How to Write Numbers in Scientific Notation (and Calculate With Them)

Scientific notation writes a number as a coefficient between 1 and 10 multiplied by a power of 10. To convert, move the decimal point until one nonzero digit is left of it, and use the number of places moved as the exponent: 93,000,000 = 9.3 × 10⁷, and 0.000452 = 4.52 × 10⁻⁴. Big numbers get positive exponents; numbers smaller than 1 get negative ones.

The coefficient a (also called the significand or mantissa) holds the meaningful digits; the exponent n holds the scale. Separating the two makes very large and very small numbers easy to read, compare and calculate with.

Converting to scientific notation

1. Place the decimal point after the first nonzero digit.
2. Count how many places it moved.
3. Write the exponent. If the original number was 10 or larger, the exponent is positive. If it was smaller than 1, the exponent is negative. For numbers from 1 to just under 10, the exponent is 0.
4. Drop zeros that only held place value, but keep any zeros that are significant.

Converting back to standard form

Reverse the process: a positive exponent moves the decimal point to the right that many places, and a negative exponent moves it to the left, filling empty places with zeros.

• 2.38 × 10⁵ → move 5 right → 238,000
• 7.1 × 10⁻³ → move 3 left → 0.0071

The scientific notation converter does both directions and also shows E notation and engineering notation.

Multiplying and dividing

Multiply or divide the coefficients, then add or subtract the exponents. Finally, renormalize if the coefficient falls outside 1 to 10.

Watch the signs when subtracting a negative exponent: 5 − (−3) = 8. These are just the exponent rules applied to powers of 10.

Adding and subtracting

Here the exponents must match first. Rewrite one number so both share the same power of 10, then add or subtract the coefficients.

If the exponents differ by a lot, the smaller number may not change the larger one at the precision you are working with. Adding 1 × 10² to 5.0 × 10⁸ still gives 5.0 × 10⁸ to two significant figures.

Powers and roots

Raise the coefficient to the power and multiply the exponent by it. For roots, make the exponent divisible by the root first.

• (2 × 10³)³ = 2³ × 10⁹ = 8 × 10⁹
• √(9 × 10⁸) = √9 × 10⁴ = 3 × 10⁴
• √(4 × 10⁷): 7 is odd, so rewrite as 40 × 10⁶. √40 × 10³ ≈ 6.32 × 10³

E notation on calculators and in code

Calculators, spreadsheets and programming languages replace “× 10^” with the letter E. On a calculator, use the EE or EXP key rather than typing × 10 ^, which can break the order of operations in some expressions.

Engineering notation and metric prefixes

Engineering notation is a variant that only allows exponents that are multiples of 3, so the coefficient ranges from 1 to just under 1,000. The benefit is that each exponent matches a metric prefix.

• 47,000 Ω = 47 × 10³ Ω = 47 kΩ
• 0.0000033 F = 3.3 × 10⁻⁶ F = 3.3 µF
• 2,400,000,000 Hz = 2.4 × 10⁹ Hz = 2.4 GHz

The metric prefix converter moves between these forms, and metric prefixes explained lists every prefix from quecto to quetta.

Significant figures in scientific notation

Every digit in the coefficient is significant, which makes scientific notation the clearest way to show precision. The number 5,000 is ambiguous, but 5 × 10³ has one significant figure and 5.000 × 10³ has four. See significant figures rules for how many digits to keep after a calculation.

Common mistakes

• A coefficient outside 1 to 10. Answers like 15 × 10¹⁰ or 0.4 × 10⁹ are correct values but not normalized. Shift the decimal one place and adjust the exponent by one: 1.5 × 10¹¹ and 4 × 10⁸.
• Moving the decimal the wrong way when renormalizing. Making the coefficient smaller (15 → 1.5) makes the exponent larger (10 → 11), and vice versa. The overall value must not change.
• Adding exponents when adding numbers. Exponents are added only when multiplying. For addition and subtraction, match the exponents and leave them alone.
• Confusing a negative exponent with a negative number. 4.52 × 10⁻⁴ is a small positive number; −4.52 × 10⁴ is a large negative one.
• Typing “× 10 ^” into a calculator. Entering 6 ÷ 2 × 10 ^ 3 for 6 ÷ (2 × 10³) gives 3,000 instead of 0.003. The EE key or E notation keeps the power of ten attached to its coefficient.

When a course asks you to “write in standard form,” check whether it means the expanded number (US usage) or scientific notation (UK usage). The standard form calculator handles both, and the exponent calculator evaluates any power of ten directly.

Real-world examples

These values are exact because they are fixed by definition: the speed of light since 1983, the Avogadro, elementary charge and Planck constants since the 2019 revision of the SI, and the astronomical unit by a 2012 resolution of the International Astronomical Union. NIST publishes the official values of the physical constants.

A quick calculation shows the notation at work: light from the Sun takes (1.496 × 10¹¹ m) ÷ (2.998 × 10⁸ m/s) ≈ 4.99 × 10² s, or about 8.3 minutes, to reach Earth.

For any calculation like this, the scientific notation calculator multiplies, divides, adds and subtracts with the work shown.