Topic focus: significant figures (sig figs), decimal places, rounding rules, exact vs. measured numbers, and unit conversions (dimensional analysis / factor-labeling).
Emphasizes that in calculations you must track precision carefully, especially when mixing addition/subtraction with multiplication/division.
Recurrent themes: units are crucial (often more important than the number itself), how to handle ratios, and practice with real-world conversion problems.
Tools introduced: line under the last decimal place to show the decimal-precision limit, guard digits to reduce rounding errors, and the practice of only rounding at the very end of a multistep calculation.
Many worked examples across sig figs, dimensional analysis, and unit conversions to develop intuition for what numbers should look like and how to manipulate units.
Sig figs and decimal places: addition and subtraction
Tip: When adding/subtracting, first determine the number with the fewest decimal places among the terms; the result should be reported with that same number of decimal places.
Distinguish between decimal places (dp) and significant figures (sig figs); decimal places count the digits to the right of the decimal point, while sig figs count all meaningful digits.
Example logic (paraphrased from class):
If a calculation yields a result with more decimal places than allowed by the term with the fewest decimal places, you round to that fewest dp.
Put a line underneath the last decimal place to indicate the precision limit in the result.
Analogy used: weakest link determines the overall strength; here, the number with the fewest decimal places determines the precision of the sum.
Important nuance: after you determine the decimal places, you then determine the number of sig figs in the answer.
Sig figs and multiplication/division
For multiplication/division, decimal places do not govern the precision directly. Instead, the number of sig figs controls the precision of the result.
The “weakest link” in a product or quotient is the factor with the fewest sig figs; the final result should have the same number of sig figs as that weakest factor.
Example workflow (as described): if you have two numbers, one with 4 sig figs and the other with 3 sig figs, the product/division should have 3 sig figs.
Practice lines under the last sig fig can be used to indicate the precision region during intermediate steps, but you should avoid rounding until the very end.
Rounding rules and practice with guard digits
Rounding rule (common in chemistry textbooks): examine the first non-significant figure to decide whether to round up or down.
If that digit is 4 or less, round down; if 5 or higher, round up.
Do not round at intermediate steps in a multi-step calculation; round only once at the very end.
Guard digits: to minimize rounding error, keep two non-sig figs (guard digits) during intermediate steps, then round only at the end.
The “two guard digits” method helps prevent error propagation when chaining several operations; after the final step, you round to the appropriate number of sig figs.
Common error to avoid: rounding twice in a sequence of steps (e.g., rounding to 3 sig figs, then again to 2 sig figs before finishing).
Example narrative: show a multi-step calculation with a calculator display and then apply a single final rounding to the correct sig figs at the