Definition:Binary Notation

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Definition

Binary notation is the positional number system whose base is $2$.

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

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

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


Motivation

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

This is because the two binary digits, $0$ and $1$, can be represented by two well-defined states of a component.


Examples

$13$ in Binary Notation

The number written in binary notation as $1101$ is expressed in decimal notation as $13$.


$19$ in Binary Notation

The number written in decimal notation as $19$ is expressed in binary notation as $10011_2$.


$23$ 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

  • Results about binary notation can be found here.


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

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