Definition:Scientific Notation
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Definition
Scientific notation is an implementation of floating-point representation for representing approximations to (usually large) numbers by presenting them in the form:
- $n \approx m \times 10^e$
where:
- $m$ is a rational number such that $1 \le m < 10$, expressed in decimal notation
- $e$ is an integer.
Base
The number $10$, in this context, is referred to as the base.
Mantissa
The number $m$ is known as the mantissa.
Exponent
The number $e$ is known as the exponent.
Also known as
Scientific notation can also be seen referred to as:
Examples
Powers of 10
Various powers of $10$ are specified in scientific notation as follows:
| \(\ds 10\) | \(=\) | \(\ds 10^1\) | ||||||||||||
| \(\ds 100\) | \(=\) | \(\ds 10^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10 \times 10\) | ||||||||||||
| \(\ds 100 \, 000\) | \(=\) | \(\ds 10^5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10 \times 10 \times 10 \times 10 \times 10\) |
Negative Powers of 10
Various powers of $10$ with negative exponent are specified in scientific notation as follows:
| \(\ds 1\) | \(=\) | \(\ds 10^0\) | ||||||||||||
| \(\ds 0 \cdotp 1\) | \(=\) | \(\ds 10^{-1}\) | ||||||||||||
| \(\ds 0 \cdotp 01\) | \(=\) | \(\ds 10^{-2}\) | ||||||||||||
| \(\ds 0 \cdotp 000 \, 01\) | \(=\) | \(\ds 10^{-5}\) |
Arbitrary Examples
Various numbers are specified in scientific notation as follows:
Example 1
| \(\ds 864 \, 000 \, 000\) | \(=\) | \(\ds 8 \cdotp 64 \times 10^8\) | ||||||||||||
| \(\ds 0 \cdotp 000 \, 034 \, 16\) | \(=\) | \(\ds 3 \cdotp 416 \times 10^{-5}\) |
Example 2
| \(\ds 48 \, 230 \, 000\) | \(=\) | \(\ds 4 \cdotp 823 \times 10^7\) |
Example 3
| \(\ds 0 \cdotp 000 \, 008 \, 4\) | \(=\) | \(\ds 8.4 \times 10^{-6}\) |
Example 4
| \(\ds 0 \cdotp 000 \, 380\) | \(=\) | \(\ds 3 \cdotp 80 \times 10^{-4}\) |
Example 5
| \(\ds 186 \, 000\) | \(=\) | \(\ds 1 \cdotp 86 \times 10^5\) |
Example 6
| \(\ds 300 \times 10^8\) | \(=\) | \(\ds 30 \, 000 \, 000 \, 000\) |
Example 7
| \(\ds 70 \, 000 \times 10^{-10}\) | \(=\) | \(\ds 0 \cdotp 000 \, 007 \, 000 \, 0\) |
Speed of Light
The speed of light is defined as:
- $c = 299 \, 792 \, 458 \text { m s}^{-1}$
In scientific notation this can be expressed as:
- $c = 2 \cdotp 99792 \, 458 \times 10^8 \text { m s}^{-1}$
Also see
- Results about scientific notation can be found here.
Sources
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Scientific Notation
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2^{86243} - 1$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): scientific notation, exponential notation or standard form
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential notation (standard form, index notation, scientific notation)
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): scientific notation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential notation (standard form, index notation, scientific notation)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): scientific notation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): scientific notation