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It is the back end of all computer First you need to convert each letter From simple mechanics to sophisticated quantum modeling, our world has evolved greatly over time Strings of 0s and 1s, Binary numbers are often used to operate computers.

But why is that? Why do While the concept behind the binary system may seem like something that was used by the earlier Register Login. Copy Clear.

Early forms of this system can be found in documents from the Fifth Dynasty of Egypt , approximately BC, and its fully developed hieroglyphic form dates to the Nineteenth Dynasty of Egypt , approximately BC.

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 BC. It is based on taoistic duality of yin and yang.

The Song Dynasty scholar Shao Yong — rearranged the hexagrams in a format that resembles modern binary numbers, although he did not intend his arrangement to be used mathematically.

The Indian scholar Pingala c. 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.

Four short syllables "" is the first pattern and corresponds to the value one. The numerical value is obtained by adding one to the sum of place values.

The residents of the island of Mangareva in French Polynesia were using a hybrid binary- decimal system before In the late 13th century Ramon Llull had the ambition to account for all wisdom in every branch of human knowledge of the time.

In Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded as scarcely visible variations in the font in any random text.

John Napier in described a system he called location arithmetic for doing binary calculations using a non-positional representation by letters.

Thomas Harriot investigated several positional numbering systems, including binary, but did not publish his results; they were found later among his papers.

The full title of Leibniz's article is translated into English as the "Explanation of Binary Arithmetic, which uses only the characters 1 and 0, with some remarks on its usefulness, and on the light it throws on the ancient Chinese figures of Fu Xi ".

An example of Leibniz's binary numeral system is as follows: [19]. Leibniz interpreted the hexagrams of the I Ching as evidence of binary calculus.

Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own religious beliefs as a Christian. He believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing.

Now one can say that nothing in the world can better present and demonstrate this power than the origin of numbers, as it is presented here through the simple and unadorned presentation of One and Zero or Nothing.

In , British mathematician George Boole published a landmark paper detailing an algebraic system of logic that would become known as Boolean algebra.

His logical calculus was to become instrumental in the design of digital electronic circuitry. In , 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.

In November , George Stibitz , then working at Bell Labs , completed a relay-based computer he dubbed the "Model K" for " K itchen", where he had assembled it , which calculated using binary addition.

Their Complex Number Computer, completed 8 January , was able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 September , 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.

The Z1 computer , which was designed and built by Konrad Zuse between and , used Boolean logic and binary floating point numbers.

Any number can be represented by a sequence of bits binary digits , which in turn may be represented by any mechanism capable of being in two mutually exclusive states.

Any of the following rows of symbols can be interpreted as the binary numeric value of The numeric value represented in each case is dependent upon the value assigned to each symbol.

In the earlier days of computing, switches, punched holes and punched paper tapes were used to represent binary values. A "positive", " yes ", or "on" state is not necessarily equivalent to the numerical value of one; it depends on the architecture in use.

In keeping with customary representation of numerals using Arabic numerals , binary numbers are commonly written using the symbols 0 and 1.

When written, binary numerals are often subscripted, prefixed or suffixed in order to indicate their base, or radix.

The following notations are equivalent:. When spoken, binary numerals are usually read digit-by-digit, in order to distinguish them from decimal numerals.

For example, the binary numeral is pronounced one zero zero , rather than one hundred , to make its binary nature explicit, and for purposes of correctness.

Since the binary numeral represents the value four, it would be confusing to refer to the numeral as one hundred a word that represents a completely different value, or amount.

Alternatively, the binary numeral can be read out as "four" the correct value , but this does not make its binary nature explicit. Counting in binary is similar to counting in any other number system.

Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar decimal counting system as a frame of reference.

Decimal counting uses the ten symbols 0 through 9. Counting begins with the incremental substitution of the least significant digit rightmost digit which is often called the first digit.

When the available symbols for this position are exhausted, the least significant digit is reset to 0 , and the next digit of higher significance one position to the left is incremented overflow , and incremental substitution of the low-order digit resumes.

This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:. Binary counting follows the same procedure, except that only the two symbols 0 and 1 are available.

Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:.

In the binary system, each digit represents an increasing power of 2, with the rightmost digit representing 2 0 , the next representing 2 1 , then 2 2 , and so on.

For example, the binary number is converted to decimal form as follows:. Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator.

Arithmetic in binary is much like arithmetic in other numeral systems. Addition, subtraction, multiplication, and division can be performed on binary numerals.

The simplest arithmetic operation in binary is addition. Adding two single-digit binary numbers is relatively simple, using a form of carrying:.

Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix 10 , the digit to the left is incremented:.

This is known as carrying. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:.

In this example, two numerals are being added together: 2 13 10 and 2 23 The top row shows the carry bits used.

The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. This time, a 1 is carried, and a 1 is written in the bottom row.

