# Definition:Binary Notation

## Definition

**Binary notation** is the technique of expressing numbers in base $2$.

That is, every number $x \in \R$ is expressed in the form:

- $\displaystyle x = \sum_{j \mathop \in \Z} r_j 2^j$

where $\forall j \in \Z: r_j \in \set {0, 1}$.

## Examples

### $23$ in Binary Notation

The number written in decimal notation as $23$ is expressed in binary notation as $10111_2$.

### $36$ in Binary Notation

The number written in decimal notation as $36$ is expressed in binary notation as $100100_2$.

### $47$ in Binary Notation

The number written in decimal notation as $47$ is expressed in binary notation as $101111_2$.

### $68$ in Binary Notation

The number written in decimal notation as $68$ is expressed in binary notation as $1000100_2$.

### $127$ in Binary Notation

The number written in decimal notation as $127$ is expressed in binary notation as $1111111_2$.

## Also see

**Binary notation**, like hexadecimal notation, has particular relevance in the field of computer science.

## Historical Note

The earliest known reference to binary notation appears to be in a Chinese book dating from approximately $3000$ B.C.E.

In Europe, binary notation was invented by Gottfried Wilhelm von Leibniz.

He associated God with $1$ and nothingness with $0$, and believed that it was mystically significant that all numbers could be built from $1$-ness and $0$-ness.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 24$ - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {1-2}$ The Basis Representation Theorem - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers: Arithmetic lives on - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**binary system**