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CHAPTER 12 Binary Arithmetic
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Setting d 1 in eq. (554), it will be seen that the equivalent decimal system values of the binary 1 digits, relative to the binary point, are binary digit: . . . 1 1 decimal value: 32 16 1 8 1 4 1 1 2 1 # : 1 :5 1 :25 1 :125 ...
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You can now verify, and commit to memory if you wish, that, including zero, the rst 16 binary numbers, with their decimal equivalents, are 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 10 1011 11 1100 12 1101 13 1110 14 1111 15
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Another important point we should be aware of is that, in any positional number system of radix R, it s very easy to multiply or divide by R raised to any positive or negative integral power. For instance, to multiply or divide a decimal number by TEN raised to any integral power n, all we need do is move the decimal point n places to the right or left, as the case may be. In the same way, to multiply or divide a binary number by TWO raised to any integral power n, all we need do is move the binary point n places to the right or left as the case may be. The following table will make this clear.
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Meaning in decimal system one ten hundred thousand ten thousand hundred thousand . . . Meaning in binary system one two four eight sixteen thirty-two . . .
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Number 1 10 100 1000 10000 100000 . . .
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Thus, while it s easy to multiply or divide by any integral power of ten in the decimal system, it s equally easy to multiply or divide by any integral power of two in the binary system. For instance, consider a binary number, N 101101; suppose, for example, that we wish to multiply or divide N by say eight, which is 1000 in the binary system. Either operation is very simply done, as follows, 101101 1000 101101000 and 101101 101:101 1000
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For the same values, the corresponding operations in the decimal system would be 45 times 8 and 45 divided by 8 which, of course, cannot be done by simply moving the decimal point to the right or the left. As you can see from the above, expressing a given value in binary notation requires the writing of many more digits than in decimal notation. Thus, while the binary system is
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admirably suited for internal use in a digital computer, it is not well suited for direct use by human beings because it requires the writing down of so many digits to express even small values. This bears out the statement made previously in regard to R 2 being the ideal choice for internal computer operation, and R 10 being a good compromise for use by human beings. Now let s consider the procedures for converting a decimal number into binary form and vice versa. In stating the rules we ll make use of the terms most signi cant digit (MSD) and least signi cant digit (LSD). The MSD is simply the digit having the most value in a number, while the LSD is the digit having the least value in a number. Thus, in the decimal number 28736, 2 is the MSD, 8 is the next-most signi cant digit, and so on, until we reach 6, which is the LSD. Or, in the binary number 110101, for example, the 1 at the left-hand end is the MSD, while, going from left to right, we nally arrive at the 1 at the right-hand end, which is the LSD. With this terminology in mind, the rules for conversion from decimal to binary, and from binary to decimal, can be summarized as follows.
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