Binary addition example worked out

Save this PDF as:

Size: px
Start display at page:

Download "Binary addition example worked out"

Transcription

1 Binary addition example worked out Some terms are given here Exercise: what are these numbers equivalent to in decimal? The initial carry in is implicitly (Carries) (Augend) (Addend) (Sum) most significant bit (MSB) least significant bit (LSB) 0

2 Basic Gates There are three basic kinds of logic gates Operation: AND of two inputs OR of two inputs NOT (complement) on one input Logic gate: Two Questions: How can we implement such switches? What can we build with Gates? And How? 1

3 Doing addition with gates Lets do simple stuff first: Can we add two numbers each with just 1 bit? Bit: binary digit 0+0 = 0, 0+1 = 1, 1+0 = 1, and 1+1 =??? 2. But 2 is not a symbol. 10 (just as is 10 in decimal) Result is 0 with 1 carried over to the next bit.. Whats 1 and 0? High and low voltage respectively. Half adder Result Carry 2

4 Half adder: result Result This circuit is so common, that it has a name an symbol as a gate by itself: Exclusive OR Exclusive OR Output is 1 iff exactly one of the 2 inputs is 1 3

5 Adding two bits A half adder is used to add two bits. The result consists of two bits: a sum (the right bit) and a carry out (the left bit) Here is the circuit and its block symbol = = = = 10 4

6 Adding three bits But what we really need to do is add three bits: the augend and addend, and the carry in from the right X Y C in C out S = = = = = = = = 11 5

7 Full adder circuit Why are these things called half adders and full adders? You can build a full adder by putting together two half adders. 6

8 A 4-bit adder Four full adders together can make a 4-bit adder There are nine total inputs to the 4-bit adder: two 4-bit numbers, A3 A2 A1 A0 and B3 B2 B1 B0 an initial carry in, CI The five outputs are: a 4-bit sum, S3 S2 S1 S0 a carry out, CO 7

9 An example of 4-bit addition Let s put our initial example into this circuit: A=1011, B= Step 1: Fill in all the inputs, including CI=0 Step 2: The circuit produces C1 and S0 ( = 01) Step 3: Use C1 to find C2 and S1 ( = 10) Step 4: Use C2 to compute C3 and S2 ( = 10) Step 5: Use C3 to compute CO and S3 ( = 11) The final answer is

10 Now that we can add, how about some memory? We want to save results computed before, and recall them in a later calculation, for example Gates help us build memory How can a circuit remember anything on its own? After all, the values on the wires are always changing, as outputs are generated in response to inputs. The basic idea is feedback: we make a loop in the circuit, so the circuit outputs are inputs as well When S and R are 0, Q is stable : whatever it was, it stays in that state. Ergo : memory. When S is 1 and R is 0, Q becomes 1 Set and Reset inputs When R is 1 and S is 0, Q becomes 0 9

11 So, we have built a calculator It is not a computer yet We have to type each step into a calculator We d like to program standard steps E.g. Add 57 numbers sitting in memory in specific places Also, support other operations (subtract..) Two new ideas and components are needed for this: Addressable memory Stored Program Addressable memory Memory organized in a bunch of locations, such that contents of specified location can be brought back to the adder when needed. Each memory location has an address (binary, of course) Stored Program: The instructions for which numbers to operate on, what operation to do (add/subtract,..) and where to store the result The instructions themselves can be represented in binary and stored in the memory! The processor must have circuits to decode and interpret these instructions 10

12 Components of a basic computer Data ALU (Arithmetic/Logic Unit: Basic operations Memory Program Control and Decoding 11

13 Summary Controllable Switches are easy to make These switches can be used to put together Logic Gates Logic Gates can be put together to make half adder, full adders and multi-bit adders So we can see they can be used for other such circtuits as well Logic Gates can be used to make circtuits that remember or store data A Computer includes, at its heart : An ALU (Arithmetic Logic Unit) Instruction Decoding and associated circuits Memory Stored Program 12

14 Number systems To get started, we ll discuss one of the fundamental concepts underlying digital computer design: Deep down inside, computers work with just 1s and 0s. Computers use voltages to represent information. In modern CPUs the voltage is usually limited to V to minimize power consumption. It s convenient for us to translate these analog voltages into the discrete, or digital, values 1 and 0. But how can two lousy digits be useful for anything? First, we ll see how to represent numbers with just 1s and 0s. Then we ll introduce special operations for computing with 1s and 0s, by treating them as the logical values true and false. 1 0 Volts

15 Rest of Today s lecture Having seen an overview last week, We will now start a more thorough study Number systems Review of binary number representation How to convert between binary and decimal representations Octal and Hex representations Basic boolean operations AND, OR and NOT The idea of Truth Table Boolean functions and expressions Truth table for Boolean expressions 14

16 Decimal review Numbers consist of a bunch of digits, each with a weight: Digits /10 1/100 1/1000 Weights The weights are all powers of the base, which is 10. We can rewrite the weights like this: Digits Weights To find the decimal value of a number, multiply each digit by its weight and sum the products. (1 x 10 2 ) + (6 x 10 1 ) + (2 x 10 0 ) + (3 x 10-1 ) + (7 x 10-2 ) + (5 x 10-3 ) =

