Significant Figures

Significant figures report how precise a measured value is. They include every digit known with confidence, plus one last estimated digit from the measurement. That is why $12.3\ \mathrm{cm}$ and $12.30\ \mathrm{cm}$ communicate different precision even though they are equal as decimals: the second reports to a smaller place value. Significant figures are about how carefully a number was measured, not how large it is.

When significant-figure rules apply

You use these rules whenever numbers come from measurement rather than exact counting. Exact values, like $12$ objects in a box or a defined conversion factor, do not limit the precision of a result; the rules govern measured values only. Common settings: lab data in chemistry and physics; measured lengths, masses, times, and temperatures; engineering calculations where reported precision matters; and scientific notation, where precision is easiest to show.

A few counts make the idea concrete:

• $45.7$ has $3$ significant figures.
• $0.0045$ has $2$, because leading zeros are not significant.
• $1002$ has $4$, because zeros between nonzero digits count.
• $3.400$ has $4$, because trailing zeros after a decimal show measured precision.

Most mistakes come from zeros, which count only when they show measured precision.

The procedure, step by step

1. Find the first meaningful digit. Start at the first nonzero digit, since leading zeros do not count.
2. Count the digits that show precision. Include zeros between nonzero digits and trailing zeros after a decimal point.
3. Match the operation to the rounding rule. Multiplication and division are limited by significant figures; addition and subtraction by decimal place.
4. Round only at the end. Keep extra digits during the calculation, then round the final answer once.

For whole numbers with trailing zeros, the count can be ambiguous: $1500$ alone does not say whether the last two zeros were measured. Scientific notation removes the doubt, since $1.5 \times 10^3$ has $2$ significant figures and $1.500 \times 10^3$ has $4$.

Full example: multiplying with significant figures

Multiply

Do the raw product first:

Now apply step 3. Here $4.56$ has $3$ significant figures and $1.4$ has $2$. For multiplication, the result takes the fewest significant figures of any factor, so $2$:

The reported answer is $6.4$. The core idea: an answer should not claim more precision than its measurements support.

Addition and subtraction follow the decimal-place rule instead. For example,

but $0.3$ is precise only to the tenths place, so the answer rounds to the tenths place:

Applying the multiplication rule here would round incorrectly. The operation sets the rule.

Where students get stuck, and how to self-check

"Do leading zeros count?" No. In $0.0028$ the zeros only place the decimal, so only $2$ and $8$ are significant: $2$ figures.

"Are all trailing zeros the same?" No. In $3.40$ the zero shows measured precision past the tenths place; in $3400$ the trailing zeros may or may not be significant without clearer notation.

"When do I round?" Only at the end. Rounding mid-calculation lets small errors accumulate, so keep extra digits and round once.

"One rule for everything?" No. Multiplication and division use fewest significant figures; addition and subtraction use the least precise decimal place. Mixing them is a top error.

A fast final check after any calculation: which input was least precise, and does my answer show more precision than that input supports? If yes, round again.

Practice this yourself

Count the significant figures in $0.04050$, then decide how many should remain in the product

For a sharper test, rewrite the same numbers in scientific notation, where the number of significant figures is usually easier to read directly.