# Base Conversion written by Cathy Saxton

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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, 10 3 = 1000, etc. Each digit in a base 10 number can have a value from 0 to 9. Base 10 numbers are commonly referred to as decimal numbers. 2. Notation As we work with numbers, the following notation will be used. The top row of the table shows the value for each place in the number. The second row shows the actual digits in the number. 253 is represented by: 100 s 10 s 1 s places digits 3. General Base Definitions Base n means that each place (x) in the number represents n x. (n 0, n 1, n 2, n 3, etc.) Valid digits in base n are 0 to n-1. When working with numbers in different bases, a subscript is usually used to indicate the base (e.g means 253 in base 10). We can represent any number in any base. The number will usually look different when expressed in another base, but its value is the same. When a number is stored in the computer s memory, it has a specific value; we can show it using base 10, base 2, base 16, etc., but the value never changes -- only the representation changes. 4. Base 2: Binary Each digit in a base 2 number can have a value of 0 or 1. Places are 1 s, 2 s, 4 s, 8 s, etc. (2 0 = 1, 2 1 = 2, 2 2 = 4, 2 3 = 8, etc.) Computers store numbers in binary, which is the common name for base 2 numbers. Each binary digit corresponds to one bit of memory. There are 8 bits (or 8 binary digits) in a byte. The computer stores values in memory by turning these bits on (1) or off (0) to correspond to the binary representation of the number Converting binary to decimal Converting from binary (base 2) to decimal (base 10) requires doing a little math with the digits in each place. Here is an example for converting a binary value of 1001 to its decimal equivalent =? 10 Remember that the places are 1 s, 2 s, 4 s, 8 s, etc: (2 3 ) (2 2 ) (2 1 ) (2 0 ) 8 s We make the following calculation to get the equivalent decimal number: = (1 * 8) + (0 * 4) + (0 * 2) + (1 * 1) = = 9 10 Here s another example: =? = (1 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1) = = Base Conversion 1

2 4.2. Converting decimal to binary Converting from decimal to binary is a little trickier. Let s consider the case of converting a decimal 5 to its binary representation =? 2 We ll calculate the appropriate digit for each place in the resulting binary number, working from the left to the right. First, we determine the largest power of 2 that s less than or equal to the number. In this case, 8 (2 3 ) is too large, so we start with the 4 (2 2 ). Therefore, we know the result will contain digits in three places -- 4 s, 2 s and 1 s:??? We need to figure out which digits should be 1 s, and which should be 0 s (since these are the only legal values for a binary digit). We can see that putting a 1 in the 4 s place will help: 1?? This accounts for 4 of the 5 units that we need to represent. That means that we can t have any 2 s (or the total would become larger than our goal of 5). So, we put a 0 in the 2 s place: 1 0? That leaves the 1 s place to fill, and we see that putting a 1 there will give us the desired result: this is the equivalent binary value 5 10 = (1 * 4) + (0 * 2) + (1 * 1) = Let s work though another example: =? 2 In this example, we need to start with the 16 s place (2 4 ); the next place (2 5 ) is the 32 s, which is too large:????? Putting a 1 in the 16 s place means that we ve accounted for 16 of 22 units. This leaves = 6 remaining: 1???? total so far: 16 remaining: 6 We can t use any 8 s since we only have 6 units remaining. (You can also observe that = 24, which is too large.) So, we put a 0 in the 8 s place: 1 0??? total so far: 16 remaining: 6 We now see whether we can use a 4. Since 4 is less than the remaining 6, the 4 will be helpful, so we put a 1 in the 4 s place. That means that we now have remaining: 6 (our previous value) - 4 (what we just accounted for) = ?? total so far: = 20 remaining: 2 Next we look at the 2 s. Since that s exactly what we still need, we put a 1 in the 2 s place. That leaves us with 0 remaining. Since we don t need anything more, we put a 0 in the 1 s place: the binary representation for = (1 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (0 * 1) = Base Conversion