17 Converting binary to decimal We can use the same trick to convert binary, or base 2, numbers to decimal. The only difference is that the weights are powers of 2. For example, here is in binary: Binary digits, or bits Weights (in base 10) The decimal value is: (1 x 2 3 ) + (1 x 2 2 ) + (0 x 2 1 ) + (1 x 2 0 ) + (0 x 2-1 ) + (1 x 2-2 ) = = Powers of 2: 2 0 = = = = = = = = = = =

18 Converting decimal to binary To convert a decimal integer into binary, keep dividing by 2 until the quotient is 0. Collect the remainders in reverse order. To convert a fraction, keep multiplying the fractional part by 2 until it becomes 0. Collect the integer parts in forward order. Example: : 162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem x 2 = x 2 = x 2 = So, =

19 Why does this work? This works for converting from decimal to any base Why? Think about converting from decimal to decimal. 162 / 10 = 16 rem 2 16 / 10 = 1 rem 6 1 / 10 = 0 rem 1 Each division strips off the rightmost digit (the remainder). The quotient represents the remaining digits in the number. Similarly, to convert fractions, each multiplication strips off the leftmost digit (the integer part). The fraction represents the remaining digits x 10 = x 10 = x 10 =

20 Base 16 is useful too The hexadecimal system uses 16 digits: A B C D E F You can convert between base 10 and base 16 using techniques like the ones we just showed for converting between decimal and binary. For our purposes, base 16 is most useful as a shorthand notation for binary numbers. Since 16 = 2 4, one hexadecimal digit is equivalent to 4 binary digits. It s often easier to work with a number like B4 instead of Hex is frequently used to specify things like 32-bit IP addresses and 24-bit colors. Decimal Binary Hex A B C D E F 19

21 Binary and hexadecimal conversions Converting from hexadecimal to binary is easy: just replace each hex digit with its equivalent 4-bit binary sequence = = To convert from binary to hex, make groups of 4 bits, starting from the binary point. Add 0s to the ends of the number if needed. Then, just convert each bit group to its corresponding hex digit = = B 4. 2 C 16 Hex Binary Hex Binary Hex Binary Hex Binary C D A 1010 E B 1011 F

22 Number Systems Summary Computers are binary devices. We re forced to think in terms of base 2. Today we learned how to convert numbers between binary, decimal and hexadecimal. We ve already seen some of the recurring themes of architecture: We use 0 and 1 as abstractions for analog voltages. We showed how to represent numbers using just these two signals. Next we ll introduce special operations for binary values and show how those correspond to circuits. 21

23 Boolean Operations So far, we ve talked about how arbitrary numbers can be represented using just the two binary values 1 and 0. Now we ll interpret voltages as the logical values true and false instead. We ll show: How functions can be defined for expressing computations How to build circuits that implement our functions in hardware 22

24 Boolean values Earlier, we used electrical voltages to represent two discrete values 1 and 0, from which binary numbers can be formed. It s also possible to think of voltages as representing two logical values, true and false. For simplicity, we often still write digits instead: 1 is true 0 is false True False We will use this interpretation along with special operations to design functions and hardware for doing arbitrary computations. Volts

25 Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y) = 2x + y = x + x + y = 2(x + y/2) =... A function table is unique but infinite x y f(x,y) We can represent logical functions in two analogous ways too: A finite, but non-unique Boolean expression. A truth table, which will turn out to be unique and finite. 24

26 Basic Boolean operations There are three basic operations for logical values. Operation: AND (product) of two inputs OR (sum) of two inputs NOT (complement) on one input Expression: xy, or x y x + y x Truth table: x y xy x y x+y x x

27 Boolean expressions We can use these basic operations to form more complex expressions: f(x,y,z) = (x + y )z + x Some terminology and notation: f is the name of the function. (x,y,z) are the input variables, each representing 1 or 0. Listing the inputs is optional, but sometimes helpful. A literal is any occurrence of an input variable or its complement. The function above has four literals: x, y, z, and x. Precedences are important, but not too difficult. NOT has the highest precedence, followed by AND, and then OR. Fully parenthesized, the function above would be kind of messy: f(x,y,z) = (((x +(y ))z) + x ) 26

28 Truth tables A truth table shows all possible inputs and outputs of a function. Remember that each input variable represents either 1 or 0. Because there are only a finite number of values (1 and 0), truth tables themselves are finite. A function with n variables has 2 n possible combinations of inputs. Inputs are listed in binary order in this example, from 000 to 111. f(x,y,z) = (x + y )z + x f(0,0,0) = (0 + 1)0 + 1 = 1 f(0,0,1) = (0 + 1)1 + 1 = 1 f(0,1,0) = (0 + 0)0 + 1 = 1 f(0,1,1) = (0 + 0)1 + 1 = 1 f(1,0,0) = (1 + 1)0 + 0 = 0 f(1,0,1) = (1 + 1)1 + 0 = 1 f(1,1,0) = (1 + 0)0 + 0 = 0 f(1,1,1) = (1 + 0)1 + 0 = 1 x y z f(x,y,z)

29 Primitive logic gates Each of our basic operations can be implemented in hardware using a primitive logic gate. Symbols for each of the logic gates are shown below. These gates output the product, sum or complement of their inputs. Operation: AND (product) of two inputs OR (sum) of two inputs NOT (complement) on one input Expression: xy, or x y x + y x Logic gate: 28

30 Expressions and circuits Any Boolean expression can be converted into a circuit by combining basic gates in a relatively straightforward way. The diagram below shows the inputs and outputs of each gate. The precedences are explicit in a circuit. Clearly, we have to make sure that the hardware does operations in the right order! (x + y )z + x 29

31 Boolean operations summary We can interpret high or low voltage as representing true or false. A variable whose value can be either 1 or 0 is called a Boolean variable. AND, OR, and NOT are the basic Boolean operations. We can express Boolean functions with either an expression or a truth table. Every Boolean expression can be converted to a circuit. Next time, we ll look at how Boolean algebra can help simplify expressions, which in turn will lead to simpler circuits. 30

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways:

Functions. Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: Boolean Algebra (1) Functions Computers take inputs and produce outputs, just like functions in math! Mathematical functions can be expressed in two ways: An expression is finite but not unique f(x,y)

More information

EE260: Digital Design, Spring n Digital Computers. n Number Systems. n Representations. n Conversions. n Arithmetic Operations.

