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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.
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 Please tell us where you read or heard it including the quote, if possible. Test Your Knowledge - and learn some interesting things along the way.
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Keep scrolling for more. Did You Know? Examples of binary in a Sentence Adjective a binary star is a system of two stars that revolve around each other under their mutual gravitation.
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First Known Use of binary Noun 15th century, in the meaning defined at sense 1 Adjective , in the meaning defined at sense 1.
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