# Definition:Scientific Notation

## 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:

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

### 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:

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