EE260: Digital Design, Spring n Digital Computers. n Number Systems. n Representations. n Conversions. n Arithmetic Operations. EE 260: Introduction to Digital Design Number Systems Yao Zheng Department of Electrical Engineering University of Hawaiʻi at Mānoa Overview n Digital Computers n Number Systems n Representations n Conversions

More information

14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System?

14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System? :33:3 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 3 Lecture #: Binary Number System Complement Number Representation X Y Why Binary Number System? Because

More information

Binary addition by hand. Adding two bits

Binary addition by hand. Adding two bits Chapter 3 Arithmetic is the most basic thing you can do with a computer We focus on addition, subtraction, multiplication and arithmetic-logic units, or ALUs, which are the heart of CPUs. ALU design Bit

More information

Chapter 1 CSCI

Chapter 1 CSCI Chapter 1 CSCI-1510-003 What is a Number? An expression of a numerical quantity A mathematical quantity Many types: Natural Numbers Real Numbers Rational Numbers Irrational Numbers Complex Numbers Etc.

More information

Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Systems I: Computer Organization and Architecture Lecture 6 - Combinational Logic Introduction A combinational circuit consists of input variables, logic gates, and output variables. The logic gates accept

More information

Latches. October 13, 2003 Latches 1

Latches. October 13, 2003 Latches 1 Latches The second part of CS231 focuses on sequential circuits, where we add memory to the hardware that we ve already seen. Our schedule will be very similar to before: We first show how primitive memory

More information

Building a Computer Adder

Building a Computer Adder Logic Gates are used to translate Boolean logic into circuits. In the abstract it is clear that we can build AND gates that perform the AND function and OR gates that perform the OR function and so on.

More information

CHAPTER1: Digital Logic Circuits Combination Circuits

CHAPTER1: Digital Logic Circuits Combination Circuits CS224: Computer Organization S.KHABET CHAPTER1: Digital Logic Circuits Combination Circuits 1 PRIMITIVE LOGIC GATES Each of our basic operations can be implemented in hardware using a primitive logic gate.

More information

Combinational Logic. By : Ali Mustafa

Combinational Logic. By : Ali Mustafa Combinational Logic By : Ali Mustafa Contents Adder Subtractor Multiplier Comparator Decoder Encoder Multiplexer How to Analyze any combinational circuit like this? Analysis Procedure To obtain the output

More information

CS1800: Hex & Logic. Professor Kevin Gold

CS1800: Hex & Logic. Professor Kevin Gold CS1800: Hex & Logic Professor Kevin Gold Reviewing Last Time: Binary Last time, we saw that arbitrary numbers can be represented in binary. Each place in a binary number stands for a different power of

More information

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering

ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN. Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering ECEN 248: INTRODUCTION TO DIGITAL SYSTEMS DESIGN Week 2 Dr. Srinivas Shakkottai Dept. of Electrical and Computer Engineering Boolean Algebra Boolean Algebra A Boolean algebra is defined with: A set of

More information

Numbers and Arithmetic

Numbers and Arithmetic Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu

More information

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit

Digital Techniques. Figure 1: Block diagram of digital computer. Processor or Arithmetic logic unit ALU. Control Unit. Storage or memory unit Digital Techniques 1. Binary System The digital computer is the best example of a digital system. A main characteristic of digital system is its ability to manipulate discrete elements of information.

More information

Hakim Weatherspoon CS 3410 Computer Science Cornell University

Hakim Weatherspoon CS 3410 Computer Science Cornell University Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by Professors Weatherspoon, Bala, Bracy, and Sirer. memory inst 32 register

More information

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary

More information

Review: Additional Boolean operations

Review: Additional Boolean operations Review: Additional Boolean operations Operation: NAND (NOT-AND) NOR (NOT-OR) XOR (exclusive OR) Expressions: (xy) = x + y (x + y) = x y x y = x y + xy Truth table: x y (xy) x y (x+y) x y x y 0 0 1 0 1

More information

THE LOGIC OF COMPOUND STATEMENTS

THE LOGIC OF COMPOUND STATEMENTS CHAPTER 2 THE LOGIC OF COMPOUND STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 2.4 Application: Digital Logic Circuits Copyright Cengage Learning. All rights reserved. Application:

More information

Numbers and Arithmetic

Numbers and Arithmetic Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu

More information

We say that the base of the decimal number system is ten, represented by the symbol

We say that the base of the decimal number system is ten, represented by the symbol Introduction to counting and positional notation. In the decimal number system, a typical number, N, looks like... d 3 d 2 d 1 d 0.d -1 d -2 d -3... [N1] where the ellipsis at each end indicates that there

More information

Contents. Chapter 2 Digital Circuits Page 1 of 30

Contents. Chapter 2 Digital Circuits Page 1 of 30 Chapter 2 Digital Circuits Page 1 of 30 Contents Contents... 1 2 Digital Circuits... 2 2.1 Binary Numbers... 2 2.2 Binary Switch... 4 2.3 Basic Logic Operators and Logic Expressions... 5 2.4 Truth Tables...

