What scientific notation is
Scientific notation writes a number as a mantissa (a value at least 1 and less than 10) multiplied by a power of ten. For 12345, the decimal moves 4 places left to give 1.2345, so the number is 1.2345 × 10⁴.
The exponent is simply a count of how many places the decimal point moved. Move it left and the exponent is positive; move it right, as with a number below 1, and it is negative. The rule that the mantissa must sit between 1 and 10 is what makes the form unique — otherwise 12345 could equally be written as 12.345 × 10³. The power of ten in the notation is an ordinary exponent, evaluated by the exponent calculator.
How to use this calculator
Enter any number, large or small, and the calculator returns it in a × 10ⁿ form, showing the mantissa and the exponent separately. Negative exponents appear for numbers smaller than 1, such as 0.0042 = 4.2 × 10⁻³.
A negative number keeps its minus sign on the mantissa, not the exponent: −12345 is −1.2345 × 10⁴. Do not confuse a negative mantissa, which means the value is below zero, with a negative exponent, which means the value is small in magnitude. Deciding how many digits to keep in the coefficient is a job for the rounding calculator.
Why scientific notation is useful
It makes very large and very small numbers easy to read and compare without counting long strings of zeros. The exponent is closely tied to the base-10 logarithm — see the logarithm calculator to explore that relationship.
It also makes arithmetic tidier. To multiply two numbers in this form, multiply the mantissas and add the exponents; to divide, divide the mantissas and subtract. Comparing magnitudes becomes a glance at the exponent.
Frequently asked questions
- What is 12345 in scientific notation?
- 1.2345 × 10⁴. The decimal moves four places left, giving a mantissa of 1.2345 and an exponent of 4.
- What is the mantissa?
- The mantissa is the leading number, at least 1 and less than 10, that multiplies the power of ten.
- How do small numbers look in scientific notation?
- They use a negative exponent. For example, 0.0042 is 4.2 × 10⁻³.
- How is the exponent related to logarithms?
- The exponent is the whole-number part of the base-10 logarithm of the number's magnitude.
- What does E notation such as 1.2345E4 mean?
- It is the same thing written for screens and keyboards, where E stands for times ten to the power of. So 1.2345E4 is 1.2345 × 10⁴, and 4.2E−3 is 4.2 × 10⁻³. Spreadsheets and programming languages display results this way.