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whenever the sum of two digits exceeds the largest single-digit number in the given number system, which is 1 in the binary system The carry is treated exactly as in the decimal system A few examples of binary addition are shown in Figure 134, with their decimal counterparts
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Binary 101 +110 1011
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3 _ _ (Note that in this example, 325 = 31 and 575 = 54) 4
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Figure 134 Examples of binary addition
The procedure for subtracting binary numbers is based on the rules of Table 133 A few examples of binary subtraction are given in Figure 135, with their decimal counterparts
Table 133 Rules for subtraction 0 0=0 1 0=1 1 1=0 0 1 = 1 (with a borrow of 1)
Decimal 9 5 4
Binary 1001 101 0100
Decimal 16 3 13
Binary 10000 11 01101
Decimal 625 450 175
Binary 11001 10010 00111
Figure 135 Examples of binary subtraction
Table 134 Rules for multiplication 0 0=0 0 1=0 1 0=0 1 1=1
Multiplication and Division Whereas in the decimal system the multiplication table consists of 102 = 100 entries, in the binary system we only have 22 = 4 entries Table 134 represents the complete multiplication table for the binary number system Division in the binary system is also based on rules analogous to those of the decimal system, with the two basic laws given in Table 135 Once again, we need be concerned with only two cases, and just as in the decimal system, division by zero is not contemplated Conversion from Decimal to Binary The conversion of a decimal number to its binary equivalent is performed by successive division of the decimal number by 2, checking for the remainder each time Figure 136 illustrates this idea with an example The result obtained in Figure 136 may be easily veri ed by performing the opposite conversion, from binary to decimal: 110001 = 25 + 24 + 20 = 32 + 16 + 1 = 49 The same technique can be used for converting decimal fractional numbers to their binary form, provided that the whole number is separated from the fractional part and each is converted to binary form (separately), with the results added at the
Table 135 Rules for division 0 1=0 1 1=1
Remainder 49 24 12 6 3 1 2 = 24 + 1 2 = 12 + 0 2= 6+0 2= 3+0 2= 1+1 2= 0+1 492 = 1100012
Figure 136 Example of conversion from decimal to binary
13
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37 18 9 4 2 1
Remainder 2 = 18 + 1 2= 9+0 2= 4+1 2= 2+0 2= 1+0 2= 0+1 3710 = 1001012
2 053 = 106 1 2 006 = 012 0 2 012 = 024 0 2 024 = 048 0 2 048 = 096 0 2 096 = 192 1 2 092 = 184 1 2 084 = 168 1 2 068 = 136 1 2 036 = 072 0 2 072 = 144 1 05310 = 010000111101
Figure 137 Conversion from decimal to binary
end Figure 137 outlines this procedure by converting the number 3753 to binary form The procedure is outlined in two steps First, the integer part is converted; then, to convert the fractional part, one simple technique consists of multiplying the decimal fraction by 2 in successive stages If the result exceeds 1, a 1 is needed to the right of the binary fraction being formed (100101 , in our example) Otherwise, a 0 is added This procedure is continued until no fractional terms are left In this case, the decimal part is 05310 , and Figure 137 illustrates the succession of calculations Stopping the procedure outlined in Figure 137 after 11 digits results in the following approximation: 375310 = 10010110000111101 Greater precision could be attained by continuing to add binary digits, at the expense of added complexity Note that an in nite number of binary digits may be required to represent a decimal number exactly Complements and Negative Numbers To simplify the operation of subtraction in digital computers, complements are used almost exclusively In practice, this corresponds to replacing the operation X Y with the operation X + ( Y ) This procedure results in considerable simpli cation, since the computer hardware need include only adding circuitry Two types of complements are used with binary numbers: the one s complement and the two s complement The one s complement of an n-bit binary number is obtained by subtracting the number itself from (2n 1) Two examples are as follows: a = 0101 One s complement of a = (24 1) a = (1111) (0101) = 1010 b = 101101 One s complement of b = (26 1) b = (111111) (101101) = 010010 The two s complement of an n-bit binary number is obtained by subtracting the number itself from 2n Two s complements of the same numbers a and b used in the preceding illustration are computed as follows: a = 0101 Two s complement of a = 24 a = (10000) (0101) = 1011 b = 101101 Two s complement of b = 26 b = (1000000) (101101) = 010011 A simple rule that may be used to obtain the two s complement directly from a binary number is the following: Starting at the least signi cant (rightmost) bit,
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