More information

ECE/CS 250 Computer Architecture

ECE/CS 250 Computer Architecture ECE/CS 250 Computer Architecture Basics of Logic Design: Boolean Algebra, Logic Gates (Combinational Logic) Tyler Bletsch Duke University Slides are derived from work by Daniel J. Sorin (Duke), Alvy Lebeck

More information

ECE 250 / CPS 250 Computer Architecture. Basics of Logic Design Boolean Algebra, Logic Gates

ECE 250 / CPS 250 Computer Architecture. Basics of Logic Design Boolean Algebra, Logic Gates ECE 250 / CPS 250 Computer Architecture Basics of Logic Design Boolean Algebra, Logic Gates Benjamin Lee Slides based on those from Andrew Hilton (Duke), Alvy Lebeck (Duke) Benjamin Lee (Duke), and Amir

More information

E40M. Binary Numbers. M. Horowitz, J. Plummer, R. Howe 1

E40M. Binary Numbers. M. Horowitz, J. Plummer, R. Howe 1 E40M Binary Numbers M. Horowitz, J. Plummer, R. Howe 1 Reading Chapter 5 in the reader A&L 5.6 M. Horowitz, J. Plummer, R. Howe 2 Useless Box Lab Project #2 Adding a computer to the Useless Box alows us

More information

of Digital Electronics

of Digital Electronics 26 Digital Electronics 729 Digital Electronics 26.1 Analog and Digital Signals 26.3 Binary Number System 26.5 Decimal to Binary Conversion 26.7 Octal Number System 26.9 Binary-Coded Decimal Code (BCD Code)

More information

CHAPTER 2 NUMBER SYSTEMS

CHAPTER 2 NUMBER SYSTEMS CHAPTER 2 NUMBER SYSTEMS The Decimal Number System : We begin our study of the number systems with the familiar decimal number system. The decimal system contains ten unique symbol 0, 1, 2, 3, 4, 5, 6,

More information

Computing via boolean logic. COS 116: 3/8/2011 Sanjeev Arora

Computing via boolean logic. COS 116: 3/8/2011 Sanjeev Arora Computing via boolean logic. COS 116: 3/8/2011 Sanjeev Arora Recap: Boolean Logic Example Ed goes to the party if Dan does not and Stella does. Choose Boolean variables for 3 events: { Each E: Ed goes

More information

Boolean Algebra and Digital Logic 2009, University of Colombo School of Computing

Boolean Algebra and Digital Logic 2009, University of Colombo School of Computing IT 204 Section 3.0 Boolean Algebra and Digital Logic Boolean Algebra 2 Logic Equations to Truth Tables X = A. B + A. B + AB A B X 0 0 0 0 3 Sum of Products The OR operation performed on the products of

More information

Cs302 Quiz for MID TERM Exam Solved

Cs302 Quiz for MID TERM Exam Solved Question # 1 of 10 ( Start time: 01:30:33 PM ) Total Marks: 1 Caveman used a number system that has distinct shapes: 4 5 6 7 Question # 2 of 10 ( Start time: 01:31:25 PM ) Total Marks: 1 TTL based devices

More information

CSE 20 Discrete Math. Algebraic Rules for Propositional Formulas. Summer, July 11 (Day 2) Number Systems/Computer Arithmetic Predicate Logic

CSE 20 Discrete Math. Algebraic Rules for Propositional Formulas. Summer, July 11 (Day 2) Number Systems/Computer Arithmetic Predicate Logic CSE 20 Discrete Math Algebraic Rules for Propositional Formulas Equivalences between propositional formulas (similar to algebraic equivalences): Associative Summer, 2006 July 11 (Day 2) Number Systems/Computer

More information

Week No. 06: Numbering Systems

Week No. 06: Numbering Systems Week No. 06: Numbering Systems Numbering System: A numbering system defined as A set of values used to represent quantity. OR A number system is a term used for a set of different symbols or digits, which

More information

Boolean Algebra & Digital Logic

Boolean Algebra & Digital Logic Boolean Algebra & Digital Logic Boolean algebra was developed by the Englishman George Boole, who published the basic principles in the 1854 treatise An Investigation of the Laws of Thought on Which to

More information

CSC9R6 Computer Design. Practical Digital Logic

CSC9R6 Computer Design. Practical Digital Logic CSC9R6 Computer Design Practical Digital Logic 1 References (for this part of CSC9R6) Hamacher et al: Computer Organization App A. In library Floyd: Digital Fundamentals Ch 1, 3-6, 8-10 web page: www.prenhall.com/floyd/

More information

Boolean Algebra, Gates and Circuits

Boolean Algebra, Gates and Circuits Boolean Algebra, Gates and Circuits Kasper Brink November 21, 2017 (Images taken from Tanenbaum, Structured Computer Organization, Fifth Edition, (c) 2006 Pearson Education, Inc.) Outline Last week: Von

