# Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)

Size: px
Start display at page:

Download "Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)"

## Transcription

1 Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people attending a class, the number of modules taken by each student and use numbers to represent grade. System can be categorized in two systems:- (a) Non-Positional System (b) Positional System Non-Positional System- In ancient times, people used to count on fingers, when the fingers became insufficient for counting, stones, pebbles or sticks were used to indicate values. But it was very difficult to perform arithmetic with such a number system as there is no symbol for zero. Positional System- In this system the value of each digit is defined not by the symbol but also by the symbol position. Positional System is used to perform arithmetic. Existing Positional number system is decimal number system. Apart from the decimal number system, there are binary number system, octal number system and hexadecimal number system. Base (Radix)- In the number system the base or radix tells the number of symbols used in the system. In the earlier days, different civilisations were using different radixes. The Egyptian used the radix 2, the Babylonians used the radix 60 and Mayans used 18 and 20. The base of a number system is indicated by a subscript (decimal number) and this will be followed by the value of the number. For example (952) 10, (456) 8, (314) 16 System that are used by the computers- Decimal System System System System Decimal System- The decimal system is the system which we use in everyday counting. The number system includes the ten digits from 0 through 9. These digits are recognized as the symbols of the decimal system. Each digit in a base ten number represents units ten times the units of the digit to its right. For example- 9542= = ( ) + ( ) + (4 10) + ( ) 1

2 System - Computers do not use the decimal system for counting and arithmetic. Their CPU and memory are made up of millions of tiny switches that can be either in ON and OFF states. 0 represents OFF and 1 represents ON. In this way we use binary system. system has two numbers 0 and 1. system has base 2 therefore the weight of n th of the number from Right Hand Side is n th 2 n-1. System- The octal system is commonly used with computers. The octal number system with its 8 digit 0,1,2,3,4,5,6, and 7 has base 8. The octal system uses a power of 8 to determine the digit of a number s position. System- is another number system that works exactly like the decimal, binary and octal number systems, except that the base is 16. Each hexadecimal represents a power of 16. The system uses 0 to 9 numbers and A to F characters to represent 10 to 15 respectively. Conversions- Any number in one number system can be converted into any other number system. There are the various methods that are used in converting numbers from one base to another. Conversions of Decimal to - The method that is used for converting of decimals into binary is known as the remainder method. We use the following steps in getting the binary number- (a) Divide the decimal number by 2. (b) Write the remainder (which is either 0 or 1) at the right most position. (c) Repeat the process of dividing by 2 until the quotient is 0 and keep writing the remainder after each step of division. (d) Write the remainders in reverse order. Example- Convert (45) 10 into binary number system Remainder Thus (45) 10 = (101101) 2 2

3 Note- In every number system- (a) The first from the right is referred as LSB (Least Significant Bit) (b) The first from the left is referred as MSB (Most Significant Bit) Conversions of Decimal Fractions to Fractions- For converting decimal fractions into binary fractions, we use multiplication. Instead of looking for a remainder we look for an integer. The following steps are used in getting the binary fractions- (a) Multiply the decimal fraction by 2. (b) If a non-zero integer is generated, record the non-zero integer otherwise record 0. (c) Remove the non-zero integer and repeat the above steps till the fraction value becomes 0. (d) Write down the number according to the occurrence. Example- Find the binary equivalent of (0.75) 10. (to be recorded) = = Thus (0.75) 10 = (0.11) 2. Moreover, we can write (45.75) 10 = ( ) 2. Remark- If the conversion is not ended and still continuing; we write the approximation in 16 s. Example- Find the binary equivalent of (0.9) 10. (to be recorded) = = = = = = = = = = = = = = = = = = Thus (0.9) 10 = ( ) 2. 3