3 5. Base 16: Hexadecimal As we look at a numbers stored by the computer, it quickly gets cumbersome to deal with long sequences of 0 s and 1 s in the binary representation. Programmers often use base 16 since it provides a convenient way to express numbers and is easily converted to and from base 2. For a base 16 number, the places are 1 s, 16 s, 256 s, etc. (16 0 = 1, 16 1 = 16, 16 2 = 256). Each digit can hold a value from 0 to 15. In order to have single characters for representing 10, 11, 12, 13, 14, and 15, we use A, B, C, D, E, and F, respectively. Thus, the valid values for a base 16 digit are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Base 16 numbers are commonly referred to as hexadecimal or hex numbers. In C/C++ programming, hex numbers often use the prefix 0x to indicate hex numbers, e.g. 0x3AC Converting hex to decimal Converting from hex (base 16) to decimal (base 10) is similar to converting binary to decimal. Here is an example for converting a hex value of 45 to its decimal representation =? 10 Remember that the places are 1 s and 16 s: (16 1 ) (16 0 ) 4 5 We can make the following calculation to get the equivalent decimal number: = (4 * 16) + (5 * 1) = = Here s another example. 3AC 16 =? 10 (16 2 ) (16 1 ) (16 0 ) 3 A (=10) C (=12) 3AC 16 = (3 * 256) + (10 * 16) + (12 * 1) = = Converting decimal to hex To convert from decimal to hex, we use the same technique as we did above when converting from decimal to binary. Let s consider the case of converting a decimal 73 to its hex equivalent =? 16 We ll calculate the appropriate digit for each place in the resulting hex number, working from the left to the right. First, we determine the largest power of 16 that s less than or equal to the number. In this case, 256 (16 2 ) is too large, so we ll start with the 16 s place. So, we know the result will contain digits in two places s and 1 s:?? 73 / 16 = 4 with a remainder of 9. So, there are four 16 s in 73. Now we know part of the result: 4? So, we ve accounted for 4 * 16 = 64 of the 73. That leaves the remainder, 9, which we take care of by putting it in the 1 s place: = (4 * 16) + (9 * 1) = Base Conversion 3

4 Let s work though another example: =? 16 In this case, we start with 256 (16 2 ). The next place is 16 3 = 4096, which is larger than our value. The resulting hex number will have 3 digits:??? Calculate the digit for the 256 s place: 940 / 256 = 3 with a remainder of 172. So, we ll put a 3 in the 256 s place. 3?? Calculate how much is left over: we ve accounted for 3 * 256 = 768; that leaves = 172. Calculate the digit for the 16 s place: 172 / 16 = 10 with a remainder of 12. So, we need ten 16 s. Recall that 10 is represented with A. 3 A? The remainder, 12, tells us how much we still need to take care of. We put that value in the 1 s place, using C to represent = (3 * 256) + (10 * 16) + (12 * 1) = 3AC Converting Between Binary and Hexadecimal Converting between base 2 and base 16 is much simpler than converting either to base 10. The minimum value that can be expressed with four binary digits is 0000, or The maximum value that can be expressed with four binary digits is 1111, or ( ). So, we see that four binary digits can be used to represent the numbers 0-15, which is exactly the range represented by a single hex digit. Here are conversions from binary to hex: = = = = = = = = = = = A = B = C = D = E = F 16 Notice that every value that can be expressed with four binary digits can be expressed with a single hex digit, and every value that can be expressed with a single hex digit can be expressed with four binary digits. Every hex digit can be represented with four binary digits. Every four binary digits can be converted to one hex digit. To convert from binary to hex, separate the binary number into groups of four digits starting on the right. Each group can be converted to a hex value. For example, can be visualized as two 4-digit groups: ; each group is converted to a hex value, and we get a result of 92: = For binary numbers like that don t divide evenly into 4-digit groups, add 0 s to the left as necessary: = Converting hex to binary is just the reverse; be careful to write four binary digits for each hex digit -- that means starting with a zero for values less than 8. B4 16 = You don t need the zeros at the beginning of the binary number: 6F 16 = _ Base Conversion

