1.1.4 · D5Measurement, Vectors & Kinematics
Question bank — Significant figures — rules for operations
True or false — justify
True or false: and have the same number of significant figures.
True — both have 3. Leading zeros in are only placeholders, so the significant digits are ; the trailing zeros after the decimal are real information.
True or false: adding a very precise number to a very rough one improves the rough number's precision.
False — absolute uncertainties add, so the sum is at best as good as the worst decimal place; the precise term cannot rescue the tenths digit that the rough term never knew.
True or false: definitely has 4 significant figures.
False — trailing zeros with no decimal point are ambiguous. It could be 2, 3, or 4 sig figs; only Scientific notation like resolves it.
True or false: a result of a multiplication can legitimately have more sig figs than either factor.
False — the answer inherits the fewest sig figs among factors, because its relative error is dominated by the worst factor. It can never be more precise than its worst ingredient.
True or false: in the "" limits your answer to 1 sig fig.
False — the is an exact/defined number with infinite sig figs; only the measured (and , which you carry to enough digits) limit the result.
True or false: the number (with a written decimal point) has 3 sig figs.
True — the explicit decimal point signals that the trailing zeros are meaningful, so all three digits count.
True or false: rounding at every intermediate step gives the same final answer as rounding once at the end.
False — early rounding discards information and accumulates rounding error; you must carry guard digits and round only the final answer.
True or false: and carry the same precision.
False as written — the plain integer has ambiguous trailing zeros, while unambiguously states 3 sig figs. Scientific notation is what makes precision explicit.
Spot the error
, keep 3 sig figs → "". Where is the slip?
The addition rule uses decimal places, not sig figs. is known only to the tens place, so the sum is uncertain in the tens: the honest answer is (-ish), not a confident .
", and since has 4 sig figs, keep 4 → ." Find the mistake.
Wrong rule chosen. Addition matches the fewest decimal places; here has only 1 decimal place, so the answer is , regardless of anyone's sig-fig count.
" from ; keep the fewest decimal places ( has 3) → ." What went wrong?
This is division, so use sig figs, not decimal places. has 2 sig figs, so the answer is , not .
A student writes . Spot the reasoning error, if any.
The verdict is fine but check the why: a square root behaves like a power (a multiplicative operation), so it keeps the sig figs of its input — has 2 sig figs, so , not . Don't be fooled by the long digit tail into keeping 3.
", and has only 1 sig fig, so I've lost precision — my answer is bad." What is the conceptual error?
The verdict "answer is bad" over-reads the situation. The subtraction is correct to 1 decimal place; the drop in sig figs (from 4 down to 1) is real and is called loss of significance, but it's an honest consequence of subtracting near-equal numbers, not an arithmetic mistake.
" trials, mean ; since has 1 sig fig, round the mean to ." Find the flaw.
is a counting number with infinite sig figs, so it never limits precision. The mean's sig figs come only from the measured data, so stays as its measurements allow.
Why questions
Why does count sig figs while counts decimal places?
Because multiplication makes relative errors add (sig figs ≈ relative precision) while addition makes absolute errors add (decimal places ≈ absolute precision) — two different kinds of precision, two different bookkeeping.
Why do leading zeros never count as significant?
They only fix the decimal point's location (they are placeholders); carries the same information as , which plainly shows just 2 significant digits.
Why does banker's rounding (round-half-to-even) exist instead of always rounding up?
Always rounding up biases many results systematically upward; rounding halves to the nearest even digit sends about half up and half down, keeping large batches of data unbiased on average.
Why can subtracting two close numbers be dangerous even though each is very precise?
Their absolute errors are small individually, but the true digits cancel while the errors do not, so the relative error of the tiny difference balloons — this is catastrophic cancellation.
Why is writing for a ruler-and-eye measurement a "lie"?
It claims the length is known to a hundred-thousandth of a cm, but the measurer only guessed the last visible digit by eye. Extra trailing zeros assert knowledge that was never measured.
Why do sig-fig rules only approximate true error propagation?
They quantise precision into whole digits, so they can't express, say, "known to 0.3%"; for careful work you must return to the actual relative/absolute uncertainty formulas in Error propagation — relative vs absolute.
Why does an order-of-magnitude estimate often need only 1 sig fig?
Estimation asks only "what power of ten?" — the leading digit dominates the answer's size, so extra digits add cost without changing the conclusion, as explored in Orders of magnitude & estimation.
Edge cases
Case: (a measured zero to one decimal). Does it have significant figures?
Yes — a measured zero to 1 decimal place means "known to be zero within the tenths", which is genuine information; its precision is 1 decimal place, even though counting sig figs of a lone zero is a special/ambiguous case best handled by decimal places for additive contexts.
Case: multiplying a measured by an exact conversion factor like (m to mm). How many sig figs survive?
2 — the exact factor has infinite sig figs and cannot shrink the result, so keeps its 2 sig figs.
Case: . How many decimals should the answer keep?
For logs, the mantissa (digits after the decimal, here ) carries the sig figs of the input, while the integer part (the ) is just the exponent. With 2 sig figs in, keep 2 in the mantissa → .
Case: a value written as vs . Same physical quantity, different notes — do they mean the same?
No — claims 1 sig fig (known to the units), claims 4 (known to thousandths). Same number line position, very different precision claims.
Case: adding numbers in different units of magnitude, e.g. . What must you do first?
Align them to a common power of ten () before applying the decimal-place rule, because "fewest decimal places" is only meaningful once the numbers share the same place-value grid.
Case: the input exactly (a defined value, e.g. reference point). Does it limit precision?
No — a defined/exact zero has infinite precision like any exact number, so it never caps the sig figs of a result.
Connections
- Significant figures — rules for operations — the parent rules these traps stress-test.
- Error propagation — relative vs absolute — the rigorous source of every "why".
- Scientific notation — the cure for trailing-zero ambiguity.
- Orders of magnitude & estimation — where 1 sig fig is enough.
- Measurement & uncertainty — what a "reliable digit" actually is.