4 Conversion of Decimal to - In converting decimal to octal, we follow the same process of converting decimal to binary. Instead of dividing the number by 2, we divide the number by 8. Example- Convert (45) 10 into octal number system Remainder Thus (45) 10 = (55) 8. Conversions of Decimal Fractions to Fractions We follow the same steps of conversions of decimal fractions to binary fractions. Here we multiply the fraction by 8 instead of 2. Example- Find the octal equivalent of (0.75) 10. (to be recorded) = Thus (0.75) 10 = (0.6) 8. And (45.75) 10 = (55.6) 8. Conversion of Decimal to We divide by 16 instead of 2 or 8. If the remainder is in between 10 to 16, then the number is represented by A to F respectively. Example- Convert (45) 10 into hexadecimal Remainder 16 2 D Thus (45) 10 = (2D) 16. Conversions of Decimal Fractions to Fractions Here we multiply the fraction by 16 instead of 2 or 8. If the non-zero integer is in between 10 to 16, then the number is represented by A to F respectively. 4

5 Example- Find the hexadecimal equivalent of (0.75) 10. (to be recorded) = C (12 = C) Thus (0.75) 10 = (0.C) 16. And (45.75) 10 = (2D.C) 16. Conversions of to Decimal In converting binary to decimal, we use the following steps- (a) Write the weight of each. (b) Get the weighted value by multiplying the weighted position with the respective. (c) Add all the weighted value to get the decimal number. Example- Convert (101101) 2 into decimal number system Thus (101101) 2 = = = 45 Conversions of Fractions to Decimal Fractions The conversions of binary fractions to the decimal fractions is similar to conversion of binary numbers to decimal numbers. Here, instead of a decimal point we have a binary point. The exponential expressions (or weight of the s) of each fractional placeholder is 2-1, 2-2 Example- Convert ( ) 2 into decimal number system Thus ( ) 2 = =

6 Conversions of to - We use the following steps in converting binary to octal- (a) Break the number into 3- sections starting from LSB to MSB. (b) If we do not have sufficient s in grouping of 3-s, we add zeros to the left of MSB so that all the groups have proper 3- number. (c) Write the 3- binary number to its octal equivalent. Example- Convert (101101) 2 into octal Thus (101101) 2 = (55) 8. Example- Convert ( ) 2 into octal Thus ( ) 2 = (155) 8. Conversions of Fractions to Fractions- We use the following steps in converting binary fractions to octal fractions- (d) Break the fraction into 3- sections starting from MSB to LSB. (e) In order to get a complete grouping of 3 s, we add trailing zeros in LSB. (f) Write the 3- binary number to its octal equivalent. Example- Convert ( ) 2 into octal Thus (101101) 2 = (55.6) 8. Conversions of to - We convert binary to hexadecimal in the similar manner as we have converted binary to octal. The only difference is that here, we form the group of 4 s. 6

7 Example- Convert (101101) 2 into hexadecimal. Decimal D Thus (101101) 2 = (2D) 16. Conversions of Fractions to Fractions - We convert binary fractions to hexadecimal fractions in the similar manner as we have converted binary fractions to octal fractions. The only difference is that here we form the group of 4 s. Example- Convert ( ) 2 into hexadecimal. Decimal D C Thus ( ) 2 = (2D.C) 16. Conversions of to Decimal- We follow the same steps of conversion of binary to decimal. The only difference is that here weight of n th is 8 n-1 instead of 2 n-1. Example- Convert (55) 8 into decimal number system Thus (55) 8 = = 45 Conversions of Fractions to Decimal Fractions- The weight of the of the fraction placeholder is 8-1, 8-2. We follow the same steps of conversion of binary fractions to decimal fractions. 7

8 Example- Convert (55.6) 8 into decimal number system Thus (55.6) 8 = = Conversions of to - We use the following steps in converting octal to binary- (a) Convert each octal digit into 3- binary equivalent. (b) Combine the 3- section by removing the spaces to get the binary number. Example- Convert (55) 8 into binary Thus (55) 8 = (101101) 2. Example- Convert (456) 8 into binary Thus (456) 8 = ( ) 2. Conversions of Fractions to Fractions- We follow the same steps of conversion of octal to binary. Example- Convert (55.6) 8 into binary Thus (55.6) 8 = ( ) 2. 8

