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:

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

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): exponential notation
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): scientific notation, exponential notation or standard form
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): standard form
• 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): index notation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): scientific notation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): standard form: 1. (of a number)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): scientific notation