More information

Carry Look Ahead Adders

Carry Look Ahead Adders Carry Look Ahead Adders Lesson Objectives: The objectives of this lesson are to learn about: 1. Carry Look Ahead Adder circuit. 2. Binary Parallel Adder/Subtractor circuit. 3. BCD adder circuit. 4. Binary

More information

Why digital? Overview. Number Systems. Binary to Decimal conversion

Why digital? Overview. Number Systems. Binary to Decimal conversion Why digital? Overview It has the following advantages over analog. It can be processed and transmitted efficiently and reliably. It can be stored and retrieved with greater accuracy. Noise level does not

More information

Chapter 3 Ctd: Combinational Functions and Circuits

Chapter 3 Ctd: Combinational Functions and Circuits Chapter 3 Ctd: Combinational Functions and Circuits 1 Value Fixing, Transferring, and Inverting Four different functions are possible as a function of single Boolean variable Transferring Inverting Value

More information

Hardware Design I Chap. 4 Representative combinational logic

Hardware Design I Chap. 4 Representative combinational logic Hardware Design I Chap. 4 Representative combinational logic E-mail: shimada@is.naist.jp Already optimized circuits There are many optimized circuits which are well used You can reduce your design workload

More information

20. Combinational Circuits

20. Combinational Circuits Combinational circuits Q. What is a combinational circuit? A. A digital circuit (all signals are or ) with no feedback (no loops). analog circuit: signals vary continuously sequential circuit: loops allowed

More information

Number System conversions

Number System conversions Number System conversions Number Systems The system used to count discrete units is called number system. There are four systems of arithmetic which are often used in digital electronics. Decimal Number

More information

ENG2410 Digital Design Introduction to Digital Systems. Fall 2017 S. Areibi School of Engineering University of Guelph

ENG2410 Digital Design Introduction to Digital Systems. Fall 2017 S. Areibi School of Engineering University of Guelph ENG2410 Digital Design Introduction to Digital Systems Fall 2017 S. Areibi School of Engineering University of Guelph Resources Chapter #1, Mano Sections 1.1 Digital Computers 1.2 Number Systems 1.3 Arithmetic

More information

We are here. Assembly Language. Processors Arithmetic Logic Units. Finite State Machines. Circuits Gates. Transistors

We are here. Assembly Language. Processors Arithmetic Logic Units. Finite State Machines. Circuits Gates. Transistors CSC258 Week 3 1 Logistics If you cannot login to MarkUs, email me your UTORID and name. Check lab marks on MarkUs, if it s recorded wrong, contact Larry within a week after the lab. Quiz 1 average: 86%

More information

CSE 241 Digital Systems Spring 2013

CSE 241 Digital Systems Spring 2013 CSE 241 Digital Systems Spring 2013 Instructor: Prof. Kui Ren Department of Computer Science and Engineering Lecture slides modified from many online resources and used solely for the educational purpose.

More information

Numbering Systems. Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary.

Numbering Systems. Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary. Numbering Systems Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary. Addition & Subtraction using Octal & Hexadecimal 2 s Complement, Subtraction Using 2 s Complement.

More information

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary

Number System. Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary Number System Decimal to binary Binary to Decimal Binary to octal Binary to hexadecimal Hexadecimal to binary Octal to binary BOOLEAN ALGEBRA BOOLEAN LOGIC OPERATIONS Logical AND Logical OR Logical COMPLEMENTATION

More information

Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary

Slide Set 3. for ENEL 353 Fall Steve Norman, PhD, PEng. Electrical & Computer Engineering Schulich School of Engineering University of Calgary Slide Set 3 for ENEL 353 Fall 2016 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2016 SN s ENEL 353 Fall 2016 Slide Set 3 slide

More information

School of Computer Science and Electrical Engineering 28/05/01. Digital Circuits. Lecture 14. ENG1030 Electrical Physics and Electronics

School of Computer Science and Electrical Engineering 28/05/01. Digital Circuits. Lecture 14. ENG1030 Electrical Physics and Electronics Digital Circuits 1 Why are we studying digital So that one day you can design something which is better than the... circuits? 2 Why are we studying digital or something better than the... circuits? 3 Why

More information

Review for Test 1 : Ch1 5

Review for Test 1 : Ch1 5 Review for Test 1 : Ch1 5 October 5, 2006 Typeset by FoilTEX Positional Numbers 527.46 10 = (5 10 2 )+(2 10 1 )+(7 10 0 )+(4 10 1 )+(6 10 2 ) 527.46 8 = (5 8 2 ) + (2 8 1 ) + (7 8 0 ) + (4 8 1 ) + (6 8

More information

Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems

Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems Menu Review of number systems >Binary math >Signed number systems Look into my... 1 Our decimal (base 10 or radix 10) number system is positional. Ex: 9437 10 = 9x10 3 + 4x10 2 + 3x10 1 + 7x10 0 We have

More information

ECE380 Digital Logic. Positional representation

ECE380 Digital Logic. Positional representation ECE380 Digital Logic Number Representation and Arithmetic Circuits: Number Representation and Unsigned Addition Dr. D. J. Jackson Lecture 16-1 Positional representation First consider integers Begin with

More information

Binary Multipliers. Reading: Study Chapter 3. The key trick of multiplication is memorizing a digit-to-digit table Everything else was just adding

Binary Multipliers. Reading: Study Chapter 3. The key trick of multiplication is memorizing a digit-to-digit table Everything else was just adding Binary Multipliers The key trick of multiplication is memorizing a digit-to-digit table Everything else was just adding 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 4 6 8 2 4 6 8 3 3 6 9 2 5 8 2 24 27 4 4 8 2 6