Proceeding like this gives the final answer 2 36 decimal. This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones.

It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of n ones where: n is any integer length , adding 1 will result in the number 1 followed by a string of n zeros.

That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s:.

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations.

In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 0 2 10 and 1 0 1 0 1 1 0 0 1 1 2 10 , using the traditional carry method on the left, and the long carry method on the right:.

Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series.

The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique.

Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 1 2 In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column. This is known as borrowing.

The principle is the same as for carrying. Subtracting a positive number is equivalent to adding a negative number of equal absolute value.

Computers use signed number representations to handle negative numbers—most commonly the two's complement notation.

Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:.

Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B , the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used.

The sum of all these partial products gives the final result. Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:.

Binary numbers can also be multiplied with bits after a binary point :. See also Booth's multiplication algorithm.

Long division in binary is again similar to its decimal counterpart. In the example below, the divisor is 2 , or 5 in decimal, while the dividend is 2 , or 27 in decimal.

The procedure is the same as that of decimal long division ; here, the divisor 2 goes into the first three digits 2 of the dividend one time, so a "1" is written on the top line.

This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit a "1" is included to obtain a new three-digit sequence:.

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:.

Thus, the quotient of 2 divided by 2 is 2 , as shown on the top line, while the remainder, shown on the bottom line, is 10 2.

In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:.

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators.

When a string of binary symbols is manipulated in this way, it is called a bitwise operation ; the logical operators AND , OR , and XOR may be performed on corresponding bits in two binary numerals provided as input.

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The image shown above might remind you of the Matrix movie series, but it still does not make sense Binary code is what all computer language is made of, and runs in the background for all computing All computer language is based in binary code.

It is the back end of all computer First you need to convert each letter From simple mechanics to sophisticated quantum modeling, our world has evolved greatly over time Strings of 0s and 1s, Binary numbers are often used to operate computers.

But why is that? Why do While the concept behind the binary system may seem like something that was used by the earlier Register Login. Copy Clear.

More on Binary Converter. Translating Text to Binary Converting text to Binary is a two step process. For example, the binary number is converted to decimal form as follows:.

Fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator. Arithmetic in binary is much like arithmetic in other numeral systems.

Addition, subtraction, multiplication, and division can be performed on binary numerals. The simplest arithmetic operation in binary is addition.

Adding two single-digit binary numbers is relatively simple, using a form of carrying:. Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column.

This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix 10 , the digit to the left is incremented:.

This is known as carrying. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:.

In this example, two numerals are being added together: 2 13 10 and 2 23 The top row shows the carry bits used. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column.

This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 2 36 decimal.

This method is generally useful in any binary addition in which one of the numbers contains a long "string" of ones.

It is based on the simple premise that under the binary system, when given a "string" of digits composed entirely of n ones where: n is any integer length , adding 1 will result in the number 1 followed by a string of n zeros.

That concept follows, logically, just as in the decimal system, where adding 1 to a string of n 9s will result in the number 1 followed by a string of n 0s:.

Such long strings are quite common in the binary system. From that one finds that large binary numbers can be added using two simple steps, without excessive carry operations.

In the following example, two numerals are being added together: 1 1 1 0 1 1 1 1 1 0 2 10 and 1 0 1 0 1 1 0 0 1 1 2 10 , using the traditional carry method on the left, and the long carry method on the right:.

Instead of the standard carry from one column to the next, the lowest-ordered "1" with a "1" in the corresponding place value beneath it may be added and a "1" may be carried to one digit past the end of the series.

The "used" numbers must be crossed off, since they are already added. Other long strings may likewise be cancelled using the same technique.

Then, simply add together any remaining digits normally. Proceeding in this manner gives the final answer of 1 1 0 0 1 1 1 0 0 0 1 2 In our simple example using small numbers, the traditional carry method required eight carry operations, yet the long carry method required only two, representing a substantial reduction of effort.

Subtracting a "1" digit from a "0" digit produces the digit "1", while 1 will have to be subtracted from the next column.

This is known as borrowing. The principle is the same as for carrying. Subtracting a positive number is equivalent to adding a negative number of equal absolute value.

Computers use signed number representations to handle negative numbers—most commonly the two's complement notation.

Such representations eliminate the need for a separate "subtract" operation. Using two's complement notation subtraction can be summarized by the following formula:.

Multiplication in binary is similar to its decimal counterpart. Two numbers A and B can be multiplied by partial products: for each digit in B , the product of that digit in A is calculated and written on a new line, shifted leftward so that its rightmost digit lines up with the digit in B that was used.

The sum of all these partial products gives the final result. Since there are only two digits in binary, there are only two possible outcomes of each partial multiplication:.

Binary numbers can also be multiplied with bits after a binary point :. See also Booth's multiplication algorithm. Long division in binary is again similar to its decimal counterpart.

