Mathematics
Scientific notation, significant figures and rounding
Type a number, however large or small: get its scientific notation, its engineering notation with the SI prefix, the number of significant figures and the rounded value.
Scientific notation and significant figures
Scientific notation writes every number as a × 10ⁿ, with a between 1 and 10: 0.000345 becomes 3.45 × 10⁻⁴, Avogadro's constant 6.022 × 10²³. The exponent shows the order of magnitude at once, and products become easy: multiply the mantissas and add the exponents.
Engineering notation uses only exponents that are multiples of 3, which match the SI prefixes: 345 × 10⁻⁶ is 345 µ, read as 345 micro. It is the notation of engineering calculators and data sheets.
Significant figures tell how precise a measured value is. Every digit from the first non-zero one counts, trailing zeros included when there is a decimal point: 0.00034560 has 5 significant figures, because the final zero says the measurement reaches that digit. Rounded to 3 it becomes 3.46 × 10⁻⁴.
Common mistakes
- Counting leading zeros as significant: 0.0045 has only 2 significant figures.
- Dropping trailing zeros after the point: 2.50 and 2.5 are not the same value, the first is more precise.
- Writing the mantissa outside 1 to 10, as in 34.5 × 10⁻⁵: correct as a value but not scientific notation.
Frequently asked questions
How many significant figures should a result keep?
In multiplication and division, as many as the least precise value; in addition and subtraction, what counts instead is the number of decimal places of the least precise value.
How does scientific notation differ from engineering notation?
In scientific notation the mantissa is between 1 and 10 and the exponent can be anything; in engineering notation the exponent is a multiple of 3 and the mantissa is between 1 and 1000, to match the prefixes kilo, mega, milli, micro.
What does 6.022e23 mean?
It is the E notation of computers and calculators: e stands for “times ten to the”, so 6.022e23 = 6.022 × 10²³.
How this calculation works
Scientific notation: x = a × 10ⁿ with 1 ≤ |a| < 10 and n the position of the first significant digit. Engineering: n a multiple of 3, 1 ≤ |a| < 1000. Significant figures: from the first non-zero digit to the last written, trailing zeros counted only when there is a point. Rounding to k figures: look at digit k + 1 and round up if it is 5 or more. The calculation works on the digits as text, with no floating-point error.