More information

Fundamentals of Digital Design

Fundamentals of Digital Design Fundamentals of Digital Design Digital Radiation Measurement and Spectroscopy NE/RHP 537 1 Binary Number System The binary numeral system, or base-2 number system, is a numeral system that represents numeric

More information

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes

Midterm Examination # 1 Wednesday, February 25, Duration of examination: 75 minutes Page 1 of 10 School of Computer Science 60-265-01 Computer Architecture and Digital Design Winter 2009 Semester Midterm Examination # 1 Wednesday, February 25, 2009 Student Name: First Name Family Name

More information

ENGIN 112 Intro to Electrical and Computer Engineering

ENGIN 112 Intro to Electrical and Computer Engineering ENGIN 112 Intro to Electrical and Computer Engineering Lecture 2 Number Systems Russell Tessier KEB 309 G tessier@ecs.umass.edu Overview The design of computers It all starts with numbers Building circuits

More information

Conversions between Decimal and Binary

Conversions between Decimal and Binary Conversions between Decimal and Binary Binary to Decimal Technique - use the definition of a number in a positional number system with base 2 - evaluate the definition formula ( the formula ) using decimal

More information

Computer Science. 19. Combinational Circuits. Computer Science COMPUTER SCIENCE. Section 6.1.

Computer Science. 19. Combinational Circuits. Computer Science COMPUTER SCIENCE. Section 6.1. COMPUTER SCIENCE S E D G E W I C K / W A Y N E PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science Computer Science An Interdisciplinary Approach Section 6.1 ROBERT SEDGEWICK

More information

Counting in Different Number Systems

Counting in Different Number Systems Counting in Different Number Systems Base 1 (Decimal) is important because that is the base that we first learn in our culture. Base 2 (Binary) is important because that is the base used for computer codes

More information

CSE 20 DISCRETE MATH. Fall

CSE 20 DISCRETE MATH. Fall CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Describe and use algorithms for integer operations based on their expansions Relate algorithms for integer

More information

S C F F F T T F T T S C B F F F F F T F T F F T T T F F T F T T T F T T T

S C F F F T T F T T S C B F F F F F T F T F F T T T F F T F T T T F T T T EECS 270, Winter 2017, Lecture 1 Page 1 of 6 Use pencil! Say we live in the rather black and white world where things (variables) are either true (T) or false (F). So if S is Mark is going to the Store

More information

UNSIGNED BINARY NUMBERS DIGITAL ELECTRONICS SYSTEM DESIGN WHAT ABOUT NEGATIVE NUMBERS? BINARY ADDITION 11/9/2018

UNSIGNED BINARY NUMBERS DIGITAL ELECTRONICS SYSTEM DESIGN WHAT ABOUT NEGATIVE NUMBERS? BINARY ADDITION 11/9/2018 DIGITAL ELECTRONICS SYSTEM DESIGN LL 2018 PROFS. IRIS BAHAR & ROD BERESFORD NOVEMBER 9, 2018 LECTURE 19: BINARY ADDITION, UNSIGNED BINARY NUMBERS For the binary number b n-1 b n-2 b 1 b 0. b -1 b -2 b

More information

Digital Logic (2) Boolean Algebra

Digital Logic (2) Boolean Algebra Digital Logic (2) Boolean Algebra Boolean algebra is the mathematics of digital systems. It was developed in 1850 s by George Boole. We will use Boolean algebra to minimize logic expressions. Karnaugh

More information

Math Lecture 18 Notes

Math Lecture 18 Notes Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,

More information

Combinational Logic Design Principles

Combinational Logic Design Principles Combinational Logic Design Principles Switching algebra Doru Todinca Department of Computers Politehnica University of Timisoara Outline Introduction Switching algebra Axioms of switching algebra Theorems

More information

Every time has a value associated with it, not just some times. A variable can take on any value within a range

Every time has a value associated with it, not just some times. A variable can take on any value within a range Digital Logic Circuits Binary Logic and Gates Logic Simulation Boolean Algebra NAND/NOR and XOR gates Decoder fundamentals Half Adder, Full Adder, Ripple Carry Adder Analog vs Digital Analog Continuous»

More information

Applications. Smartphone, tablet, game controller, antilock brakes, microprocessor, Wires

Applications. Smartphone, tablet, game controller, antilock brakes, microprocessor, Wires COMPUTER SCIENCE Combinational circuits Q. What is a combinational circuit? A. A digital circuit (all signals are or ) with no feedback (no loops). analog circuit: signals vary continuously sequential

More information

Computer Science. 20. Combinational Circuits. Computer Science COMPUTER SCIENCE. Section

Computer Science. 20. Combinational Circuits. Computer Science COMPUTER SCIENCE. Section COMPUTER SCIENCE S E D G E W I C K / W A Y N E Computer Science 20. Combinational Circuits Computer Science An Interdisciplinary Approach Section 6.1 ROBERT SEDGEWICK K E V I N WAY N E http://introcs.cs.princeton.edu

More information

Digital Systems Overview. Unit 1 Numbering Systems. Why Digital Systems? Levels of Design Abstraction. Dissecting Decimal Numbers