5 Memory is just a continuous span of 1 s and 0 s. Modern computers typically like to retrieve 32 bits at a time. While we often use all 32 bits to represent a single value, it is also common to use that 32-bit space to store 2 or 4 values. Consider a 32-bit number such as: The following table shows how we can interpret that memory -- as a single 32-bit value, two 16-bit values, or four 8-bit values: Binary Representation Hex Decimal One Value: A Two Values: , , A023 69, Four Values: , , , , 45, A0, 23 0, 69, 160, 35 Notice that by using the hex representation for the 32-bit value, you can easily determine the 2 or 4 values that are held in the 32 bits of memory. Ultimately, the computer uses binary to store numbers. Unfortunately, representing numbers in binary is very verbose and hard to read. Using a base that s a power of 2 retains the binary nature of computer math, but yields more compact and readable numbers. Base 4 isn t much of an improvement. Base 8 is good, but represents binary digits in groups of three, while computers generally use groups of eight. Base 256 would require too many symbols. Base 16 represents binary digits in convenient groups of four, and only requires 16 symbols. That is why hex is commonly used when programming. 7. Other Interesting Facts When we put a 0 at the end of a base 10 number, it has the effect of multiplying the value by 10; e.g. adding a 0 after 37 gives us 370, or 37 * 10. The same works for other bases, with the multiplier being the base number; e.g (5 10 ) becomes (10 10 = 5 [the original value] * 2 [the base]). Dropping the last digit does an integer divide by the base; e.g. 368 / 10 = 36. Base 8 s common name is octal. In C or C++, if you start an integer with a 0, it will be interpreted as an octal value. Conversion from binary to octal is like converting to hex, but using groups of three binary digits; e.g = When converting between hex and octal, the easiest way is to start by representing the value in binary! 8. Summary base common name places valid digit values 10 decimal 1, 10, 100, 1000, etc. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 2 binary 1, 2, 4, 8, 16, 32, etc. 0, 1 16 hexadecimal, hex 1, 16, 256, 4096, etc. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Base Conversion 5

### 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

### Number System. Some important number systems are as follows. Decimal number system Binary number system Octal number system Hexadecimal number system

Number System When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand positional number system where there are only

### Data Representation. Why Study Data Representation?

Why Study Data Representation? Computers process and store information in binary format For many aspects of programming and networking, the details of data representation must be understood C Programming

### NUMBER SYSTEMS CHAPTER 19-1

NUMBER SYSTEMS 19.1 The Decimal System 19. The Binary System 19.3 Converting between Binary and Decimal Integers Fractions 19.4 Hexadecimal Notation 19.5 Key Terms and Problems CHAPTER 19-1 19- CHAPTER

### 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

### UNIT 2 : NUMBER SYSTEMS

UNIT 2 : NUMBER SYSTEMS page 2.0 Introduction 1 2.1 Decimal Numbers 2 2.2 The Binary System 3 2.3 The Hexadecimal System 5 2.4 Number Base Conversion 6 2.4.1 Decimal To Binary 6 2.4.2 Decimal to Hex 7

### Number Systems and Base Conversions

Number Systems and Base Conversions As you know, the number system that we commonly use is the decimal or base- 10 number system. That system has 10 digits, 0 through 9. While it's very convenient for

### 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

### 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...

### Number Systems Richard E. Haskell

NUMBER SYSTEMS D Number Systems Richard E. Haskell Data inside a computer are represented by binary digits or bits. The logical values of these binary digits are denoted by and, while the corresponding

### 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

### Common Number Systems Number Systems

5/29/204 Common Number Systems Number Systems System Base Symbols Used by humans? Used in computers? Decimal 0 0,, 9 Yes No Binary 2 0, No Yes Octal 8 0,, 7 No No Hexadecimal 6 0,, 9, A, B, F No No Number

### CHAPTER 3 Number System and Codes

CHAPTER 3 Number System and Codes 3.1 Introduction On hearing the word number, we immediately think of familiar decimal number system with its 10 digits; 0,1, 2,3,4,5,6, 7, 8 and 9. these numbers are called

