Definition:Scientific Notation

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Definition

Scientific notation is a technique 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:

exponential notation
standard form
index notation


Examples

Powers of 10

Various powers of $10$ are specified in scientific notation as follows:

\(\ds 10\) \(=\) \(\ds 10^1\)
\(\ds 100\) \(=\) \(\ds 10^2 = 10 \times 10\)
\(\ds \) \(=\) \(\ds 10 \times 10\)
\(\ds 100 \, 000\) \(=\) \(\ds 10^5\)
\(\ds \) \(=\) \(\ds 10 \times 10 \times 10 \times 10 \times 10\)


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:

$c = 2 \cdotp 99792 \, 458 \times 10^8 \text { m s}^{-1}$


Sources