the dropped digit is 5 or greater, round the previous digit up; if 4 or lower, leave it.
Scientific Notation
- Scientific notation expresses numbers as a coefficient (1 ≤ x < 10) times 10 to a power.
- Positive exponents indicate values greater than one; negative exponents indicate values less than one.
- The exponent shows how many places the decimal is moved to create the coefficient.
- Scientific notation clarifies significant figures and is useful for very large or small numbers.
Key Terms & Definitions
- Significant Figures (Sig Figs/SF) — all known digits in a measurement plus the last, estimated digit.
- Exact Number — a value from counting or a definition; no uncertainty.
- Inexact Number — a measured value containing some uncertainty.
- Scientific Notation — a method to express numbers as a product of a number (1 to <10) and a power of ten.
- Placeholder Zero — zeros used only to position the decimal point, not considered significant.
Action Items / Next Steps
- Review place value concepts (ones, tenths, hundredths, etc.).
- Practice identifying significant figures and converting to/from scientific notation.
- Prepare for in-class exercises on these topics.
- Contact instructor for additional explanations or resources if needed.