Digital Systems Overview. Unit 1 Numbering Systems. Why Digital Systems? Levels of Design Abstraction. Dissecting Decimal Numbers Unit Numbering Systems Fundamentals of Logic Design EE2369 Prof. Eric MacDonald Fall Semester 2003 Digital Systems Overview Digital Systems are Home PC XBOX or Playstation2 Cell phone Network router Data

More information

COMBINATIONAL LOGIC FUNCTIONS

COMBINATIONAL LOGIC FUNCTIONS COMBINATIONAL LOGIC FUNCTIONS Digital logic circuits can be classified as either combinational or sequential circuits. A combinational circuit is one where the output at any time depends only on the present

More information

SAU1A FUNDAMENTALS OF DIGITAL COMPUTERS

SAU1A FUNDAMENTALS OF DIGITAL COMPUTERS SAU1A FUNDAMENTALS OF DIGITAL COMPUTERS Unit : I - V Unit : I Overview Fundamentals of Computers Characteristics of Computers Computer Language Operating Systems Generation of Computers 2 Definition of

More information

The goal differs from prime factorization. Prime factorization would initialize all divisors to be prime numbers instead of integers*

The goal differs from prime factorization. Prime factorization would initialize all divisors to be prime numbers instead of integers* Quantum Algorithm Processor For Finding Exact Divisors Professor J R Burger Summary Wiring diagrams are given for a quantum algorithm processor in CMOS to compute, in parallel, all divisors of an n-bit

More information

Digital Systems and Information Part II

Digital Systems and Information Part II Digital Systems and Information Part II Overview Arithmetic Operations General Remarks Unsigned and Signed Binary Operations Number representation using Decimal Codes BCD code and Seven-Segment Code Text

More information

Student Responsibilities Week 8. Mat Section 3.6 Integers and Algorithms. Algorithm to Find gcd()

Student Responsibilities Week 8. Mat Section 3.6 Integers and Algorithms. Algorithm to Find gcd() Student Responsibilities Week 8 Mat 2345 Week 8 Reading: Textbook, Section 3.7, 4.1, & 5.2 Assignments: Sections 3.6, 3.7, 4.1 Induction Proof Worksheets Attendance: Strongly Encouraged Fall 2013 Week

More information

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.)

DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) DIGITAL CIRCUIT LOGIC BOOLEAN ALGEBRA (CONT.) 1 Learning Objectives 1. Apply the laws and theorems of Boolean algebra to to the manipulation of algebraic expressions to simplifying an expression, finding

More information

14:332:231 DIGITAL LOGIC DESIGN. 2 s-complement Representation

14:332:231 DIGITAL LOGIC DESIGN. 2 s-complement Representation 4:332:23 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 203 Lecture #3: Addition, Subtraction, Multiplication, and Division 2 s-complement Representation RECALL

More information

Digital electronics form a class of circuitry where the ability of the electronics to process data is the primary focus.

Digital electronics form a class of circuitry where the ability of the electronics to process data is the primary focus. Chapter 2 Digital Electronics Objectives 1. Understand the operation of basic digital electronic devices. 2. Understand how to describe circuits which can process digital data. 3. Understand how to design

More information

UNIVERSITI TENAGA NASIONAL. College of Information Technology

UNIVERSITI TENAGA NASIONAL. College of Information Technology UNIVERSITI TENAGA NASIONAL College of Information Technology BACHELOR OF COMPUTER SCIENCE (HONS.) FINAL EXAMINATION SEMESTER 2 2012/2013 DIGITAL SYSTEMS DESIGN (CSNB163) January 2013 Time allowed: 3 hours

More information

Binary addition (1-bit) P Q Y = P + Q Comments Carry = Carry = Carry = Carry = 1 P Q

Binary addition (1-bit) P Q Y = P + Q Comments Carry = Carry = Carry = Carry = 1 P Q Digital Arithmetic In Chapter 2, we have discussed number systems such as binary, hexadecimal, decimal, and octal. We have also discussed sign representation techniques, for example, sign-bit representation

More information

Four Important Number Systems

Four Important Number Systems Four Important Number Systems System Why? Remarks Decimal Base 10: (10 fingers) Most used system Binary Base 2: On/Off systems 3-4 times more digits than decimal Octal Base 8: Shorthand notation for working

More information

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure.

XOR - XNOR Gates. The graphic symbol and truth table of XOR gate is shown in the figure. XOR - XNOR Gates Lesson Objectives: In addition to AND, OR, NOT, NAND and NOR gates, exclusive-or (XOR) and exclusive-nor (XNOR) gates are also used in the design of digital circuits. These have special

More information

Intro To Digital Logic

Intro To Digital Logic Intro To Digital Logic 1 Announcements... Project 2.2 out But delayed till after the midterm Midterm in a week Covers up to last lecture + next week's homework & lab Nick goes "H-Bomb of Justice" About

More information

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table.

Boolean algebra. Examples of these individual laws of Boolean, rules and theorems for Boolean algebra are given in the following table. The Laws of Boolean Boolean algebra As well as the logic symbols 0 and 1 being used to represent a digital input or output, we can also use them as constants for a permanently Open or Closed circuit or

More information

Numbers & Arithmetic. Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University. See: P&H Chapter , 3.2, C.5 C.

