Scientific Notation Calculator

Scientific notation

Scientific notation writes a number as a coefficient multiplied by a power of ten, a × 10ⁿ, where the coefficient's absolute value is at least 1 and less than 10 and n is an integer. It makes very large and very small numbers compact and easy to compare. The speed of light, 299,792,458 m/s, becomes about 2.998 × 10⁸ m/s, and the mass of an electron is about 9.109 × 10⁻³¹ kg.

To convert, move the decimal point until one non-zero digit is left of it, and count the moves. Moving left gives a positive exponent and moving right gives a negative one. The default number, 1,568,938, needs six moves to the left, so it is 1.568938 × 10⁶. For 0.00052, four moves to the right give 5.2 × 10⁻⁴. The coefficient also shows the significant figures plainly: 3.00 × 10² claims three significant figures in a way that 300 cannot.

Calculations with scientific notation

Multiplication and division are the easy operations. Multiply the coefficients and add the exponents, then renormalize if the coefficient leaves the 1 to 10 range. With the default values, (1.23 × 10⁷) × (3.45 × 10²) = 4.2435 × 10⁹. To divide, divide the coefficients and subtract the exponents: (1.23 × 10⁷) ÷ (3.45 × 10²) = 0.356521… × 10⁵, which renormalizes to 3.56521… × 10⁴ by moving the decimal one place right and lowering the exponent by one.

Addition and subtraction need matching exponents first. Rewrite the number with the smaller exponent so that it uses the larger one, then add or subtract the coefficients and keep the shared exponent. To add 3.45 × 10² to 1.23 × 10⁷, write it as 0.0000345 × 10⁷; the sum is 1.2300345 × 10⁷. When the exponents are far apart, the smaller number barely changes the larger one, which is why adding a grain of sand to a beach does not change its weight to any meaningful precision.

Powers and roots apply to both parts. Squaring 1.23 × 10⁷ squares the coefficient and doubles the exponent: 1.5129 × 10¹⁴. To take a square root, make the exponent even first, then halve it and take the root of the coefficient: √(1.23 × 10⁷) = √(12.3 × 10⁶) ≈ 3.5071 × 10³.

Keep track of significant figures. A product or quotient should be rounded to as many significant figures as the least precise input, and a sum or difference to the decimal place of the least precise input. The calculator shows results to the precision you choose, so round the final answer to match your measurements.

Engineering notation

Engineering notation is a variant in which the exponent is always a multiple of three, and the coefficient may be anywhere from 1 up to (but not including) 1,000. The point is that each exponent matches a metric prefix: 10³ is kilo, 10⁶ is mega, 10⁻³ is milli and 10⁻⁶ is micro. A resistance of 47,000 ohms is 47 × 10³ Ω, which reads directly as 47 kΩ, and a capacitance of 0.000 002 2 farads is 2.2 × 10⁻⁶ F, or 2.2 µF.

The default number, 1,568,938, already has an exponent of 6, so its engineering form is the same as its scientific form, 1.568938 × 10⁶, or about 1.57 mega-units. A number like 45,600 would be 4.56 × 10⁴ in scientific notation but 45.6 × 10³ in engineering notation.

E-notation

E-notation replaces × 10 with the letter E or e, which is how calculators, spreadsheets and programming languages display scientific notation in plain text. 1.568938e6 means 1.568938 × 10⁶, and 5.2E-4 means 5.2 × 10⁻⁴. The letter has nothing to do with Euler's number e ≈ 2.718; it simply stands for exponent.

Most software accepts E-notation as input too, so you can type 6.022e23 into a spreadsheet cell or into this calculator. Be careful when importing data: some spreadsheets automatically convert identifiers that happen to look like E-notation, such as a part number 12E5, into the number 1,200,000.

Frequently asked questions