Binary Arithmetic

The binary Numeral System (''' Base 2 numerals''') represents numeric values using two symbols, typically 0 and 1 . More specifically, binary is a Positional Notation with a Radix of Two . Owing to its relatively straightforward implementation in Electronic Circuit ry, the binary system is used internally by virtually all modern Computer s. HISTORY The ancient Indian Mathematician Pingala presented the first known description of a binary numeral system in the 3rd Century BCE , which coincided with his discovery of the concept of Zero . The modern binary number system was fully documented by Gottfried Leibniz in the 17th Century in his article '' Explication De L'Arithmétique Binaire ''. Leibniz's uses 0 and 1, like the modern binary numeral system. In 1854 , British mathematician George Boole published a landmark paper detailing a system of Logic that would become known as Boolean Algebra . His logical system proved instrumental in the development of the binary system, particularly in its implementation in electronic circuitry. In 1937 , Claude Shannon produced his master's thesis at MIT that implemented Boolean Algebra and binary arithmetic using electronic relays and switches for the first time in history. Entitled '' A Symbolic Analysis Of Relay And Switching Circuits '', Shannon's thesis essentially founded practical Digital Circuit design. In November of 1937 , George Stibitz , then working at Bell Labs , completed a relay-based computer he dubbed the "Model K" (for "Kitchen", where he had assembled it), which calculated using binary addition. Bell Labs thus authorized a full research program in late 1938 with Stibitz at the helm. Their Complex Number Computer , completed January 8 , 1940 , was able to calculate Complex Numbers . In a demonstration to the American Mathematical Society conference at Dartmouth College on September 11 , 1940 , Stibitz was able to send the Complex Number Calculator remote commands over telephone lines by a Teletype . It was the first computing machine ever used remotely over a phone line. Some participants of the conference who witnessed the demonstration were John Von Neumann , John Mauchly , and Norbert Wiener , who wrote about it in his memoirs. REPRESENTATION A binary number can be represented by any sequence of Bit s (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states. The following sequences of symbols could all be interpreted as the same binary numeric value: 1 0 1 0 0 1 1
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rac{12_{10}}{17_{10}}

rac{1100_2}{10001_2}

= 0.10110100 10110100 10110100...₂

The phenomenon that the binary representation of any rational is either terminating or recurring also occurs in other radix-based numeral systems. See, for instance, the explanation in Decimal . Another similarity is the existence of alternative representations for any terminating representation, relying on the fact that 0.111111... is the sum of the Geometric Series 2^-1 + 2^-2 + 2^-3 + ... which is 1.

Binary numerals which neither terminate nor recur represent Irrational Number s. For instance,

0.10100100010000100000100.... does have a pattern, but it is not a fixed-length recurring pattern, so the number is irrational

1.0110101000001001111001100110011111110... is the binary representation of $\sqrt{2}$ , the Square Root of 2, another irrational. It has no discernible pattern, although a proof that $\sqrt{2}$ is irrational requires more than this. See Irrational Number .

BINARY HUMOR

"Binary is as easy as 1, 10, 11."

"I'm just 10 people short of a threesome!"

"There are 10 kinds of people in the world—those who understand binary, and those who don't."

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Binary Arithmetic