Numbers & Arithmetic. Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University. See: P&H Chapter , 3.2, C.5 C. Numbers & Arithmetic Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University See: P&H Chapter 2.4-2.6, 3.2, C.5 C.6 Example: Big Picture Computer System Organization and Programming

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Simple Processor CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Digital

More information

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1 Digital Systems Digital systems have such a prominent role in everyday life The digital age The technology around us is ubiquitous, that is we don t even notice it anymore Digital systems are used in:

More information

CISC 1400 Discrete Structures

CISC 1400 Discrete Structures CISC 1400 Discrete Structures Chapter 2 Sequences What is Sequence? A sequence is an ordered list of objects or elements. For example, 1, 2, 3, 4, 5, 6, 7, 8 Each object/element is called a term. 1 st

More information

Mat Week 8. Week 8. gcd() Mat Bases. Integers & Computers. Linear Combos. Week 8. Induction Proofs. Fall 2013

Mat Week 8. Week 8. gcd() Mat Bases. Integers & Computers. Linear Combos. Week 8. Induction Proofs. Fall 2013 Fall 2013 Student Responsibilities Reading: Textbook, Section 3.7, 4.1, & 5.2 Assignments: Sections 3.6, 3.7, 4.1 Proof Worksheets Attendance: Strongly Encouraged Overview 3.6 Integers and Algorithms 3.7

More information

CS61C : Machine Structures

CS61C : Machine Structures CS 61C L15 Blocks (1) inst.eecs.berkeley.edu/~cs61c/su05 CS61C : Machine Structures Lecture #15: Combinational Logic Blocks Outline CL Blocks Latches & Flip Flops A Closer Look 2005-07-14 Andy Carle CS

More information

Part 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life...

Part 1: Digital Logic and Gates. Analog vs. Digital waveforms. The digital advantage. In real life... Part 1: Digital Logic and Gates Analog vs Digital waveforms An analog signal assumes a continuous range of values: v(t) ANALOG A digital signal assumes discrete (isolated, separate) values Usually there

More information

Implementation of Boolean Logic by Digital Circuits

Implementation of Boolean Logic by Digital Circuits Implementation of Boolean Logic by Digital Circuits We now consider the use of electronic circuits to implement Boolean functions and arithmetic functions that can be derived from these Boolean functions.

More information

MC9211 Computer Organization

MC9211 Computer Organization MC92 Computer Organization Unit : Digital Fundamentals Lesson2 : Boolean Algebra and Simplification (KSB) (MCA) (29-2/ODD) (29 - / A&B) Coverage Lesson2 Introduces the basic postulates of Boolean Algebra

More information

convert a two s complement number back into a recognizable magnitude.

convert a two s complement number back into a recognizable magnitude. 1 INTRODUCTION The previous lesson introduced binary and hexadecimal numbers. In this lesson we look at simple arithmetic operations using these number systems. In particular, we examine the problem of

More information

Combinational Logic Design Arithmetic Functions and Circuits

Combinational Logic Design Arithmetic Functions and Circuits Combinational Logic Design Arithmetic Functions and Circuits Overview Binary Addition Half Adder Full Adder Ripple Carry Adder Carry Look-ahead Adder Binary Subtraction Binary Subtractor Binary Adder-Subtractor

More information

Chapter 4. Combinational: Circuits with logic gates whose outputs depend on the present combination of the inputs. elements. Dr.

Chapter 4. Combinational: Circuits with logic gates whose outputs depend on the present combination of the inputs. elements. Dr. Chapter 4 Dr. Panos Nasiopoulos Combinational: Circuits with logic gates whose outputs depend on the present combination of the inputs. Sequential: In addition, they include storage elements Combinational

More information

Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS

Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS Department of Electrical & Electronics EE-333 DIGITAL SYSTEMS 1) Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X -Y and (b) Y - X using 2's complements. a) X = 1010100

More information

Computer organization

Computer organization Computer organization Levels of abstraction Assembler Simulator Applications C C++ Java High-level language SOFTWARE add lw ori Assembly language Goal 0000 0001 0000 1001 0101 Machine instructions/data

More information

Computer Science 324 Computer Architecture Mount Holyoke College Fall Topic Notes: Digital Logic

Computer Science 324 Computer Architecture Mount Holyoke College Fall Topic Notes: Digital Logic Computer Science 324 Computer Architecture Mount Holyoke College Fall 2007 Topic Notes: Digital Logic Our goal for the next few weeks is to paint a a reasonably complete picture of how we can go from transistor

More information

Combinational Logic. Lan-Da Van ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C.

Combinational Logic. Lan-Da Van ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Combinational Logic ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Fall, 2017 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Combinational Circuits

More information

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks

ECE 545 Digital System Design with VHDL Lecture 1. Digital Logic Refresher Part A Combinational Logic Building Blocks ECE 545 Digital System Design with VHDL Lecture Digital Logic Refresher Part A Combinational Logic Building Blocks Lecture Roadmap Combinational Logic Basic Logic Review Basic Gates De Morgan s Law Combinational

More information

CS61C : Machine Structures

CS61C : Machine Structures inst.eecs.berkeley.edu/~cs61c/su05 CS61C : Machine Structures Lecture #15: Combinational Logic Blocks 2005-07-14 CS 61C L15 Blocks (1) Andy Carle Outline CL Blocks Latches & Flip Flops A Closer Look CS

More information

Logic Theory in Designing of Digital Circuit & Microprocessor

Logic Theory in Designing of Digital Circuit & Microprocessor Logic Theory in Designing of Digital Circuit & Microprocessor Prof.Vikram Mahendra Kakade Assistant Professor, Electronics & Telecommunication Engineering Department, Prof Ram Meghe College of Engineering

More information