Scientific notation is often introduced as a way to write very large and very small numbers compactly. Its second, less celebrated job is just as important: it shows exactly how precise a measurement is. Writing 4.50 × 10³ says something different from 4.5 × 10³, and in lab reports, exams and engineering documents that difference matters. This guide explains how significant figures work in scientific notation and how to keep them right through a calculation.
What significant figures are
Significant figures (sig figs) are the digits in a number that carry meaningful information about its precision. The usual rules are:
- All non-zero digits are significant. 382 has three.
- Zeros between non-zero digits are significant. 4,007 has four.
- Leading zeros are never significant. 0.0052 has two; the zeros only locate the decimal point.
- Trailing zeros after a decimal point are significant. 2.300 has four.
- Trailing zeros in a whole number are ambiguous. Does 1,500 have two, three or four significant figures? Without more information, you can't tell.
That fifth rule is where scientific notation comes to the rescue.
Removing the ambiguity
In scientific notation, every digit in the coefficient (the number before the × 10ⁿ) is significant. So you can say exactly what you mean:
- 1.5 × 10³ has two significant figures
- 1.50 × 10³ has three
- 1.500 × 10³ has four
All three equal 1,500, but they describe measurements of different quality. A distance "1.5 × 10³ m" might be a rough pacing; "1.500 × 10³ m" suggests a surveyed measurement accurate to the metre.
Small numbers work the same way: 0.000470 becomes 4.70 × 10⁻⁴, and the trailing zero stays because it was measured.
Converting while keeping precision
To convert a number to scientific notation without losing or inventing precision:
- Count the significant figures in the original number.
- Move the decimal point so one non-zero digit is to its left.
- Keep exactly the significant digits in the coefficient.
- Set the exponent to the number of places you moved the point (positive if you moved it left, negative if you moved it right).
Example: 0.0030600 has five significant figures (3, 0, 6, 0, 0). Moving the point three places right gives 3.0600 × 10⁻³. Writing 3.06 × 10⁻³ would throw away two measured digits.
Rules for calculations
Multiplication and division: the answer keeps as many significant figures as the input with the fewest significant figures.
(3.2 × 10⁴) × (2.115 × 10⁻²) = 676.8, which must be rounded to two significant figures: 6.8 × 10².
Addition and subtraction: the answer is limited by the decimal place of the least precise value, not the count of significant figures. It's easiest to rewrite the numbers with the same exponent first:
4.52 × 10³ + 3.1 × 10² = 4.52 × 10³ + 0.31 × 10³ = 4.83 × 10³
Here 0.31 × 10³ is known to the hundredths place in the coefficient, the same as 4.52, so the result keeps three significant figures.
Exact numbers such as counted objects (12 test tubes) or defined constants (100 cm in a metre) have unlimited significant figures and never limit your answer.
Rounding correctly
Round only at the end of a calculation. If you round after every step, errors accumulate. Keep one or two extra digits in intermediate results, then round the final answer. When the digit after your last kept digit is 5 or more, round up; otherwise round down. (Some fields use "round half to even" to avoid bias, so follow the convention your course or lab uses.)
Common mistakes
- Dropping trailing zeros that were measured. A balance reading 2.50 g gives three significant figures; don't write 2.5 g.
- Copying every digit from a calculator. A display showing 6.7682 × 10² isn't more accurate than your data.
- Applying the multiplication rule to addition. Addition cares about decimal places, not counts.
- Treating exponents as significant. Only the coefficient's digits count; the power of ten just sets the scale.
Checking your work
Once you've done the working by hand, it's worth checking your counts with a sig figs calculator, especially for numbers with confusing zeros. Use it to confirm your own reasoning rather than replace it, because exams expect you to apply the rules yourself.
Summary
Scientific notation isn't just shorthand; it's a precision statement. Keep every measured digit in the coefficient, use the fewest-sig-figs rule for multiplying and dividing, the decimal-place rule for adding and subtracting, and round once, at the end.
