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Binary numbers

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binary numbers

In mathematics and digital electronicsa binary number is a number expressed in the binary numeral system or base-2 numeral system which represents numeric values using two different symbols: typically zero and one.

The base numbers is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gatesthe binary system is used internally by almost all modern computers and computer-based devices. Each digit is referred to as a bit. Leibniz was specifically inspired by the Chinese I Ching.

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 numbers Horusalthough this has been disputed. In this method, multiplying one number by a second is performed by a sequence binary 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 Papyruswhich dates to around BC. The base-2 system utilized in geomancy had long been widely applied in sub-Saharan Africa. In Francis Bacon discussed a system whereby letters of the alphabet could be reduced to sequences of binary digits, which could then be encoded numbers scarcely visible variations in the font in any random text. 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.

InBritish 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.

Their Complex Number Computer, completed 8 Januarywas able to calculate complex numbers. In a demonstration to the American Mathematical Society conference at Dartmouth College on 11 SeptemberStibitz was able to send the Complex Number Calculator remote commands over telephone numbers 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 NeumannJohn Mauchly and Norbert Wienerwho wrote about it in his memoirs.

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 a computer, the numeric values may be represented by two different voltages ; on a magnetic diskmagnetic polarities may be used.

A "positive", " yes ", or "on" state is not binary equivalent to the numerical value of one; it depends on the architecture in use. Numbers keeping with customary representation of numerals using Arabic numeralsbinary numbers are commonly written using the symbols and 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 zerorather than one hundredto 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 valuebut 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 through Counting begins with the incremental substitution of the least binary 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 0and the next digit of higher significance one position to the left is incremented overflowand incremental substitution of the low-order digit resumes.

This method of reset and overflow is repeated for each digit of significance. Counting progresses binary follows: Binary counting binary the same procedure, except that only the two symbols and 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 representingthe next representingthenand so on. The equivalent decimal representation of a binary number is sum of the powers of 2 which each digit represents.

For example, the binary number is converted to decimal form as follows: Fractions in binary only terminate if the denominator has as the only prime factor. 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 binary in decimal when certain single-digit numbers binary added together; if the result equals or exceeds the value of the radixthe digit to the left is incremented: This is known as carrying. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix that is, to the left, adding it to the next positional value.

This is correct since the next position has a weight that is higher by a factor equal to the radix. Proceeding like this gives the final answer decimal. A simplification for many binary addition problems is the Long Carry Method or Brookhouse Method of Binary Addition. This method is generally useful in any binary addition where 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 lengthadding 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 numbers a string of n 0s Binary Decimal likewise ——————————— ——————————— 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. 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 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.

This is known as borrowing. The principle is the same as for carrying. Such representations eliminate the need for a separate "subtract" operation. Two numbers A and B can be multiplied by partial products: for each digit in Bthe 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 numbers B that was used. The sum of all these partial products gives the final result.

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

The process of taking a binary square root digit by digit is the same as for a decimal square root, and is explained here. When a string of binary symbols is manipulated in this way, it is called a bitwise operation ; the logical operators BinaryORand 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 numbers, 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-10 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. 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 to decimal: The result is Note that 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. 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. It is for this reason that many are surprised to discover thatadditions differs from 1 in floating point arithmetic.

In fact, the only binary fractions with terminating expansions are of the form of an integer divided by a power of 2, which is not. 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 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 numbers, it is divided by kwhere 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 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 somewhat more easily. 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.

For example: 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,so it takes exactly three binary digits to represent an octal digit.

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

Other rational numbers have binary representation, but instead of terminating, they recurwith a finite sequence of digits repeating indefinitely.

For instance The binary 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. Binary numerals binary neither terminate nor recur represent irrational numbers. Leibnizens Novissima Sinica von Internationales Symposium, Berlin 4. Oktober Stuttgart: Franz Steiner Verlag. Boca Raton, Florida: CRC Press. Binary of the National Academy binary Sciences. Gerhardt, Berlinvol.

Smith Leibniz: What Kind of Rationalist? New York: Cambridge University Press. Computer History Association of California. By using this site, you agree to the Terms of Use and Privacy Policy.

binary numbers

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