Lecture Digital logic design - Lecture 1: Number systems

The main contents of the chapter consist of the following: Binary numbers are made of binary digits (bits); binary and octal number systems; conversion between number systems; addition, subtraction, and multiplication in binary.

Digital Logic Design

° Digital

- Concerned with the interconnection among digital

components and modules

» Best Digital System example is General Purpose  Computer

° Logic Design

- Deals with the basic concepts and tools used to design digital hardware consisting of logic circuits

» Circuits to perform arithmetic operations (+, ­, x, ÷)

° Decimal values are difficult to represent in electrical

systems It is easier to use two voltage values than ten.

° Digital Signals have two basic states:

1 (logic “high”, or H, or “on”)

0 (logic “low”, or L, or “off”)

° Digital values are in a binary format Binary means

Digital Logic Design

° Bits and Pieces of DLD History

° George Boole

- Mathematical Analysis of Logic (1847)

- An Investigation of Laws of Thoughts; Mathematical Theories of Logic and Probabilities (1854)

° Claude Shannon

- Rediscovered the Boole

- “ A Symbolic Analysis of Relay and Switching Circuits “

- Boolean Logic and Boolean Algebra were Applied to Digital Circuitry

- Beginning of the Digital Age and/or Computer Age

World War II

Computers as Calculating Machines

Arlington (State Machines) “ Control “

Objectives

° Number System, Their Uses, Conversions

° Basic Building Blocks of Digital System

° Minimization

° Combinational And Sequential Logic

° Digital System/Circuit Analysis and Design

° State Minimizations

° Integrated Circuits

° Simulations

“Logic and Computer Design Fundamentals” By M

Morris Mano & Charles R Kime

Digital Logic Design

Lecture 1

Number Systems

Number Systems

° Decimal is the number system that we use

° Binary is a number system that computers use

° Octal is a number system that represents groups of

binary numbers (binary shorthand) It is used in

digital displays, and in modern times in conjunction with file permissions under Unix systems

° Hexadecimal (Hex) is a number system that

represents groups of binary numbers (binary

shorthand) Hex is primarily used in computing as the most common form of expressing a human-

readable string representation of a byte (group of 8 bits)

Overvie

w

° The design of computers

• It all starts with numbers

• Building circuits

• Building computing machines

° Digital systems

° Understanding decimal numbers

° Binary and octal numbers

• The basis of computers!

° Conversion between different number systems

• Digital

Try replicating ON or OFF

°Two discrete values are used in digital systems.

°How are discrete elements represented?

• Signals are the physical quantities used to represent discrete  elements of information in a digital system.

°Electric signals used:

• Voltage

• Current

14

Digital Computer Systems

° Digital systems consider discrete amounts of data.

° Examples

• 26 letters in the alphabet

• 10 decimal digits

° Larger quantities can be built from discrete values:

• Words made of letters

• Numbers made of decimal digits (e.g 239875.32)

° Computers operate on binary values (0 and 1)

° Easy to represent binary values electrically

• Voltages and currents.

• Can be implemented using circuits

• Create the building blocks of modern computers

Understanding Decimal Numbers

° Decimal numbers are made of decimal digits:

Understanding Octal Numbers

° Octal numbers are made of octal digits:

° Octal numbers don’t use digits 8 or 9

° Who would use octal number, anyway?

Understanding Binary Numbers

° Binary numbers are made of binary digits (bits):

° Possible to tolerate noise.

° Easy to transmit data

° Easy to build binary circuits

AND Gate 1

° Binary number has base 2

° Each digit is one of two

numbers: 0 and 1

° Each digit is called a bit

° Eight binary bits make a byte

° All 256 possible values of a

byte can be represented using

2 digits in hexadecimal notation.

Binary as a Voltage

° Voltages are used to represent logic values:

° A voltage present (called Vcc or Vdd) = 1

° Zero Volts or ground (called gnd or Vss) = 0

A simple switch can provide a logic high or a logic low.

Binary digits

Bit: single binary digit

Byte: 8 binary digits

Bit

Byte

Radix

° Learn to convert between bases.

° Already demonstrated how to convert

from binary to decimal.

° Hexadecimal described in next

lecture.

Number Systems

Used by humans?

Used in computers?

Convert an Integer from Decimal to Another

Base

1 Divide decimal number by the base (e.g 2)

2 The remainder is the lowest-order digit

3 Repeat first two steps until no divisor remains.

For each digit position:

Example for (13)10:

Integer Quotient

13/2 = 6 + ½ a0 = 1 6/2 = 3 + 0 a1 = 0 3/2 = 1 + ½ a2 = 1 1/2 = 0 + ½ a3 = 1

Convert an Fraction from Decimal to Another

Base

1 Multiply decimal number by the base (e.g 2)

2 The integer is the highest-order digit

3 Repeat first two steps until fraction becomes

Binary

Addition

° Binary addition is very simple.

° This is best shown in an example of adding two

1 1

carries

Binary Multiplication

° Binary multiplication is much the same as decimal

multiplication, except that the multiplication

operations are much simpler…

1 1 1 0 0 1 1 0

Convert an Integer from Decimal to

Octal

1 Divide decimal number by the base (8)

2 The remainder is the lowest-order digit

3 Repeat first two steps until no divisor remains.

For each digit position:

Example for (175)10:

Integer Quotient

175/8 = 21 + 7/8 a0 = 7 21/8 = 2 + 5/8 a1 = 5 2/8 = 0 + 2/8 a2 = 2

Convert an Fraction from Decimal to

Octal

1 Multiply decimal number by the base (e.g 8)

2 The integer is the highest-order digit

3 Repeat first two steps until fraction becomes

Summary

° Binary numbers are made of binary digits (bits)

° Binary and octal number systems

° Conversion between number systems

° Addition, subtraction, and multiplication in binary