### Lab 1: Information Representation I -- Number Systems

Unit 1: Computer Systems, pages 1 of 7 - Department of Computer and Mathematical Sciences CS 1408 Intro to Computer Science with Visual Basic 1 Lab 1: Information Representation I -- Number Systems Objectives:

### Lab 1: Information Representation I -- Number Systems

Unit 1: Computer Systems, pages 1 of 7 - Department of Computer and Mathematical Sciences CS 1410 Intro to Computer Science with C++ 1 Lab 1: Information Representation I -- Number Systems Objectives:

### 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

### 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

### Chapter 2 Numeric Representation.

Chapter 2 Numeric Representation. Most of the things we encounter in the world around us are analog; they don t just take on one of two values. How then can they be represented digitally? The key is that

### Grade 6 Math Circles October 25 & 26, Number Systems and Bases

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 25 & 26, 2016 Number Systems and Bases Numbers are very important. Numbers

### 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

### Chapter 3: Number Systems

Slide 1/40 Learning Objectives In this chapter you will learn about: Non-positional number system Positional number system Decimal number system Binary number system Octal number system Hexadecimal number

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

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

### MATH 2420 Discrete Mathematics Lecture notes Numbering Systems

MATH 2420 Discrete Mathematics Lecture notes Numbering Systems Objectives: Introduction: 1. Represent a binary (hexadecimal, octal) number as a decimal number. 2. Represent a decimal (hexadecimal, octal)

### Binary Numbers The Computer Number System

Binary Numbers The Computer Number System Number systems are simply ways to count things. Ours is the base-0 or radix-0 system. Note that there is no symbol for 0 or for the base of any system. We count,2,3,4,5,6,7,8,9,

### Understanding Binary Numbers. Different Number Systems. Conversion: Bin Hex. Conversion MAP. Binary (0, 1) Hexadecimal 0 9, A(10), B(11),, F(15) :

Understanding Binary Numbers 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

### Number Representation and Arithmetic in Various Numeral Systems

1 Number Representation and Arithmetic in Various Numeral Systems Computer Organization and Assembly Language Programming 203.8002 Adapted by Yousef Shajrawi, licensed by Huong Nguyen under the Creative

### The Mathematics Driving License for Computer Science- CS10410

The Mathematics Driving License for Computer Science- CS10410 Approximating Numbers, Number Systems and 2 s Complement by Nitin Naik Approximating Numbers There are two kinds of numbers: Exact Number and

### CSC 1103: Digital Logic. Lecture Six: Data Representation

CSC 1103: Digital Logic Lecture Six: Data Representation Martin Ngobye mngobye@must.ac.ug Mbarara University of Science and Technology MAN (MUST) CSC 1103 1 / 32 Outline 1 Digital Computers 2 Number Systems

### Information Science 1

Information Science 1 - Representa*on of Data in Memory- Week 03 College of Information Science and Engineering Ritsumeikan University Topics covered l Basic terms and concepts of The Structure of a Computer

### Number Systems. Decimal Number System. Number Systems Week 3

Number Systems When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand positional number system where there are only

### Number Systems (2.1.1)

Computer Studies 0 Syllabus Number Systems (..) Representation of numbers in binary. Conversion between decimal and binary, and between binary and hexadecimal. Use of subscripts, 0 and 6 for bases. The

### DIT250 / BIT180 - MATHEMATICS. Number Bases. Decimal numbers (base ten), Decimal numbers (base ten), Decimal numbers (base ten), 3/24/2014

DIT250 / BIT180 - MATHEMATICS ASSESSMENT CA 40% 3 TESTS EXAM 60% LECTURE SLIDES www.lechaamwe.weebly.com Lecture notes DIT250/BIT180 Number Bases In this lesson we shall discuss different Number Bases,

### Grade 6 Math Circles October 25 & 26, Number Systems and Bases

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 6 Math Circles October 25 & 26, 2016 Number Systems and Bases Numbers are very important. Numbers

### Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

### Binary Which powers of 2 are we adding? Decimal answer 10101

1 CS 105 Lab #1 representation of numbers numbers are the dough we use to represent all information inside the computer. The purpose of this lab is to practice with some binary number representations.

