Scientific Notation Calculator
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Other tools you may find usefulScientific notation calculator - working with very large and very small numbers
In the exact sciences, such as physics, chemistry, astronomy and computer science, researchers operate on values of extreme orders of magnitude on a daily basis. For example, the mass of the Earth is approximately 5,970,000,000,000,000,000,000,000 kg, and the elementary charge of an electron is only 0.00000000000000000016 coulomb. Writing such numbers in the traditional decimal format is not only impractical, but also leads to cardinal errors when copying zeros. The solution to this problem is scientific notation (exponential form), which allows you to write any number as the product of a number in the range \([1, 10)\) and a power of 10. Our online scientific notation calculator allows you to instantly convert values in both directions.
Just enter a standard decimal number and the calculator will present its exponential form and the so-called E scientific notation (used in spreadsheets and physics calculators). The tool also supports reverse operations - it expands the exponential notation to full decimal form along with information about the order of magnitude and the matching SI unit prefix.
SI unit prefixes and their exponential notation
The International System of Units (SI) defines special prefixes to help specify orders of magnitude. The table below shows the most common prefixes and their exponents:
| Unit prefix | Symbol | Multiplier (Exponential form) | Decimal value | Application example |
|---|---|---|---|---|
| giga | G | \(10^9\) | 1,000,000,000 | Gigabyte (GB) – disk capacity. |
| mega | M | \(10^6\) | 1,000,000 | Megawatt (MW) – power plant capacity. |
| kilo | k | \(10^3\) | 1 000 | Kilometer (km) – distance. |
| milli | m | \(10^{-3}\) | 0.001 | Millimeter (mm) – small lengths. |
| micro | µ | \(10^{-6}\) | 0.000 001 | Micrometer (µm) – sizes of biological cells. |
| nano | n | \(10^{-9}\) | 0.000 000 001 | Nanometer (nm) – semiconductor technology, processors. |
Recording format and the so-called scientific notation E (E-notation)
The general notation formula in scientific notation is: \(a \times 10^n\), where \(a\) (mantissa) satisfies the condition \(1 \le |a| < 10\), and \(n\) (feature/exponent) is an integer. Due to the limitations of old computer displays, an alternative notation was introduced called E-notation:
- E-notation:Replaces the "\(\times 10^n\)" with the letter "e" or "E" (forexponent) followed by an exponent (e.g. \(3.0 \times 10^8\) is written as
3e8or3.0E+8). - Positive exponents:Denotes very large numbers. For example, the speed of light in a vacuum is approximately \(3 \times 10^8\) m/s, or
3e8. - Negative exponents:They mean fractional values, close to zero. For example, Planck's constant is approximately \(6.626 \times 10^{-34}\) J s, which in E notation is written as
6.626e-34.
How to convert a number to an exponential form? Step-by-step algorithm
Conversion is based on moving the decimal point and counting the steps taken:
- For large numbers (greater than 10):Move the decimal point to the left until there is only one non-zero digit left before the decimal point. The number of shifts is your positive exponent \(n\). (e.g. for the number 5200: move the decimal point 3 places to the left -> \(5.2 \times 10^3\)).
- For small numbers (less than 1):Move the decimal point to the right until you pass the first non-zero digit. The number of shifts is your negative exponent \(n\). (e.g. for the number 0.0045: move the decimal point 3 places to the right -> \(4.5 \times 10^{-3}\)).
- Set the mantissa value:The mantissa must be in the range \([1, 10)\). Write it down and multiply it by \(10^n\).
Frequently asked questions (FAQ)
Are scientific notation and exponential notation the same thing?
Yes. These terms are used interchangeably. They define the same mathematical way of writing numbers as the product of a mantissa and a power of ten (\(a \times 10^n\)).
How to enter exponential form in Excel?
In Microsoft Excel, you can type an exponential form directly in a cell using the letter E. For example, typing1.5E+6will automatically be interpreted as 1,500,000. You can also change the cell formatting to "Scientific".
Why does the mantissa have to be less than 10?
This is a requirement to standardize scientific notation. If the mantissa could be greater than 10 (e.g. \(15 \times 10^2\)), the same number could be written in multiple ways (e.g. \(1.5 \times 10^3\)), making it difficult to quickly compare orders of magnitude of numbers by eye.
What is "Order of Magnitude"?
Order of magnitude is the approximate value of a number rounded to the nearest power of ten. Two numbers have the same order of magnitude if their exponents to powers of 10 in scientific notation are identical. This allows for quick estimation of proportions in space or the microworld.
Does your calculator round periodic numbers correctly?
Yes. The calculator automatically recognizes and correctly converts periodic fractions and allows you to choose the number of decimal places (precision) in the output exponential notation.