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Digital Arithmetic and Arithmetic Circuits, Lecture notes of Engineering

Wayne State CollegeEngineering

System level design of digital logic circuits using hardwired and programmable logic devices. ROMs, PROMs, and PLAs. Synchronous and asynchronous circuit design and analysis

Typology: Lecture notes

2021/2022

Available from 12/22/2022

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Digital Arithmetic 1
Digital Arithmetic and Arithmetic Circuits
(Chapter 7 Dueck)
•Signed Binary Number:
A binary number of fixed length whose sign (+/–)
is represented by one bit (usually MSB) and its
magnitude by the remaining bits.
•Unsigned Binary Number:
A binary number of fixed length whose sign is not
specified by a bit. All bits are magnitude and the
sign is assumed +.
Signed/Unsigned Binary Numbers
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(Chapter 7 Dueck)

  • Signed Binary Number: A binary number of fixed length whose sign (+/–) is represented by one bit (usually MSB) and its magnitude by the remaining bits.
  • Unsigned Binary Number: A binary number of fixed length whose sign is not specified by a bit. All bits are magnitude and the sign is assumed +.

Signed/Unsigned Binary Numbers

(Chapter 7 Dueck)

  • Sum:
    • Result of an Addition Operation of two (or more) binary numbers (operands).
  • Carry:
    • A digit (or bit) that is carried over to the next most significant bit during an n-Bit addition operation.
  • The carry bit is a 1 if the result was too large to

be expressed in n bits.

Unsigned Binary Arithmetic

(Chapter 7 Dueck)

Signed Binary Numbers – 1

  • Sign Bit:
    • A bit (usually the MSB) that indicates

whether a number is positive (= 0) or

negative (= 1).

  • Magnitude Bits:
    • The bits of a signed binary number that tell

how large it is in value.

(Chapter 7 Dueck)

Signed Binary Numbers – 2

  • True-Magnitude Form:
    • A form of signed binary whose magnitude

bits are the TRUE binary form (not

complements).

(Chapter 7 Dueck)

Signed Binary Addition (8-Bit)

  • Signed Addition Positive (S = 0)
  • Similar to binary addition with a sign

bit.

0110 1001

75 0100 1011

30 0001 1110

(Chapter 7 Dueck)

Signed Binary Numbers – 3

  • 1’s Complement:
    • A form of signed binary in which negative numbers are created by complementing all bits.
  • 2’s Complement:
    • A form of signed binary in which the negative numbers are created by complementing all the bits and adding a 1 (1’s Complement + 1).

(Chapter 7 Dueck)

  • Used in MPU (PC) Arithmetic

2’s Complement Form

1011 1000

1

  • 72 1011 0111

72 0100 1000

1011 0111

1

  • 57 1100 0110

57 0011 1001

(Chapter 7 Dueck)

  • Add 2’s Complement to Minuend.

DiscardCarryBitfromResult

100001111

65 10111111 10111111

80 01010000 01010000

80 65?

2’s Complement Subtraction

10111111

65 10111110 1

65 01000001

2'scomplimentof 65

(Chapter 7 Dueck)

Excess-3 Code

  • A BCD Code formed by adding 3 (0011) to its

true 4-bit binary value.

  • Excess-3 is a self-complementing code:
    • A negative code equivalent can be found by inverting the binary bits of the positive code
  • Inverting the bits of the Excess-3 digit yields

9’s Complement of the decimal equivalent.

(Chapter 7 Dueck)

Excess-3 Examples

  • 3 = 0011 + 0011 = 0110 = 6 in E3.
  • 1 = 0001 + 0011 = 0100 = 4 in E3.
  • If we complement 1 = 1011 in E3, this

is the code for an 8.

  • 9’s Complement of 1 = (9 – 1) = 8 (Self-

Complement)