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# how many digits are used in binary number system

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## Binary number

Place-value notation

show Sign-value notation

List of numeral systems

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A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implemention.[1]

## Contents

1 History 1.1 Egypt 1.2 China 1.3 India 1.4 Other cultures

1.5 Western predecessors to Leibniz

1.6 Leibniz and the I Ching

1.7 Later developments

2 Representation

3 Counting in binary

3.1 Decimal counting

3.2 Binary counting 4 Fractions 5 Binary arithmetic 5.1 Addition

5.1.1 Long carry method

5.2 Subtraction 5.3 Multiplication

5.3.1 Multiplication table

5.4 Division 5.5 Square root

6 Bitwise operations

7 Conversion to and from other numeral systems

7.1 Decimal to Binary

7.2 Binary to Decimal

8 Representing real numbers

## History

The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. Leibniz was specifically inspired by the Chinese I Ching.

### Egypt

Arithmetic values thought to have been represented by parts of the Eye of Horus

The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions (not related to the binary number system) and Horus-Eye fractions (so called because many historians of mathematics believe that the symbols used for this system could be arranged to form the eye of Horus, although this has been disputed).[2] Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. Early forms of this system can be found in documents from the Fifth Dynasty of Egypt, approximately 2400 BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt, approximately 1200 BC.[3]

The method used for ancient Egyptian multiplication is also closely related to binary numbers. In this method, multiplying one number by a second is performed by a sequence of steps in which a value (initially the first of the two numbers) is either doubled or has the first number added back into it; the order in which these steps are to be performed is given by the binary representation of the second number. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, which dates to around 1650 BC.[4]

### China

Daoist Bagua

The I Ching dates from the 9th century BC in China.[5] The binary notation in the is used to interpret its quaternary divination technique.[6]

It is based on taoistic duality of yin and yang.[7] Eight trigrams (Bagua) and a set of 64 hexagrams ("sixty-four" gua), analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.[5]

The Song Dynasty scholar Shao Yong (1011–1077) rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.[6] Viewing the least significant bit on top of single hexagrams in Shao Yong's square and reading along rows either from bottom right to top left with solid lines as 0 and broken lines as 1 or from top left to bottom right with solid lines as 1 and broken lines as 0 hexagrams can be interpreted as sequence from 0 to 63. [8]

### India

The Indian scholar Pingala (c. 2nd century BC) developed a binary system for describing prosody.[9][10] He used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code.[11][12] They were known as (light) and (heavy) syllables.

Pingala's Hindu classic titled Chandaḥśāstra (8.23) describes the formation of a matrix in order to give a unique value to each meter. "Chandaḥśāstra" literally translates to in Sanskrit. The binary representations in Pingala's system increases towards the right, and not to the left like in the binary numbers of the modern positional notation.[11][13] In Pingala's system, the numbers start from number one, and not zero. Four short syllables "0000" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values.[14]

स्रोत : en.wikipedia.org

## binary number system

binary number system, in mathematics, positional numeral system employing 2 as the base and so requiring only two different symbols for its digits, 0 and 1, instead of the usual 10 different symbols needed in the decimal system. The numbers from 0 to 10 are thus in binary 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, and 1010. The importance of the binary system to information theory and computer technology derives mainly from the compact and reliable manner in which 0s and 1s can be represented in electromechanical devices with two states—such as “on-off,” “open-closed,” or “go–no go.”

## binary number system

mathematics

Alternate titles: base-2 number system

By The Editors of Encyclopaedia Britannica • Edit History

Key People: Gottfried Wilhelm Leibniz

Related Topics: byte bit number system positional numeral system

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binary number system, in mathematics, positional numeral system employing 2 as the base and so requiring only two different symbols for its digits, 0 and 1, instead of the usual 10 different symbols needed in the decimal system. The numbers from 0 to 10 are thus in binary 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, and 1010. The importance of the binary system to information theory and computer technology derives mainly from the compact and reliable manner in which 0s and 1s can be represented in electromechanical devices with two states—such as “on-off,” “open-closed,” or “go–no go.” (See numerals and numeral systems: The binary system.)

स्रोत : www.britannica.com