Understanding Integer Representations in Computer Arithmetic: ALU, Unsigned, Sign, Lecture notes of Logic

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Computer

Arithmetic:

Arithmetic and

Logic Unit (ALU)

Arithmetic & Logic Unit (ALU)

• Part of the computer that actually performs arithmetic and logical operations on data
• All of the other elements of the computer system are there mainly to bring data into the ALU for it to process and then to take the results back out
• Based on the use of simple digital logic devices that can store binary digits and perform simple Boolean logic operations

Integer Representations

• In the binary number system arbitrary numbers can be represented with: - The digits zero and one - The minus sign (for negative numbers) - The period, or radix point (for numbers with a fractional component)
• For purposes of computer storage and processing we do not have the benefit of special symbols for the minus sign and radix point
• Only binary digits (0,1) may be used to represent numbers

Integer Representations

• There are 4 commonly known (1 not common) integer representations.
• All have been used at various times for various reasons. 1. Unsigned 2. Sign Magnitude 3. One’s Complement 4. Two’s Complement 5. Biased (not commonly known)

1. Unsigned (Cont’d.)

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2. Sign-Magnitude

conventions used to represent^ There are several alternative negative as well as positive integers Sign-magnitude representation is the simplest form that employs a sign bit

Drawbacks:

Because of these drawbacks, sign-magnitude representation is rarely used in implementing the integer portion of the ALU

• All of these alternatives involve treating themost significant (leftmost) bit in the word as
• a sign bitIf the sign bit is 0 the number is positive
• If the sign bit is 1 the number is negative
  • Addition and subtraction require aconsideration of both the signs of the numbers and their relative magnitudes tocarry out the required operation
• There are two representations of 0

2. Sign-Magnitude (Cont’d.)

• Left most bit is sign bit
• 0 means positive
• 1 means negative
• +18 = 00010010
• -18 = 10010010

2-

3. One’s Complement

• Used to get two’s complement integers.
• Nowadays, it is not being applied to any of the

machines.

• Stated in this slide for historical purpose.

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5. Biased

• an integer representation that skews the bit

patterns so as to look just like unsigned but actually represent negative numbers.

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Table 10. Alternative Representations for 4-Bit Integers

Range of Numbers

• 8 bit 2s complement
  • +127 = 01111111 = 2^7 -
  • -128 = 10000000 = -2^7
• 16 bit 2s complement
  • +32767 = 011111111 11111111 = 2^15 - 1
  • -32768 = 100000000 00000000 = -2^15

2-

Negation

• Twos complement operation
  • Take the Boolean complement of each bit of the integer (including the sign bit)
  • Treating the result as an unsigned binary integer, add 1
• The negative of the negative of that number is itself:

+18 = 00010010 (twos complement) bitwise complement = 11101101

• 1 11101110 = -

-18 = 11101110 (twos complement) bitwise complement = 00010001

• 1 00010010 = +

Negation Special Case 2

-128 = 10000000 (twos complement) Bitwise complement = 01111111 Add 1 to LSB + 1 Result 10000000 So: -(-128) = -128 X Monitor MSB (sign bit) It should change during negation

OVERFLOW RULE:

If two numbers are added,

and they are both positive or

both negative, then overflow

occurs if and only if the

result has the opposite sign.