Scientific notation makes very large and very small measurements easier to read, while significant figures communicate how precisely those measurements are known. This guide explains how to convert between ordinary and scientific notation, count significant figures, choose the correct rounding rule for each operation, and check a final answer before using it in a lab report, homework problem, or scientific calculation.
Overview
Scientific notation expresses a number as a coefficient multiplied by a power of ten:
a × 10n
The coefficient, a, is usually at least 1 but less than 10. The exponent, n, records how far the decimal point moved. For example, 450,000 becomes 4.5 × 105, while 0.00045 becomes 4.5 × 10−4.
Significant figures, also called significant digits, are the digits that convey measurement precision. They include all certain digits and one estimated digit. A result with too many digits can suggest a level of precision that the original measurements do not support.
Scientific notation conversion table
| Ordinary notation | Scientific notation | Decimal movement |
|---|---|---|
| 72,000 | 7.2 × 104 | 4 places left |
| 3,060,000 | 3.06 × 106 | 6 places left |
| 0.0081 | 8.1 × 10−3 | 3 places right |
| 0.00000052 | 5.2 × 10−7 | 7 places right |
A positive exponent generally indicates a number greater than 1; a negative exponent indicates a number between 0 and 1. When converting back to ordinary notation, move the decimal right for a positive exponent and left for a negative exponent.
Rules for counting significant figures
- Nonzero digits are always significant. The number 582 has three significant figures.
- Zeros between nonzero digits are significant. Both zeros in 5.008 are significant.
- Leading zeros are not significant. In 0.0047, only 4 and 7 count.
- Trailing zeros after a decimal point are significant. The number 6.20 has three significant figures.
- Trailing zeros in a whole number can be ambiguous. Scientific notation removes that ambiguity: 4.0 × 103 has two significant figures, while 4 × 103 has one.
When the intended precision matters, write the value in scientific notation or state the number of significant figures explicitly.
How to estimate
A reliable significant figures calculation follows three stages: preserve the input precision, perform the operation without premature rounding, and round only the final result. A significant figures calculator or scientific notation converter can speed up the arithmetic, but it cannot decide whether an input was measured, exact, or copied with an ambiguous number of digits.
Multiplication and division
For multiplication and division, the final answer should have the same number of significant figures as the input with the fewest significant figures.
Example: (3.42 × 104) ÷ (1.2 × 102) = 2.85 × 102 before rounding. The first value has three significant figures and the second has two, so the reported result is 2.9 × 102.
Addition and subtraction
For addition and subtraction, round according to decimal places, not the total number of significant figures. The result should not extend beyond the least precise decimal place.
Example: 12.47 + 3.2 + 0.086 = 15.756. The least precise input is 3.2, which reaches only the tenths place. The reported result is 15.8.
Mixed operations
Use the correct rule at each stage, but avoid rounding intermediate values more than necessary. Keep extra digits in your calculator memory, then apply the final rounding rule to the completed calculation.
Example: For (8.25 − 1.4) ÷ 2.00, first subtract: 8.25 − 1.4 = 6.85, which is limited to the tenths place, so the exact unrounded working value should remain available if possible. Divide by 2.00, then report the final value according to the precision supported by the calculation. If a class, laboratory, or publication specifies a particular convention for chained operations, follow that convention consistently.
Rounding checklist
- Identify the place or digit where the answer must stop.
- Look at the next digit.
- If the next digit is less than 5, leave the retained digit unchanged.
- If the next digit is 5 or greater, increase the retained digit by one.
- Remove following digits and preserve trailing zeros when they communicate precision.
- Check that the result is still reasonable in size and units.
For example, rounding 0.006784 to three significant figures gives 0.00678, or 6.78 × 10−3. Rounding 9.996 to three significant figures gives 10.0, where the trailing zero shows that three significant figures are intended.
Inputs and assumptions
Before entering values into a scientific notation converter or significant figures calculator, check what each number represents.
- Measured values: Apply significant figure limits based on the recorded measurement.
- Exact counted values: A count such as 12 samples or 4 containers is normally treated as exact, so it does not limit the precision of a calculation in the same way a measurement does.
- Defined conversions: Some unit relationships are defined exactly, while measured conversion factors may carry uncertainty. Follow the convention required by the course, laboratory, or technical standard.
- Units: Convert units before combining quantities that must match. Do not use significant figures to hide a unit mismatch.
- Ambiguous zeros: Rewrite unclear whole numbers in scientific notation. For instance, 1.50 × 103 clearly indicates three significant figures.
Do not confuse decimal places with significant figures. The value 0.00450 has three significant figures but five digits after the decimal point. Conversely, 4500 may have two, three, or four significant figures unless its notation makes the intended precision clear.
For related reference work, a periodic table lookup guide can help identify atomic masses and unit context before you begin a chemistry calculation.
Worked examples
Converting ordinary notation
Convert 0.0000390 to scientific notation. Move the decimal five places to the right to obtain 3.90 × 10−5. The final zero is retained because it indicates three significant figures.
Multiplying scientific notation
Calculate (2.5 × 103)(4.20 × 10−2).
- Multiply the coefficients: 2.5 × 4.20 = 10.5.
- Add the exponents: 3 + (−2) = 1.
- Rewrite 10.5 × 101 as 1.05 × 102.
- The least precise input has two significant figures, so report 1.1 × 102.
Adding values with different decimal precision
Calculate 4.682 m + 0.7 m + 12.04 m. The arithmetic gives 17.422 m. Because 0.7 m is precise only to the tenths place, the reported result is 17.4 m.
Checking an implausible result
If a calculation produces 6.2 × 108 when the inputs are values near 102 and 103 being multiplied, revisit the exponent arithmetic. Multiplication adds exponents; division subtracts them. A quick order-of-magnitude check often catches misplaced decimal points before rounding makes the error harder to see.
When to recalculate
Recalculate whenever an input changes, a unit conversion is corrected, or the required reporting precision changes. This is especially important when you reuse a result in a later equation: an intermediate value rounded too early can affect the final digit.
Before submitting a lab report or worksheet, use this final check:
- Confirm that every quantity has compatible units.
- Identify which inputs are measured and which are exact.
- Count significant figures again, paying special attention to zeros.
- Use decimal-place rules for addition and subtraction.
- Use significant-figure rules for multiplication and division.
- Round once at the end when practical.
- Write scientific notation when it removes ambiguity.
- Compare the result with a rough estimate.
Notation conventions can differ slightly between classrooms, laboratories, and technical publications, particularly for rounding a trailing 5 or handling logarithms. If you are preparing formal work, check the applicable instructions or reference standard. For broader science reading, precise numerical interpretation also matters when examining satellite measurements, telescope images, or environmental data; the same habits of checking units, scale, and stated precision apply.
