Scientific notation expresses a number as a value multiplied by a power of ten. It makes very large and very small quantities easier to read, compare, and calculate, especially in science, engineering, and mathematics. The standard form is a × 10ⁿ, where the absolute value of a is at least 1 and less than 10, and n is an integer.

Scientific notation formula

N = a × 10ⁿ

Move the decimal point until one non-zero digit remains to its left. Count how many places it moved: that count determines the exponent. Moving the decimal left to make a number smaller than 10 produces a positive exponent for a large original number; moving it right for a small decimal produces a negative exponent.

Examples

For 45,000, move the decimal four places left to get 4.5 × 10⁴. For 0.0032, move the decimal three places right to get 3.2 × 10⁻³. To return to ordinary notation, move the decimal right for a positive exponent and left for a negative exponent.

How to convert a number

  1. Write the original number and locate its decimal point.
  2. Move the decimal until the leading value is between 1 and 10 in absolute value.
  3. Count the moves and record the direction.
  4. Use a positive exponent for a large number and a negative exponent for a non-zero value smaller than 1.
  5. Convert back to ordinary notation to check the result.

Calculating with scientific notation

When multiplying values in scientific notation, multiply their leading numbers and add the exponents. When dividing, divide the leading numbers and subtract the exponents. For addition and subtraction, first express both values with the same power of ten, then combine the leading numbers.

Common mistakes

The exponent records decimal movement, not the number of digits. Check its sign carefully: 0.004 is 4 × 10⁻³, not 4 × 10³. The leading value should also be normalized to the required range.

Frequently asked questions

Can the leading number be 12?

It can appear in an unnormalized expression, but standard scientific notation uses a leading value from 1 up to, but not including, 10 in absolute value.

Can the exponent be negative?

Yes. Negative exponents represent fractions of a power of ten.

Does scientific notation change the number?

No. It is an equivalent way to write the same value.