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:
- $\ds x = \sum_{j \mathop \in \Z} r_j 2^j$
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): Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm
- 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): binary system