A scientific notation calculator rewrites a number as a coefficient between 1 and 10 multiplied by a power of ten, and converts back. 123,456 becomes 1.23456 × 10^5 and 0.00042 becomes 4.2 × 10^−4.
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How to use the scientific notation calculator
- Choose standard to scientific or scientific to standard.
- Enter the number, or the coefficient and exponent.
- Read the result.
How the scientific notation calculator works
Scientific notation writes a number as a coefficient times a power of ten: a × 10^n, where the coefficient a is at least 1 and less than 10 and the exponent n is an integer. The scientific notation calculator has two modes. In standard-to-scientific mode it finds n as the floor of the base-10 logarithm of the absolute value and divides by 10^n to get the coefficient. In scientific-to-standard mode it multiplies the coefficient by 10 to the exponent you enter.
Scientific notation calculator worked example
To write 123,456 in scientific notation, move the decimal point five places left to get 1.23456, so the number is 1.23456 × 10^5. For 0.00042, move the point four places right to get 4.2, so the number is 4.2 × 10^−4. The speed of light, 299,792,458 meters per second, is 2.99792458 × 10^8. The scientific notation calculator gives the same results instantly.
| Standard form | Scientific notation |
|---|---|
| 123456 | 1.23456 × 10^5 |
| 0.00042 | 4.2 × 10^−4 |
| 299792458 | 2.99792458 × 10^8 |
| 6020 | 6.02 × 10^3 |
| 0.5 | 5 × 10^−1 |
| 1 | 1 × 10^0 |
Reading the exponent
A positive exponent means a large number: count that many places to the right of the first digit. A negative exponent means a small number between 0 and 1. An exponent of zero means the number is between 1 and 10. Each step of one in the exponent is a factor of ten, so 10^6 is a million and 10^−6 is a millionth. SI prefixes name some of these powers: kilo is 10^3, mega 10^6, milli 10^−3 and micro 10^−6.
Multiplying and dividing
Scientific notation makes arithmetic with very large and small numbers easier. To multiply, multiply the coefficients and add the exponents: (3 × 10^4) × (2 × 10^5) = 6 × 10^9. To divide, divide the coefficients and subtract the exponents. If the new coefficient falls outside 1 to 10, shift the decimal point and adjust the exponent: 12 × 10^3 becomes 1.2 × 10^4.
Significant figures
The digits in the coefficient show the precision of a measurement. Writing 6.02 × 10^3 says the value is known to three significant figures, while 6020 alone is ambiguous about whether the final zero is meaningful. Round the coefficient to the precision of your data. When rounding, check that the coefficient has not become 10.0; if it has, write 1.0 and add one to the exponent.
E notation on calculators and computers
Calculators, spreadsheets and programming languages often show scientific notation as E notation: 1.23456E5 or 1.23456e+5 means 1.23456 × 10^5, and 4.2E-4 means 4.2 × 10^−4. If you paste a number from a spreadsheet into the scientific notation calculator, split it at the E into the coefficient and the exponent.
Where it is used
Scientists use scientific notation for the mass of an electron, the distance to stars and the number of molecules in a mole. Engineers use a related form, engineering notation, where the exponent is always a multiple of three to match SI prefixes. Finance and data storage use it for very large totals.
A quick check for any answer from the scientific notation calculator: the exponent should equal the number of places you moved the decimal point, positive if you moved it left and negative if you moved it right. If the coefficient has more than one digit before the decimal point, the exponent is off by at least one.
Tips
- Keep the coefficient between 1 and 10.
- Use negative exponents for numbers below 1.
- Match significant figures to your data.
- Zero cannot be written in scientific notation with this rule.
Related converters and calculators
Related calculators:
- Exponent calculator: powers
- SI prefix converter: prefixes
- Big number calculator: large numbers
- Unit converter: units
Sources
OpenStax Prealgebra 2e, 10.5: Integer exponents and scientific notation; NIST Guide to the SI, Appendix B.8: Factors for units listed alphabetically.
Formula and method
to_sci: exp = floor(log10(|value|)), coef = value / 10^exp. from_sci: coef × 10^exp.
Base-10 scientific notation helper. Rejects zero in to_sci mode.
Questions about this calculator
What is scientific notation?
A number written as a × 10^n with 1 ≤ |a| < 10.
What does a negative exponent mean?
The number is between 0 and 1.
What is E notation?
A computer form: 4.2E-4 means 4.2 × 10^−4.
How do I multiply?
Multiply coefficients and add exponents.
Can zero be written this way?
No standard form exists for zero.
Check the assumptions and units before relying on a result. Financial, tax and health estimates do not replace advice for your individual circumstances.
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