ALU OPERATIONS ALU operations in the CPU include calculations of integers  and/or  fractions. All  the  computations  are performed using the binary number system. ALU operations also include  signed arithmetic   operations. First we discuss how the binary equivalents of decimal numbers are represented in  fixed-point representation (integers), then   we   discuss   floating-point representation  (fractional).  Fixed-  and  floating-point operations are important for the computer. They make the computer versatile when performing arithmetic and logical types of ALU operations. Fixed-Point  Operations Fixed-point  arithmetic  operations  are  performed  on integral or whole numbers where the binary point is assumed to be to the right of the least significant bit (bit 0). For example, if we have an 8-bit register, we may express integer decimal numbers between 0 and 28 minus 1 (or 255), by converting the decimal number to its binary equivalent. If we have a 16-bit register, we can store integer decimal numbers between 0 and 216 minus 1 (or 65535). Because the binary point is fixed and always to the right of the least significant digit, fractions  are  not  represented.  The  magnitude  or absolute value of the number is always represented by 2N minus 1 where N is the number of bits within the register or memory cell where the number is being stored. In  fixed-point  operations,  the  computer  can perform calculations on signed numbers (positive and negative). The most significant bit (msb) is used as a sign bit. A zero (0) in the msb indicates a positive or true form number, and a one (1) in the msb indicates a negative  or  one’s  complement/radix-minus-1  form number. When dealing with binary numbers, we can take this one step further; we find the two’s complement or radix-minus-2  of  the  number.  It  is  important  to understand   the   concepts   behind   1’s   and   2’s complement. It is the basis by which the computer performs arithmetic and logical calculations. Now if you want to accommodate an equal amount of positive and  negative  numbers,  a  16-bit  register  can  contain numbers from –32768 to +32767 or –215 to 215 minus 1. The reason they are not both 215 is because one combination is taken up for the zero value. This is more easily  seen  if  we  examine  a  4-bit  register.  The combinations are shown in table 5-2. 5-20 Table 5-2.—Binary and Decimal Values of a 4-Bit Register That is, there are 23 or 2N combinations and one combination is for the number zero. Negative numbers are represented by their two’s complement and the most significant bit (regardless of the word or operand size) is the sign bit. Fixed-point  operations  can  include double-length arithmetic operations, where operands contain 64 bits and bit 263 is the sign bit. Floating-Point Operations Floating-point operations are used to simplify the addition,  subtraction,  multiplication,  and  division  of fractional numbers. They are used when dealing with fractional numbers, such as 5.724 or a very large number   and   signed   fractional   numbers.   When performing  arithmetic  operations  involving  fractions  or very large numbers, it is necessary to know the location of the binary (radix) point and to properly align this point   before   the   arithmetic   operation.   For floating-point  operations,  the  location  of  the  binary point will depend on the  format of the computer. All numbers are placed in this format before the arithmetic operation. The factional portion of the number is called the mantissa and the whole integer portion, indicating the scaled factor or exponent, is called the characteristic.


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