In the example below, the divisor is 2 , or 5 in decimal, while the dividend is 2 , or 27 in decimal. The procedure is the same as that of decimal long division ; here, the divisor 2 goes into the first three digits 2 of the dividend one time, so a "1" is written on the top line.

This result is multiplied by the divisor, and subtracted from the first three digits of the dividend; the next digit a "1" is included to obtain a new three-digit sequence:.

The procedure is then repeated with the new sequence, continuing until the digits in the dividend have been exhausted:. Thus, the quotient of 2 divided by 2 is 2 , as shown on the top line, while the remainder, shown on the bottom line, is 10 2.

In decimal, this corresponds to the fact that 27 divided by 5 is 5, with a remainder of 2. Aside from long division, one can also devise the procedure so as to allow for over-subtracting from the partial remainder at each iteration, thereby leading to alternative methods which are less systematic, but more flexible as a result.

The process of taking a binary square root digit by digit is the same as for a decimal square root and is explained here. An example is:.

Though not directly related to the numerical interpretation of binary symbols, sequences of bits may be manipulated using Boolean logical operators.

When a string of binary symbols is manipulated in this way, it is called a bitwise operation ; the logical operators AND , OR , and XOR may be performed on corresponding bits in two binary numerals provided as input.

The logical NOT operation may be performed on individual bits in a single binary numeral provided as input. Sometimes, such operations may be used as arithmetic short-cuts, and may have other computational benefits as well.

For example, an arithmetic shift left of a binary number is the equivalent of multiplication by a positive, integral power of 2.

To convert from a base integer to its base-2 binary equivalent, the number is divided by two. The remainder is the least-significant bit. The quotient is again divided by two; its remainder becomes the next least significant bit.

This process repeats until a quotient of one is reached. The sequence of remainders including the final quotient of one forms the binary value, as each remainder must be either zero or one when dividing by two.

For example, 10 is expressed as 2. Conversion from base-2 to base simply inverts the preceding algorithm. The bits of the binary number are used one by one, starting with the most significant leftmost bit.

Beginning with the value 0, the prior value is doubled, and the next bit is then added to produce the next value. This can be organized in a multi-column table.

For example, to convert 2 to decimal:. The result is The first Prior Value of 0 is simply an initial decimal value. This method is an application of the Horner scheme.

The fractional parts of a number are converted with similar methods. They are again based on the equivalence of shifting with doubling or halving. In a fractional binary number such as 0.

Double that number is at least 1. This suggests the algorithm: Repeatedly double the number to be converted, record if the result is at least 1, and then throw away the integer part.

Thus the repeating decimal fraction 0. This is also a repeating binary fraction 0. It may come as a surprise that terminating decimal fractions can have repeating expansions in binary.

It is for this reason that many are surprised to discover that 0. The final conversion is from binary to decimal fractions. The only difficulty arises with repeating fractions, but otherwise the method is to shift the fraction to an integer, convert it as above, and then divide by the appropriate power of two in the decimal base.

For example:. For very large numbers, these simple methods are inefficient because they perform a large number of multiplications or divisions where one operand is very large.

A simple divide-and-conquer algorithm is more effective asymptotically: given a binary number, it is divided by 10 k , where k is chosen so that the quotient roughly equals the remainder; then each of these pieces is converted to decimal and the two are concatenated.

Given a decimal number, it can be split into two pieces of about the same size, each of which is converted to binary, whereupon the first converted piece is multiplied by 10 k and added to the second converted piece, where k is the number of decimal digits in the second, least-significant piece before conversion.

Binary may be converted to and from hexadecimal more easily. This is because the radix of the hexadecimal system 16 is a power of the radix of the binary system 2.

To convert a hexadecimal number into its binary equivalent, simply substitute the corresponding binary digits:.

To convert a binary number into its hexadecimal equivalent, divide it into groups of four bits. If the number of bits isn't a multiple of four, simply insert extra 0 bits at the left called padding.

To convert a hexadecimal number into its decimal equivalent, multiply the decimal equivalent of each hexadecimal digit by the corresponding power of 16 and add the resulting values:.

Binary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two namely, 2 3 , so it takes exactly three binary digits to represent an octal digit.

The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary is equivalent to the octal digit 0, binary is equivalent to octal 7, and so forth.

Converting from octal to binary proceeds in the same fashion as it does for hexadecimal :. Non-integers can be represented by using negative powers, which are set off from the other digits by means of a radix point called a decimal point in the decimal system.

For example, the binary number Other rational numbers have binary representation, but instead of terminating, they recur , with a finite sequence of digits repeating indefinitely.

For instance. 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.

Binary numerals which neither terminate nor recur represent irrational numbers. For instance,. From Wikipedia, the free encyclopedia.

Number expressed though the symbols 0 and 1.

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