Significant figures and rounding rules: Question bank

Every trap below targets one of the four ideas the topic is built on:

Which zeros are real (measured) vs which are placeholders (just locate the decimal).
Relative vs absolute uncertainty — the reason multiply/divide and add/subtract use different rules.
Exact numbers never limiting precision.
Honest rounding, including the half-to-even ("banker's") rule.

True or false — justify

True or false:

0.0042

and

0.00420

have the same number of significant figures.

False.

0.0042

has 2 sig figs; the extra trailing

0

0.00420

is after the decimal and deliberately written, so it counts — giving 3. The leading zeros are placeholders in both.

True or false: writing more digits always makes a measurement more accurate.

False. Extra digits you didn't measure are fake precision — a lie about how sure you are. Accuracy is about being close to the true value (see Accuracy vs precision); padding zeros changes neither accuracy nor precision.

True or false: the number

1500

definitely has 4 significant figures.

False. Trailing zeros in a whole number with no decimal are ambiguous —

1500

could be 2, 3, or 4 sig figs. Only scientific notation settles it, e.g.

1.50 \times 1 0^{3}

is unambiguously 3.

True or false:

205

and

2.05

both have 3 significant figures.

True. The zero is sandwiched between non-zero digits in both — you can't reach the last digit without passing through it, so it is genuinely measured.

True or false: exact numbers like "12 in a dozen" have exactly 2 significant figures.

False. Counted or defined numbers are infinitely precise — there is no uncertainty in "exactly 12." They never cap the sig figs of a calculation.

True or false: in

2 π r

, the factor

2

limits the answer to 1 sig fig.

False. The

2

is exact (a definition of a diameter's relation to radius), so it has infinite sig figs. Only the measured

r

(and the precision you use for

π

) limits the result.

True or false: the same rounding rule (fewest sig figs) applies to both

\times

and

+

False. Multiply/divide → match the fewest significant figures; add/subtract → match the fewest decimal places. They differ because

\times / \div

stacks relative error while

+ / -

stacks absolute error.

True or false:

100. + 0.1 = 100.1

has fewer sig figs than

100.

alone.

False.

100.

has 3 sig figs; the sum

100.1

has 4. Addition is governed by decimal places, not sig figs — the answer keeps 1 decimal place (the fewest of the inputs), which happens to give 4 sig figs.

True or false: rounding after every step gives the same final answer as rounding once at the end.

False. Intermediate rounding injects small errors that snowball. Always carry guard digits and round only the final answer.

Spot the error

A student says: "

0.0500

has 4 sig figs because there are four digits after the point." Spot the error.

The two leading zeros (

0.0

) are placeholders and don't count. Only the

5

and the two trailing zeros are significant → 3 sig figs, cleaner as

5.00 \times 1 0^{- 2}

A student computes

3.24 \times 1 0^{2} + 5 = 3.29 \times 1 0^{2}

and reports 3 sig figs. Spot the error.

For addition you compare decimal places on the same scale.

324 + 5 = 329

, and both are integers (0 decimal places), so the answer

329

is fine — but the reasoning "3 sig figs from the sig-fig rule" is wrong; addition uses decimal places, and it lands at 3 sig figs only by coincidence.

A student rounds

2.45

to one decimal using banker's rounding and gets

2.4

. Spot the error.

Banker's rounding only applies when the dropped part is exactly

5

with nothing after. Here the dropped part is "

5

" but there's a nonzero digit context — wait: dropping just the final

5

from

2.45

leaves "

5

" as the decider with nothing after → nearest even →

2.4

. That is actually correct; the trap is thinking you must always round

5

up.

A student says

4.203/2.1 = 2.0014

so the density is

2.0014

g/mL. Spot the error.

They kept fake precision. Division caps sig figs at

min (4, 2) = 2

, set by the 2-sig-fig volume, so the honest answer is

2.0

g/mL. See Density and derived quantities.

A student writes "

1500

metres to 3 sig figs is

1500

." Spot the error.

1500

as written is ambiguous, so it can't be the 3-sig-fig form. Write

1.50 \times 1 0^{3}

m to make the 3-sig-fig promise unmistakable.

A student says

0.030

has only 1 sig fig "because zeros never count." Spot the error.

Leading zeros don't count, but the trailing zero after the decimal does. So

0.030

has 2 sig figs (

3

and the final

0

Why questions

Why do leading zeros never count, no matter how many you write?

They are placeholders that only fix where the decimal point sits.

0.0042 = 4.2 \times 1 0^{- 3}

is the same measurement; the zeros vanish in scientific notation because they carry no precision information.

Why does multiplication track significant figures but addition tracks decimal places?

Multiplying scales relative (percent) uncertainty, and sig figs are a relative measure of precision. Adding stacks absolute uncertainty in fixed units, and decimal places measure absolute position. See Uncertainty and error propagation.

Why do scientists prefer writing numbers in scientific notation?

Because in

N \times 1 0^{n}

, every digit of $N$ is significant — there are no ambiguous trailing zeros, so your precision promise is exact and unmistakable.

Why is "round half to even" (banker's rounding) fairer than "always round 5 up"?

Always rounding up pushes averages systematically upward, creating a bias. Sending half the borderline 5's up and half down keeps the mean rounding error near zero.

Why can't a calculated answer be more precise than its least-precise input?

Because uncertainty propagates — mixing a fuzzy number with a sharp one gives a fuzzy result, like adding clean water to mud. The weakest measurement caps how much you honestly know.

Why do exact numbers never limit a calculation's precision?

They carry zero uncertainty (counted or defined), so they contribute no fuzziness to propagate. Only measured values, which have uncertain last digits, can cap the result.

Edge cases

Edge case: how many sig figs does

0

(plain zero) have?

A bare

0

has no significant figures in the usual sense — it's a placeholder for "nothing measured." A measured zero must be written with a decimal, e.g.

0.0

(1 sig fig) or

0.00

(2), to declare its precision.

Edge case: round

0.9996

to 3 sig figs — what happens to the digit count?

The dropped

6

rounds the

9

up, cascading:

0.9996 \to 1.00

. The result is written

1.00

to keep 3 sig figs; dropping the trailing zeros would falsely lower the precision.

Edge case: does

1.0 \times 1 0^{0}

have the same sig figs as

1

No.

1.0 \times 1 0^{0}

explicitly has 2 sig figs (the

1

and the

0

), while a lone

1

has just 1. Scientific notation lets you state precision that a bare integer can't.

Edge case: subtract

10.01 - 10.00 = 0.01

. Why is "catastrophic cancellation" a sig-fig trap here?

Both inputs had 4 sig figs but the result

0.01

has only 1. Subtracting nearly equal numbers destroys significant figures — the leading digits cancel, leaving only the uncertain tail. This is a key case in Uncertainty and error propagation.

Edge case: is the leading zero in

0.5

significant?

No. It's purely a placeholder that makes the decimal point easy to see; only the

5

is significant, so

0.5

has 1 sig fig.

Edge case: how many sig figs in a conversion factor like

1 in = 2.54 cm

used in Units and dimensional analysis?

Infinite — this is a defined exact relationship, so

2.54

here does not limit precision, unlike a measured

2.54

cm which would have 3.

Recall One-line summaries to lock in

Zeros: lead = placeholder, sandwiched = real, trailing-after-decimal = real, trailing-whole = ambiguous. Which digit ::: bare "yes"/"no" is never a complete answer — always attach the reason (relative vs absolute error, placeholder vs measured, exact vs measured).