What scientific notation means
Scientific notation writes a number as a coefficient multiplied by a power of ten. The coefficient is at least 1 and below 10, and the exponent is a whole number that says how many places the decimal point has moved. It is a compact way to write very large and very small values, such as distances in astronomy, the size of a molecule or a rate of one failure in a billion, and it makes the size of a number obvious at a glance. The same number can be written several ways, and the one with a single non-zero digit before the decimal point is called normalised. This calculator converts any number to that form and shows the equivalent E notation, engineering notation and the number written out in full.
Convert and normalise
To normalise a number, move the decimal point until exactly one non-zero digit is to its left. Count the places you moved it. If you moved left, the exponent is that count; if you moved right, it is negative. The number 1568938 becomes 1.568938 × 10⁶ after six moves to the left, so the calculator reports 1.568938e6. A small number goes the other way: 0.0004212 needs four moves to the right, giving 4.212 × 10⁻⁴. The sign belongs to the coefficient, so −5,000,000,000 is -5 × 10⁹.
You can type a number as a plain decimal, in E notation such as 2.3e11, or as 3.5x10^-12. Zero is written 0 × 10⁰ by convention, because it has no first non-zero digit. Commas are not accepted, since a comma could be a thousands separator or a decimal comma, and the calculator would rather ask than guess.
Add and subtract
Addition and subtraction need matching exponents, because you can only add things measured in the same unit. Rewrite both numbers with the larger power of ten, add or subtract the coefficients, then normalise again. For 1.432 × 10² minus 8 × 10¹, write 8 × 10¹ as 0.8 × 10², subtract 1.432 − 0.8 to get 0.632, and normalise to 6.32 × 10¹. Adding 1.23 × 10⁷ and 3.45 × 10² gives 1.2300345 × 10⁷, which is 12,300,345 written out. The smaller number has little effect when the exponents differ by a lot, but the calculator keeps every digit and rounds only at the end, so nothing is lost on the way.
Multiply, divide and use powers
Multiplication and division are easier, because the exponents do not need to match. Multiply or divide the coefficients and add or subtract the exponents. For 1.23 × 10⁷ times 3.45 × 10², the coefficients give 4.2435 and the exponents add to 9, so the product is 4.2435 × 10⁹, or 4,243,500,000. Dividing 1.432 × 10² by 8 × 10¹ gives 1.79 × 10⁰. Squaring doubles the exponent and squares the coefficient, so (1.2 × 10³)² is 1.44 × 10⁶. A square root halves the exponent: the root of 9 × 10⁶ is 3 × 10³, and when the exponent is odd the calculator first moves one factor of ten into the coefficient.
General powers raise the coefficient to the power and multiply the exponent by it. For example, 2 to the power 10 is 1.024 × 10³. A negative base can only be raised to a whole-number power, because a fractional power of a negative number is not a real number. Zero to the power 0 and zero to a negative power are undefined, and so are division by zero and the square root of a negative number. In those cases the calculator explains the problem instead of showing zero or infinity.
E notation versus engineering notation
E notation replaces the × 10 with the letter e, so 1.568938 × 10⁶ is 1.568938e6. It is what spreadsheets and programming languages print when a number is too large or small for a normal display. Here the e means times ten to the power, and has nothing to do with Euler's number. A handheld calculator may show the same thing as EE or EXP, and this page does not imitate a particular device.
Engineering notation restricts the exponent to multiples of three so that it lines up with metric prefixes such as kilo, mega and milli. The coefficient then lies from 1 up to, but not including, 1000. The number 123,400,000 is 1.234 × 10⁸ in scientific notation and 123.4 × 10⁶ in engineering notation, and 0.0004212 becomes 421.2 × 10⁻⁶. All four formats on the page show the same rounded value.
| Number | Scientific | E notation | Engineering |
|---|---|---|---|
| 1,568,938 | 1.568938 × 10⁶ | 1.568938e6 | 1.568938 × 10⁶ |
| 0.0004212 | 4.212 × 10⁻⁴ | 4.212e-4 | 421.2 × 10⁻⁶ |
| −5,000,000,000 | -5 × 10⁹ | -5e9 | -5 × 10⁹ |
| 123,400,000 | 1.234 × 10⁸ | 1.234e8 | 123.4 × 10⁶ |
Precision and supported syntax
Results are rounded half-up to the number of significant digits you choose, from 1 to 50, with 12 as the default. Addition, subtraction, multiplication and squaring are computed exactly before rounding. Divisions and roots that do not end are labelled approximate. If rounding changes the value, the result keeps all the chosen digits to show the precision: 9.999 rounded to 3 digits is 1.00 × 10¹. Inputs may have up to 100 significant digits and exponents from −1000 to 1000, and results are limited to exponents from −10000 to 10000. A very long expanded decimal is withheld, and the other formats are still shown. Only single numbers are accepted, not full expressions with brackets.
Worked examples and questions
One third is a good test of precision. With 12 significant digits, 1 ÷ 3 is 3.33333333333e-1, which is labelled approximate because the digits never end. Scientific notation also appears in statistics, where very small probabilities are written this way, so see the normal distribution calculator for tail probabilities. To work with percentages of very large or small values, use the percentage calculator, and the sample size calculator shows numbers in everyday units.
How do I convert 0.00042 to scientific notation?
Move the decimal point four places to the right to reach 4.2, so the exponent is −4: 4.2 × 10⁻⁴.
What does the letter e mean in 2.3e11?
It means times ten to the power: 2.3e11 is 2.3 × 10¹¹. It is not Euler's number.
Do I add the exponents when I add two numbers?
No. For addition and subtraction, first rewrite both numbers with the same exponent, then combine the coefficients. You add exponents only when multiplying.
What is engineering notation?
Scientific notation with the exponent forced to a multiple of three, so the coefficient runs from 1 to 999.999… and matches SI prefixes.
Why was my result rounded?
The calculator rounds to the number of significant digits you choose. Raise the setting, up to 50, to see more digits.
Can the numbers be negative?
Yes. A leading minus sign belongs to the coefficient. Square roots need a non-negative number, and a negative base can only take a whole-number power.