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First, the following steps can be taken to convert a WHOLE decimal number into binary form. 1. 2. Divide the decimal number by 2; this produces a quotient plus a remainder of either 1 or 0. The remainder, 1 or 0, is the LSD in the equivalent binary number. Divide the quotient found in step (1) by 2; this produces a second quotient plus a remainder of 1 or 0. This remainder, 1 or 0, is the second least signi cant in the binary number. Continue on in this fashion, dividing each quotient by 2, until the quotient is equal to zero plus the nal remainder of 1 or 0, which is the MSD in the binary number. Example 1
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Convert the decimal number 105 to binary form.
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Solution 105 divided by 2 52; plus remainder 1; the LSD 52 divided by 2 26; plus remainder 0; second LSD 26 divided by 2 13; plus remainder 0; third LSD 13 divided by 2 6; plus remainder 1; fourth LSD 6 divided by 2 3; plus remainder 0; fifth LSD 3 divided by 2 1; plus remainder 1; sixth LSD 1 divided by 2 0; plus remainder 1; the MSD Since positional numbers are always written from left to right, with the MSD at the lefthand end, we have that 105 decimal 1101001 binary; answer Next let us consider the conversion of a decimal fraction into its equivalent binary fraction. Actually the procedure is very simple, but takes a lot of words to describe. Let
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CHAPTER 12 Binary Arithmetic
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us therefore rst try to describe the procedure in words, and then, by means of an example, show that the procedure is really very easy. Begin by multiplying the given decimal fraction by 2; if the product is greater than 1 the binary fraction begins as 0.1, but if the product is less than 1 the binary fraction begins as 0.0. 2. If the product found in step (1) is greater than 1, then subtract 1 from the product and then multiply the result by 2; if the result is greater than 1, the binary fraction is now of the form 0.11, but if less than 1 it is of the form 0.10. If, however, the product found in step (1) is less than 1, then multiply it by 2; if the result is greater than 1 the binary fraction is now of the form 0.01, but if less than 1 it is of the form 0.00. We continue on in this fashion to any degree of accuracy required. Example 2
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Convert the decimal fraction 0.403 into binary form.
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Solution Here, multiply and subtract are abbreviated as mult and sub. The given decimal fraction: 0:403 mult by 2: 0:806 mult by 2: 1:612 sub 1; then mult by 2: 1:224 sub 1; then mult by 2: 0:448 mult by 2: 0:896 mult by 2: 1:792 sub 1; then mult by 2: 1:584 sub 1; then mult by 2: 1:168 sub 1; then mult by 2: 0:336 binary: 0:0 0:01 0:011 0:0110 0:01100 0:011001 0:0110011 0:01100111 0:011001110
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because 0:806 < 1 because 1:612 > 1 because 1:224 > 1 because 0:448 < 1 because 0:896 < 1 because 1:792 > 1 because 1:584 > 1 because 1:168 > 1 because 0:336 < 1
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and so on, to whatever accuracy is required. To test the accuracy of the last result, above, let us make use of eq. (554). Since we re dealing entirely with a binary fraction, we need only use the terms with negative exponents in eq. (554); doing this, we nd that 0:011001110 2 2 2 3 2 6 2 7 2 8 0:40234 . . . which may or may not be close enough to 0.403, depending upon accuracy requirements. If you wish to continue with the above example, you can verify, for instance, that 0:0110011100101 0:40295; which is of course closer to 0:403 than before: Although such conversions are very time-consuming when done using pencil and paper, they present no such di culty when done internally in a digital computer; this is because the computer can execute millions of such routine steps per second. Also, in regard to decimal-to-binary conversion in general, it should be pointed out that an integral decimal number always has an exact equivalent in the binary system, but a decimal fraction may or may not have exact representation in the binary system. (However, the binary equivalent of a decimal fraction can always be determined to any desired degree of accuracy.)
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