9 Conversions of to - The conversion involves the following steps- (a) Convert each octal digit to 3 binary form. (b) Combine all the 3- binary numbers. (c) Group them in 4- binary form by starting from MSB to LSB. (d) Convert these 4- blocks into their hexadecimal symbols. Example- Convert (55) 8 into hexadecimal Combining the 3- binary block, we have Grouping them in 4 binary form- Symbol D Thus (55) 8 = (2D) 16. Conversions of Fractions to Fractions- The method of conversion is based on the same procedure that we have discussed in conversions of octal to hexadecimal. Example- Convert (55.6) 8 into hexadecimal Combining the 3- binary block, we have Grouping them in 4 binary form- Symbol D C Thus (55) 8 = (2D.C) 16. Conversions of to Decimal- We do the conversion of hexadecimal to decimal as we have done the conversion of binary to decimal. Here weight of n th is 16 n-1 instead of 2 n-1. 9

10 Example- Convert (2D) 16 into decimal. 2 D(=13) Thus (2D) 16 = = 45. Conversions of Fractions to Decimal Fractions- We do the conversion of hexadecimal fractions to decimal fractions in the similar manner as we have done the conversion of binary fractions to decimal fractions. Here weight of is 16-1, Example- Convert (2D.C) 16 into decimal. 2 D(=13) C(=12) Thus (2D.C) 16 = = Conversions of to -We use the following steps- (a) Convert each hexadecimal digit to its 4- binary equivalent. (b) Combine all the binary numbers. Example- Convert (2D) 16 into binary. 2 D(=13) Thus (2D) 16 = ( ) 2 = (101101) 2. Conversions of Fractions to Fractions -We use the same steps of hexadecimal to binary conversion. 10

11 Example- Convert (2D.C) 16 into binary. 2 D(=13) C(=12) Thus (2D) 16 = ( ) 2 = ( ) 2. Conversions of to - We convert each hexadecimal digit in binary. Combine all the binary numbers. Again group them into 3- form. Convert the 3- block in octal. Example- Convert (2D) 16 into octal. 2 D(=13) Combining the binary number, we get = Grouping the binary number in Thus (2D) 16 = (55) 8. Conversions of Fractions to Fractions We follow the same steps of hexadecimal to octal conversion. Example- Convert (2D.C) 16 into octal. 2 D(=13) C(=12) Combining the binary number, we get = Grouping the binary number in Thus (2D.C) 16 = (55.6) 8. 11

### Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

### Goals. Unary Numbers. Decimal Numbers. 3,148 is. 1000 s 100 s 10 s 1 s. Number Bases 1/12/2009. COMP370 Intro to Computer Architecture 1

Number Bases //9 Goals Numbers Understand binary and hexadecimal numbers Be able to convert between number bases Understand binary fractions COMP37 Introduction to Computer Architecture Unary Numbers Decimal

### 3. Convert a number from one number system to another

3. Convert a number from one number system to another Conversion between number bases: Hexa (16) Decimal (10) Binary (2) Octal (8) More Interest Way we need conversion? We need decimal system for real

### 2 Number Systems. Source: Foundations of Computer Science Cengage Learning. Objectives After studying this chapter, the student should be able to:

2 Number Systems 2.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish

### NUMBER SYSTEMS. 1.1 Introduction

NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.

### Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8

ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also

### Useful Number Systems

Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2

### COMPSCI 210. Binary Fractions. Agenda & Reading

COMPSCI 21 Binary Fractions Agenda & Reading Topics: Fractions Binary Octal Hexadecimal Binary -> Octal, Hex Octal -> Binary, Hex Decimal -> Octal, Hex Hex -> Binary, Octal Animation: BinFrac.htm Example

### CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

### Lecture 11: Number Systems

Lecture 11: Number Systems Numeric Data Fixed point Integers (12, 345, 20567 etc) Real fractions (23.45, 23., 0.145 etc.) Floating point such as 23. 45 e 12 Basically an exponent representation Any number

### NUMBER SYSTEMS. William Stallings

NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html

### CPEN 214 - Digital Logic Design Binary Systems

CPEN 4 - Digital Logic Design Binary Systems C. Gerousis Digital Design 3 rd Ed., Mano Prentice Hall Digital vs. Analog An analog system has continuous range of values A mercury thermometer Vinyl records

### Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 04 Digital Logic II May, I before starting the today s lecture

### Binary, Hexadecimal, Octal, and BCD Numbers

23CH_PHCalter_TMSETE_949118 23/2/2007 1:37 PM Page 1 Binary, Hexadecimal, Octal, and BCD Numbers OBJECTIVES When you have completed this chapter, you should be able to: Convert between binary and decimal

### EE 261 Introduction to Logic Circuits. Module #2 Number Systems

EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook

### Number and codes in digital systems

Number and codes in digital systems Decimal Numbers You are familiar with the decimal number system because you use them everyday. But their weighted structure is not understood. In the decimal number

### Unsigned Conversions from Decimal or to Decimal and other Number Systems

Page 1 of 5 Unsigned Conversions from Decimal or to Decimal and other Number Systems In all digital design, analysis, troubleshooting, and repair you will be working with binary numbers (or base 2). It

### Section 1.4 Place Value Systems of Numeration in Other Bases

Section.4 Place Value Systems of Numeration in Other Bases Other Bases The Hindu-Arabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason

### The string of digits 101101 in the binary number system represents the quantity

Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for

### Number Systems and Radix Conversion

Number Systems and Radix Conversion Sanjay Rajopadhye, Colorado State University 1 Introduction These notes for CS 270 describe polynomial number systems. The material is not in the textbook, but will

### 2011, The McGraw-Hill Companies, Inc. Chapter 3

Chapter 3 3.1 Decimal System The radix or base of a number system determines the total number of different symbols or digits used by that system. The decimal system has a base of 10 with the digits 0 through

### 6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

### CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012

CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6-3-1-1 and Excess-3? Data Representation (1/2) Each numbering

### Chapter 2. Binary Values and Number Systems

Chapter 2 Binary Values and Number Systems Numbers Natural numbers, a.k.a. positive integers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative numbers A

### LSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (0-9) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 10-1 10-2 10-3

### Positional Numbering System

APPENDIX B Positional Numbering System A positional numbering system uses a set of symbols. The value that each symbol represents, however, depends on its face value and its place value, the value associated

### THE BINARY NUMBER SYSTEM

THE BINARY NUMBER SYSTEM Dr. Robert P. Webber, Longwood University Our civilization uses the base 10 or decimal place value system. Each digit in a number represents a power of 10. For example, 365.42

### Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

### A Step towards an Easy Interconversion of Various Number Systems

A towards an Easy Interconversion of Various Number Systems Shahid Latif, Rahat Ullah, Hamid Jan Department of Computer Science and Information Technology Sarhad University of Science and Information Technology

### Number Systems. Introduction / Number Systems

Number Systems Introduction / Number Systems Data Representation Data representation can be Digital or Analog In Analog representation values are represented over a continuous range In Digital representation

### Number Representation

Number Representation CS10001: Programming & Data Structures Pallab Dasgupta Professor, Dept. of Computer Sc. & Engg., Indian Institute of Technology Kharagpur Topics to be Discussed How are numeric data

### 4.3 TABLE 3 TABLE 4. 1342 five 1 125 3 25 4 5 2 1 125 75 20 2 222.

.3 Conversion Between Number Bases 169.3 Conversion Between Number Bases Although the numeration systems discussed in the opening section were all base ten, other bases have occurred historically. For

### Chapter 1: Digital Systems and Binary Numbers

Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching

### Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x

### Binary Numbers. Binary Octal Hexadecimal

Binary Numbers Binary Octal Hexadecimal Binary Numbers COUNTING SYSTEMS UNLIMITED... Since you have been using the 10 different digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 all your life, you may wonder how

### COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

### Binary Representation

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must tbe able to handle more than just values for real world problems

### Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.

Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems

### Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems

Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems This assignment is designed to familiarize you with different numbering systems, specifically: binary, octal, hexadecimal (and decimal)

### Chapter Binary, Octal, Decimal, and Hexadecimal Calculations

Chapter 5 Binary, Octal, Decimal, and Hexadecimal Calculations This calculator is capable of performing the following operations involving different number systems. Number system conversion Arithmetic

### To convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:

Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents

### CS321. Introduction to Numerical Methods

CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in

### Machine Architecture and Number Systems. Major Computer Components. Schematic Diagram of a Computer. The CPU. The Bus. Main Memory.

1 Topics Machine Architecture and Number Systems Major Computer Components Bits, Bytes, and Words The Decimal Number System The Binary Number System Converting from Decimal to Binary Major Computer Components

### Recall the process used for adding decimal numbers. 1. Place the numbers to be added in vertical format, aligning the decimal points.

2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added

### PROGRAMMABLE LOGIC CONTROLLERS Unit code: A/601/1625 QCF level: 4 Credit value: 15 TUTORIAL OUTCOME 2 Part 1

UNIT 22: PROGRAMMABLE LOGIC CONTROLLERS Unit code: A/601/1625 QCF level: 4 Credit value: 15 TUTORIAL OUTCOME 2 Part 1 This work covers part of outcome 2 of the Edexcel standard module. The material is

### plc numbers - 13.1 Encoded values; BCD and ASCII Error detection; parity, gray code and checksums

plc numbers - 3. Topics: Number bases; binary, octal, decimal, hexadecimal Binary calculations; s compliments, addition, subtraction and Boolean operations Encoded values; BCD and ASCII Error detection;

### Numbering Systems. InThisAppendix...

G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal

### CS201: Architecture and Assembly Language

CS201: Architecture and Assembly Language Lecture Three Brendan Burns CS201: Lecture Three p.1/27 Arithmetic for computers Previously we saw how we could represent unsigned numbers in binary and how binary

### Digital Design. Assoc. Prof. Dr. Berna Örs Yalçın

Digital Design Assoc. Prof. Dr. Berna Örs Yalçın Istanbul Technical University Faculty of Electrical and Electronics Engineering Office Number: 2318 E-mail: siddika.ors@itu.edu.tr Grading 1st Midterm -

### Figure 1. A typical Laboratory Thermometer graduated in C.

SIGNIFICANT FIGURES, EXPONENTS, AND SCIENTIFIC NOTATION 2004, 1990 by David A. Katz. All rights reserved. Permission for classroom use as long as the original copyright is included. 1. SIGNIFICANT FIGURES

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### Numeral Systems. The number twenty-five can be represented in many ways: Decimal system (base 10): 25 Roman numerals:

Numeral Systems Which number is larger? 25 8 We need to distinguish between numbers and the symbols that represent them, called numerals. The number 25 is larger than 8, but the numeral 8 above is larger

### Lecture 2. Binary and Hexadecimal Numbers

Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations

### Divide: Paper & Pencil. Computer Architecture ALU Design : Division and Floating Point. Divide algorithm. DIVIDE HARDWARE Version 1

Divide: Paper & Pencil Computer Architecture ALU Design : Division and Floating Point 1001 Quotient Divisor 1000 1001010 Dividend 1000 10 101 1010 1000 10 (or Modulo result) See how big a number can be

