Notes on Significant Figures, Dimensional Analysis, and SI Prefixes
Significant Figures, Rounding Rules, and Dimensional Analysis
Overview from the transcript
Significance of reporting measurements correctly: If you report the wrong number of significant figures, the instructor may line through the page and annotate it as a problem, indicating the issue is with reporting, not necessarily the math.
There are rules for how many significant figures to report depending on the operation (addition/subtraction vs. multiplication/division) and on how the measurements were made (which have the fewest decimal places or the fewest significant figures).
The session covers: what counts as significant digits, how to handle addition/subtraction, how to handle multiplication/division, scientific notation, exact numbers, and dimensional analysis with SI prefixes.
What counts as a significant figure
Non-zero digits are always significant.
Zeros between nonzero digits are significant.
Leading zeros (zeros to the left of the first nonzero digit) are not significant.
Trailing zeros:
If a decimal point is present, trailing zeros are significant (e.g., has four sig figs; has five).
If no decimal point is present, trailing zeros are ambiguous; often only one significant figure (the nonzero digit) is assumed unless otherwise indicated.
In decimal notation and in scientific notation, the digits that you write relative to the decimal point indicate sig figs.
Scientific notation: the digits in the mantissa are all significant. Example: has 4 significant figures.
Exact numbers (counted values or defined quantities) have infinite significant figures (no rounding needed for sig fig purposes).
Examples discussed in class
801.5 → 4 significant figures.
801.5 vs. 801.50 → depends on decimal places; extra trailing zero after decimal increases sig figs.
100 (no decimal) → typically 1 sig fig; 100. (with a decimal) → 3 sig figs.
A zero between nonzero digits is significant (e.g., the zero in 1607 is significant, giving 4 sig figs).
0.04 has 1 sig fig (the 4), since leading zeros are not significant.
Exact numbers (like counting items or defined constants) have unlimited sig figs.
Significance in practice: two main rules to apply depending on the operation
Addition and subtraction (report to the least precise decimal place)
You look at decimal places for each measurement; report the result with the fewest decimal places among the operands.
If you have measurements with 3, 2, and 1 decimal places, you must round the result to 1 decimal place.
The decimal places determine round-off, not the magnitude of the numbers themselves.
The rule applies regardless of the numbers involved; the limiting factor is the least precise decimal placement.
Rounding: after summing, examine the first digit beyond the specified decimal place; if it is 5 or greater, round up the last reported digit.
Multiplication and division (report to the fewest significant figures among the factors)
The result should have as many significant figures as the factor with the smallest number of sig figs.
Example idea described: if you multiply/divide numbers where the factors have, say, 4, 3, and 3 sig figs, the result should have 3 sig figs.
Rounding is performed to preserve that fewest sig figs in the final answer.
The instructor emphasized that the number of sig figs in the result is determined by the limiting factor, not by the decimal places alone.
Scientific notation and units
In scientific notation, the digits in the mantissa determine significant figures; the exponent does not affect sig figs.
Angstrom units were introduced as an example: the Angstrom symbol Å; 1 Å = 10^{-10} m.
Dimensional analysis and SI prefixes (base units and prefixes)
The SI system uses prefixes to indicate powers of ten on base units (e.g., meter, gram, second).
Common prefixes (left to right, larger to smaller):
kilo (k): 10^3
mega (M): 10^6
giga (G): 10^9
nano (n): 10^{-9}
micro (µ): 10^{-6}
milli (m): 10^{-3}
centi (c): 10^{-2}
The speaker illustrated the idea of moving decimal places by the prefix exponent when converting between units. For example, converting from mega (10^6) meters to millimeters (10^{-3} m):
1 \, ext{Mm} = 10^{6} \, ext{m}
1 \, ext{m} = 1000 \, ext{mm} = 10^{3} \, ext{mm}
Therefore, 1 \, ext{Mm} = 10^{6} imes 10^{3} \, ext{mm} = 10^{9} \, ext{mm}
In a concrete example: 536 \, ext{Mm} = 536 imes 10^{9} \, ext{mm} = 5.36 imes 10^{11} \, ext{mm}
Another example discussed: moving from nano to kilo involves moving across 12 exponent steps (from 10^{-9} to 10^{3}, a difference of 12), which corresponds to multiplying/dividing by 10^{12} as appropriate.
Dimensional analysis approach (units first): construct a conversion factor that cancels the current unit and leaves the desired unit.
