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Scientific Notation Calculator

Scientific notation expresses numbers in the form a x 10^n, where 1 <= |a| < 10 and n is an integer. This compact format is essential for working with very large numbers (the speed of light is 3 x 10^8 m/s) and very small numbers (a hydrogen atom's radius is about 5.3 x 10^-11 meters). For example, the number 4,500,000 becomes 4.5 x 10^6 in scientific notation. This calculator converts numbers to and from scientific notation and performs multiplication and division while keeping results in proper scientific form.

Quick Answer

The number 4,500,000 in scientific notation is 4.5 x 10^6. The number 0.00032 in scientific notation is 3.2 x 10^-4.

Convert to Scientific Notation

Enter any number (e.g. 4500000 or 0.00032)

Convert from Scientific Notation

Multiply or Divide

Common Examples

Input Result
4500000 4.5 x 10^6
0.00032 3.2 x 10^-4
(3 x 10^4) x (2 x 10^3) 6 x 10^7
(8.4 x 10^6) / (2.1 x 10^2) 4 x 10^4
6.022 x 10^23 602,200,000,000,000,000,000,000

How It Works

The Format

Scientific notation expresses a number as:

a x 10^n

where a (the coefficient or significand) satisfies 1 <= a < 10, and n (the exponent) is an integer. The exponent indicates how many places to move the decimal point: positive exponents move it right (large numbers), negative exponents move it left (small numbers).

Converting to Scientific Notation

  1. Move the decimal point so that there is exactly one nonzero digit to its left.
  2. Count how many places the decimal point moved. This count becomes the exponent n.
  3. If the decimal moved left (the original number was large), n is positive. If it moved right (the original number was small), n is negative.

For example: 4,500,000 becomes 4.5 x 10^6 (decimal moved 6 places left). 0.00032 becomes 3.2 x 10^-4 (decimal moved 4 places right).

Multiplying in Scientific Notation

To multiply (a x 10^m) by (b x 10^n): multiply the coefficients and add the exponents, then normalize if the new coefficient is outside the range [1, 10).

(a x 10^m) x (b x 10^n) = (a x b) x 10^(m + n)

Dividing in Scientific Notation

To divide (a x 10^m) by (b x 10^n): divide the coefficients and subtract the exponents, then normalize.

(a x 10^m) / (b x 10^n) = (a / b) x 10^(m - n)

Worked Example

To convert 67,800 to scientific notation: move the decimal 4 places left to get 6.78. The result is 6.78 x 10^4.

To multiply (3 x 10^4) by (2 x 10^3): multiply coefficients 3 x 2 = 6, add exponents 4 + 3 = 7. Result: 6 x 10^7. Since 6 is between 1 and 10, no normalization is needed. As a standard number, that is 60,000,000.

To divide (8.4 x 10^6) by (2.1 x 10^2): divide coefficients 8.4 / 2.1 = 4, subtract exponents 6 - 2 = 4. Result: 4 x 10^4, which is 40,000.

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Frequently Asked Questions

What is the difference between scientific notation and standard notation?
Standard notation writes a number in its full decimal form (like 4,500,000), while scientific notation uses a compact coefficient-and-exponent form (4.5 x 10^6). Scientific notation is preferred when numbers are extremely large or small, because it makes them easier to read, compare, and compute with.
Why must the coefficient be between 1 and 10?
The convention of keeping the coefficient between 1 (inclusive) and 10 (exclusive) ensures a unique, standard representation for every number. Without this rule, the same number could be written many different ways (45 x 10^5, 4.5 x 10^6, 0.45 x 10^7). The normalized form makes it easy to compare magnitudes by looking at the exponent.
What is engineering notation?
Engineering notation is a variation of scientific notation where the exponent is always a multiple of 3 (corresponding to SI prefixes like kilo, mega, giga, milli, micro, nano). For example, 4,500,000 is written as 4.5 x 10^6 in both scientific and engineering notation, but 45,000 would be 4.5 x 10^4 in scientific notation and 45 x 10^3 in engineering notation.
How do I add or subtract numbers in scientific notation?
To add or subtract, the exponents must match first. Adjust one number so both have the same exponent, then add or subtract the coefficients. For example, (3 x 10^4) + (5 x 10^3) = (3 x 10^4) + (0.5 x 10^4) = 3.5 x 10^4. This is more involved than multiplication or division, which is why this calculator focuses on those operations.
Where is scientific notation used?
Scientific notation is standard in physics (speed of light: 3 x 10^8 m/s), chemistry (Avogadro's number: 6.022 x 10^23), astronomy (distances in light-years), biology (cell sizes), and engineering. It appears wherever measurements span many orders of magnitude. Most scientific calculators and programming languages support it using 'E notation' (e.g., 4.5E6 means 4.5 x 10^6).