### Design and FPGA Implementation of a Novel Square Root Evaluator based on Vedic Mathematics

International Journal of Information & Computation Technology. ISSN 0974-2239 Volume 4, Number 15 (2014), pp. 1531-1537 International Research Publications House http://www. irphouse.com Design and FPGA

### BINARY CODED DECIMAL: B.C.D.

BINARY CODED DECIMAL: B.C.D. ANOTHER METHOD TO REPRESENT DECIMAL NUMBERS USEFUL BECAUSE MANY DIGITAL DEVICES PROCESS + DISPLAY NUMBERS IN TENS IN BCD EACH NUMBER IS DEFINED BY A BINARY CODE OF 4 BITS.

### Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6

Everything you wanted to know about using Hexadecimal and Octal Numbers in Visual Basic 6 Number Systems No course on programming would be complete without a discussion of the Hexadecimal (Hex) number

### Memory is implemented as an array of electronic switches

Memory Structure Memory is implemented as an array of electronic switches Each switch can be in one of two states 0 or 1, on or off, true or false, purple or gold, sitting or standing BInary digits (bits)

### Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

### 6 The Hindu-Arabic System (800 BC)

6 The Hindu-Arabic System (800 BC) Today the most universally used system of numeration is the Hindu-Arabic system, also known as the decimal system or base ten system. The system was named for the Indian

### CS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS

### Grade 6 Math Circles. Binary and Beyond

Faculty of Mathematics Waterloo, Ontario N2L 3G1 The Decimal System Grade 6 Math Circles October 15/16, 2013 Binary and Beyond The cool reality is that we learn to count in only one of many possible number

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### Chapter 1. Binary, octal and hexadecimal numbers

Chapter 1. Binary, octal and hexadecimal numbers This material is covered in the books: Nelson Magor Cooke et al, Basic mathematics for electronics (7th edition), Glencoe, Lake Forest, Ill., 1992. [Hamilton

### Binary Division. Decimal Division. Hardware for Binary Division. Simple 16-bit Divider Circuit

Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend -480 41-36 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until

### Bachelors of Computer Application Programming Principle & Algorithm (BCA-S102T)

Unit- I Introduction to c Language: C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating

### Solution to Exercise 2.2. Both m and n are divisible by d, som = dk and n = dk. Thus m ± n = dk ± dk = d(k ± k ),som + n and m n are divisible by d.

[Chap. ] Pythagorean Triples 6 (b) The table suggests that in every primitive Pythagorean triple, exactly one of a, b,orc is a multiple of 5. To verify this, we use the Pythagorean Triples Theorem to write

### COMBINATIONAL CIRCUITS

COMBINATIONAL CIRCUITS http://www.tutorialspoint.com/computer_logical_organization/combinational_circuits.htm Copyright tutorialspoint.com Combinational circuit is a circuit in which we combine the different

### Chapter 5. Binary, octal and hexadecimal numbers

Chapter 5. Binary, octal and hexadecimal numbers A place to look for some of this material is the Wikipedia page http://en.wikipedia.org/wiki/binary_numeral_system#counting_in_binary Another place that

### Verilog - Representation of Number Literals

Verilog - Representation of Number Literals... And here there be monsters! (Capt. Barbossa) Numbers are represented as: value ( indicates optional part) size The number of binary

### Decimal to Binary Conversion

Decimal to Binary Conversion A tool that makes the conversion of decimal values to binary values simple is the following table. The first row is created by counting right to left from one to eight, for

### Chapter 4: Computer Codes

Slide 1/30 Learning Objectives In this chapter you will learn about: Computer data Computer codes: representation of data in binary Most commonly used computer codes Collating sequence 36 Slide 2/30 Data

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### 2010/9/19. Binary number system. Binary numbers. Outline. Binary to decimal

2/9/9 Binary number system Computer (electronic) systems prefer binary numbers Binary number: represent a number in base-2 Binary numbers 2 3 + 7 + 5 Some terminology Bit: a binary digit ( or ) Hexadecimal

### Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

### Playing with Numbers

PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also

### The BBP Algorithm for Pi

The BBP Algorithm for Pi David H. Bailey September 17, 2006 1. Introduction The Bailey-Borwein-Plouffe (BBP) algorithm for π is based on the BBP formula for π, which was discovered in 1995 and published

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC

Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n

### Computers. Hardware. The Central Processing Unit (CPU) CMPT 125: Lecture 1: Understanding the Computer

Computers CMPT 125: Lecture 1: Understanding the Computer Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 3, 2009 A computer performs 2 basic functions: 1.

### Math Released Set 2015. Algebra 1 PBA Item #13 Two Real Numbers Defined M44105

Math Released Set 2015 Algebra 1 PBA Item #13 Two Real Numbers Defined M44105 Prompt Rubric Task is worth a total of 3 points. M44105 Rubric Score Description 3 Student response includes the following

### Number of bits needed to address hosts 8

Advanced Subnetting Example 1: Your ISP has assigned you a Class C network address of 198.47.212.0. You have 3 networks in your company with the largest containing 134 hosts. You need to figure out if

### Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

### Streaming Lossless Data Compression Algorithm (SLDC)

Standard ECMA-321 June 2001 Standardizing Information and Communication Systems Streaming Lossless Data Compression Algorithm (SLDC) Phone: +41 22 849.60.00 - Fax: +41 22 849.60.01 - URL: http://www.ecma.ch

### Multiplying and Dividing Signed Numbers. Finding the Product of Two Signed Numbers. (a) (3)( 4) ( 4) ( 4) ( 4) 12 (b) (4)( 5) ( 5) ( 5) ( 5) ( 5) 20

SECTION.4 Multiplying and Dividing Signed Numbers.4 OBJECTIVES 1. Multiply signed numbers 2. Use the commutative property of multiplication 3. Use the associative property of multiplication 4. Divide signed

### DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION

DIGITAL-TO-ANALOGUE AND ANALOGUE-TO-DIGITAL CONVERSION Introduction The outputs from sensors and communications receivers are analogue signals that have continuously varying amplitudes. In many systems

### 26 Integers: Multiplication, Division, and Order

26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue

### Cyber Security Workshop Encryption Reference Manual

Cyber Security Workshop Encryption Reference Manual May 2015 Basic Concepts in Encoding and Encryption Binary Encoding Examples Encryption Cipher Examples 1 P a g e Encoding Concepts Binary Encoding Basics

### The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

### Lecture 8: Binary Multiplication & Division

Lecture 8: Binary Multiplication & Division Today s topics: Addition/Subtraction Multiplication Division Reminder: get started early on assignment 3 1 2 s Complement Signed Numbers two = 0 ten 0001 two

### COURSE TITLE: INTRODUCTION TO COMPUTER DESIGN

NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: CIT344 COURSE TITLE: COURSE GUIDE COURSE GUIDE CIT344 Course Team Adaora Obayi (Developer/Writer) - NOUN Dr. Oyebanji (Programme

### Counting in base 10, 2 and 16

Counting in base 10, 2 and 16 1. Binary Numbers A super-important fact: (Nearly all) Computers store all information in the form of binary numbers. Numbers, characters, images, music files --- all of these

### 1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12

C5 Solutions 1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12 To Compute 0 0 = 00000000 To Compute 1 Step 1. Convert 1 to binary 00000001 Step 2. Flip the bits 11111110

### Solution for Homework 2

Solution for Homework 2 Problem 1 a. What is the minimum number of bits that are required to uniquely represent the characters of English alphabet? (Consider upper case characters alone) The number of

### Logic Reference Guide

Logic eference Guide Advanced Micro evices INTOUCTION Throughout this data book and design guide we have assumed that you have a good working knowledge of logic. Unfortunately, there always comes a time