Example: Convert 536 \, ext{mm} to \, ext{m}. Use the factor: (1 \, ext{m} / 1000 \, ext{mm}). Then: 536 \, ext{mm} × (1 \, ext{m} / 1000 \, ext{mm}) = 0.536 \, ext{m}.
The non-base units appear in the denominator or numerator to cancel appropriately; the “1” in the factor is used to balance units.
A common explicit relation: 1 \, ext{cm}^3 = 1 \, ext{mL}.
The base unit is the unit you’re converting to; the other unit is connected through a prefix; the factor that is not the base unit is the one you use to connect to the base.
The “not base” unit often becomes a multiplier (or divisor) with a numeric factor equal to the inverse of the prefix power when you cancel units.
Practical application: example walkthroughs described
Example: Convert mega-meters (Mm) to millimeters (mm)
1 \, ext{Mm} = 10^6 \, ext{m}
1 \, ext{m} = 1000 \, ext{mm} = 10^3 \, ext{mm}
Thus, 1 \, ext{Mm} = 10^{6} imes 10^{3} \, ext{mm} = 10^{9} \, ext{mm}
Therefore, 536 \, ext{Mm} = 5.36 imes 10^{11} \, ext{mm}
Example: nano to kilo (scaling across many prefixes)
Difference in prefix exponent: from -9 (nano) to +3 (kilo) is 12 steps
Moving the decimal accordingly, or multiplying by 10^{12} in the appropriate direction to obtain the desired unit scale
Example: hours, minutes, seconds
1 hour = 60 minutes; 1 minute = 60 seconds; 1 day = 24 hours
Therefore, 1 day = 24 \, ext{h} = 24 \, ext{h} imes 60 \, ext{min/h} imes 60 \, ext{s/min} = 86400 \, ext{s}
If you combine these in a calculation, you must consider sig figs: if the initial quantities are limited in sig figs, the result should reflect the fewest sig figs (e.g., if you start with a value having 3 sig figs, report the final answer with 3 sig figs, not more).
Practice notes: the worksheet includes three addition/subtraction problems (to apply decimal-place rules) followed by three multiplication/division problems (to apply sig figs rules).
Common points of confusion and clarifications from the transcript
What is your "base" unit in a conversion? The base unit is the unit you want to end up with; other units are connected via prefixes.
The statement about the element that is not your base being “1” refers to the conversion factors: you place a factor of 1 in the appropriate position to balance units, e.g., 1 \, ext{m} = 1000 \, ext{mm} so that mm cancels when converting to meters.
Angstroms and metric vs. English (imperial) units were touched on; the emphasis is on consistency in using SI prefixes for dimension analysis and conversions.
Quick reference rules (summary)
Significant figures:
Nonzero digits are significant.
Zeros between significant digits are significant.
Leading zeros are not significant.
Trailing zeros: significant if a decimal point is present; otherwise ambiguous.
Exact numbers have unlimited sig figs.
Addition/Subtraction: report to the least precise decimal place among the operands.
Multiplication/Division: report to the same number of significant figures as the factor with the fewest sig figs.
Dimensional analysis: cancel units using appropriate conversion factors; use prefixes to move between scales; 1 cm³ = 1 mL.
SI prefixes: kilo (10^3), mega (10^6), giga (10^9), nano (10^-9), micro (10^-6), milli (10^-3), centi (10^-2); Angstrom (Å) = 10^-10 m.
Examples to remember: 1 Mm = 10^9 mm; 1 day = 86400 s; 1 cm^3 = 1 mL.
Notes for the upcoming worksheet and study strategy
For significant figures problems, identify the operation type first (add/subtract vs. multiply/divide) and apply the corresponding rule before performing rounding.
For dimensional analysis, practice converting between common prefixes (mm, cm, m, km, μm, nm, pm, etc.) and between time units (s, min, h, day) to build fluency with unit cancellation.
Be comfortable with the Angstrom and SI prefix chart as a quick reference during problems.
When in doubt, write out the units fully first (e.g., meters, seconds) and only then simplify to the final unit to ensure correct cancellation and correct rounding.
Standalone reminders from lecture style
If you get a unit mismatch or a decimal place error, the instructor will correct the reporting, not necessarily the math; re-check the decimal places and sig figs, then adjust.
Exact numbers are not subject to rounding; treat them as having infinite sig figs for the purpose of the calculation.
When converting from one unit to another, think in terms of power-of-ten shifts (the exponents of ten) to understand how many places you must move the decimal point.