### 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

### NUMBERING SYSTEMS C HAPTER 1.0 INTRODUCTION 1.1 A REVIEW OF THE DECIMAL SYSTEM 1.2 BINARY NUMBERING SYSTEM

12 Digital Principles Switching Theory C HAPTER 1 NUMBERING SYSTEMS 1.0 INTRODUCTION Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk,

### 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

### 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

### The Power of Binary 0, 1, 10, 11, 100, 101, 110,

The Power of Binary 0, 1, 10, 11, 100, 101, 110, 111... What is Binary? a binary number is a number expressed in the binary numeral system, or base-2 numeral system, which represents numeric values using

### Digital Electronics. 1.0 Introduction to Number Systems. Module

Module 1 www.learnabout-electronics.org Digital Electronics 1.0 Introduction to What you ll learn in Module 1 Section 1.0. Recognise different number systems and their uses. Section 1.1 in Electronics.

### CHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems

CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,

### 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

### 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

### 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

### Lecture 1: Digital Systems and Binary Numbers

Lecture 1: Digital Systems and Binary Numbers Matthew Shuman September 30th, 2009 1 Digital Systems 1-1 in Text 1.1 Analog Systems Analog systems are continuous. Look at the analog clock in figure 1. The

### 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

### Data Representation in Computers

Chapter 3 Data Representation in Computers After studying this chapter the student will be able to: *Learn about binary, octal, decimal and hexadecimal number systems *Learn conversions between two different

### 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

### 23 1 The Binary Number System

664 Chapter 23 Binary, Hexadecimal, Octal, and BCD Numbers 23 The Binary Number System Binary Numbers A binary number is a sequence of the digits 0 and, such as 000 The number shown has no fractional part

### Collin's Lab: Binary & Hex

Collin's Lab: Binary & Hex Created by Collin Cunningham Last updated on 2014-08-21 01:30:12 PM EDT Guide Contents Guide Contents Video Transcript Learn More Comparison Converters Mobile Web Desktop 2 3

### Digital Logic. The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer.

Digital Logic 1 Data Representations 1.1 The Binary System The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer. The system we

### Lecture 1: Digital Systems and Number Systems

Lecture 1: Digital Systems and Number Systems Matthew Shuman April 3rd, 2013 The Digital Abstraction 1.3 in Text Analog Systems Analog systems are continuous. Look at the analog clock in figure 1. The

### Lecture 8 Binary Numbers & Logic Operations The focus of the last lecture was on the microprocessor

Lecture 8 Binary Numbers & Logic Operations The focus of the last lecture was on the microprocessor During that lecture we learnt about the function of the central component of a computer, the microprocessor

### 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

### 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

### Introduction ToNumber Systems

Introduction ToNumber Systems Collected by: Zainab Alkadhem Page 1 of 25 Digital systems have such a prominent role in everyday life that we refer to the present technological period as the digital age.

### 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)

### 1. True or False? A natural number is the number 0 or any number obtained by adding 1 to a natural number.

CS Illuminated, 5 th ed. Chapter 2 Review Quiz 1. True or False? A natural number is the number 0 or any number obtained by adding 1 to a natural number. 2. True or False? The category of numbers called

### Encoding Systems: Combining Bits to form Bytes

Encoding Systems: Combining Bits to form Bytes Alphanumeric characters are represented in computer storage by combining strings of bits to form unique bit configuration for each character, also called

### NUMBERSYSTEMS hundreds 8 tens 5 ones Thus,the numeral 485 represents the number four-hundred eighty-five and can be written in expanded formas

NUMBERSYSTEMS We noted in several places that a binary scheme having only the two binary digits 0 and 1 is used to represent information in a computer. In PART OF THE PIC- TURE: Data Representation in

