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CHAPTER 12 BINARY-TO-DECIMAL CONVERSION
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Binary Arithmetic
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To convert an integral (whole) binary number to decimal form we move left to right through the binary number, from the MSD to the LSD, as follows. Multiply the MSD by 2, then add on the next digit. Multiply the result by 2, then add on the next digit. Multiply this result by 2, then add on the next digit. Continue on until the last digit to the right (the LSD) is included in the conversion. Example 3
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Convert the binary number 1101010 to decimal form.
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Solution Beginning at the left-hand end of the given binary number and using the above procedure, we can chart the results as follows: binary: decimal: 1 2 1 3 0 6 1 13 0 26 1 53 0 106
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Thus; 1101010 bi 106 dec; answer Next, to convert a BINARY FRACTION into a DECIMAL FRACTION we move through the binary fraction from right to left toward the binary point in the following manner. Divide the right-hand digit by 2, then add to this the next digit and divide the result by 2. Now add, to the last result, the next digit and divide the result by 2. Continue on in this way until the binary point is reached. Example 4
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Convert the binary fraction 0.011011 to decimal form.
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Solution Beginning at the right-hand end, and following the above procedure, we can chart the results as follows. binary: :0 1 1 0 1 1 0:5 0:75 0:375 0:6875 0:84375 0:421875; thus 0:011011 bi 0:421875 dec; answer CHECK: 0:011011 2 2 2 3 2 5 2 6 1 1 1 1 0:421875 4 8 32 64
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Now let us continue on, and study the basic arithmetic ( air ith MET ik ) operations of addition, subtraction, multiplication, and division in both the decimal and binary systems. Let us begin with addition, rst in the familiar decimal system, as follows.
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CHAPTER 12 Binary Arithmetic
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As we know, the decimal system employs the radix ten, making use of the ten digits 0; 1; 2; . . . ; 9. In the decimal system the number ten is denoted by 10. Now, in regard to the sum of any two of the ten digits, two possibilities arise; thus 1. 2. the sum of the two digits is less than the radix ten, or the sum of the two digits is equal to or greater than the radix ten.
In case (1) no di culty arises; thus, 2 5 7, for example. In case (2), however, a problem does arise in taking a sum such as 8 8, for example, because no single digit exists in the decimal system to denote the quantity sixteen. Instead, to indicate in number form the quantity sixteen we write 16, in which the 1 now has the value ten. This is the carry operation that we are all familiar with in the decimal system; if (in the decimal system) the sum of two digits is equal to or greater than the radix ten, we write down the required digit in the units column and carry the 1 to the left into the tens column where the 1 now has the value ten. If the numbers consist of more than one digit we use the same procedure of carrying a 1 into the next higher valued column; for example 6 6 5 8 9 6 2 6 1 6 2 8 4 Now consider the addition of two digits in the binary system. Here there are only two possible two-digit sums, 1 0 1, and 1 1 two. In the binary system, however, there is no single digit to represent the value two ; hence we carry a 1 into the twos column, which is the next column to the left, and write the result of 1 1 in the form 1 1 10 where the 1 in the 10 now represents the value two. Now consider, as an example, the addition of the two binary numbers 1011 and 1101, as indicated below to the left eights column fours column twos column units column 1 0 1 1 1 1 1 0 1 1 0 11 dec 13 dec 24 dec ! ! ! 0 1 ! 1 1 1.
The procedure for performing the addition indicated to the left above is: the sum of the digits in the units column is 1 1 10; hence we write down 0 and carry 1 into the twos column, 2. the sum of the digits now in the twos column is 1 1 10; hence write down 0 and carry a 1 into the fours column, 3. the sum of the digits now in the fours column is 1 1 10; hence we write 0 and carry a 1 into the eights column,
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