### 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

### CSCC85 Spring 2006: Tutorial 0 Notes

CSCC85 Spring 2006: Tutorial 0 Notes Yani Ioannou January 11 th, 2006 There are 10 types of people in the world, those who understand binary, and those who don t. Contents 1 Number Representations 1 1.1

### 1 Number systems 1.1 DECIMAL SYSTEM

A programmable logical controller uses the binary system rather than the decimal system to process memory cells, inputs, outputs, timers, flags etc.. DECIMAL SYSTEM In order to understand the binary number

### 2 Number Systems 2.1. Foundations of Computer Science Cengage Learning

2 Number Systems 2.1 Foundations of Computer Science Cengage Learning 2.2 Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish between

### Here 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).

Number System Introduction Number systems provide the basis for all operations in information processing systems. In a number system the information is divided into a group of symbols; for example, 26

### Chapter II Binary Data Representation

Chapter II Binary Data Representation The atomic unit of data in computer systems is the bit, which is actually an acronym that stands for BInary digit. It can hold only 2 values or states: 0 or 1, true

### Chapter 2: Number Systems

Chapter 2: Number Systems Logic circuits are used to generate and transmit 1s and 0s to compute and convey information. This two-valued number system is called binary. As presented earlier, there are many

### NUMBER REPRESENTATIONS IN THE COMPUTER for COSC 120

NUMBER REPRESENTATIONS IN THE COMPUTER for COSC 120 First, a reminder of how we represent base ten numbers. Base ten uses ten (decimal) digits: 0, 1, 2,3, 4, 5, 6, 7, 8, 9. In base ten, 10 means ten. Numbers

### 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

### Binary Representation and Computer Arithmetic

Binary Representation and Computer Arithmetic The decimal system of counting and keeping track of items was first created by Hindu mathematicians in India in A.D. 4. Since it involved the use of fingers

### By the end of the lecture, you should be able to:

Extra Lecture: Number Systems Objectives - To understand: Base of number systems: decimal, binary, octal and hexadecimal Textual information stored as ASCII Binary addition/subtraction, multiplication

### 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)

### Digital Fundamentals

Digital Fundamentals with PLD Programming Floyd Chapter 2 29 Pearson Education Decimal Numbers The position of each digit in a weighted number system is assigned a weight based on the base or radix of

### A B C

Data Representation Module 2 CS 272 Sam Houston State University Dr. Tim McGuire Copyright 2001 by Timothy J. McGuire, Ph.D. 1 Positional Number Systems Decimal (base 10) is an example e.g., 435 means

### Note that the exponents also decrease by 1 with each column move to the right, so the

Base Systems Jacqueline A. Jones People use the decimal number system to perform arithmetic operations. Computers, on the other hand, use the binary system, which contains only two digits: 0 and 1. We

### 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

### Computer Architecture CPIT 210 LAB 1 Manual. Prepared By: Mohammed Ghazi Al Obeidallah.

Computer Architecture CPIT 210 LAB 1 Manual Prepared By: Mohammed Ghazi Al Obeidallah malabaidallah@kau.edu.sa LAB 1 Outline: 1. Students should understand basic concepts of Decimal system, Binary system,

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

Digital System Design Prof. D. Roychoudhury Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 03 Digital Logic - I Today, we will start discussing on Digital

### Activity 1: Bits and Bytes

ICS3U (Java): Introduction to Computer Science, Grade 11, University Preparation Activity 1: Bits and Bytes The Binary Number System Computers use electrical circuits that include many transistors and

### How to count like a computer. 2010, Robert K. Moniot 1

Binary and Hex How to count like a computer 2010, Robert K. Moniot 1 Why Binary? The basic unit of storage in a computer is the bit (binary digit), which can have one of just two values: 0 or 1. This is

### Integer Numbers. The Number Bases of Integers Textbook Chapter 3

Integer Numbers The Number Bases of Integers Textbook Chapter 3 Number Systems Unary, or marks: /////// = 7 /////// + ////// = ///////////// Grouping lead to Roman Numerals: VII + V = VVII = XII Better:

### 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

### 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

### 1.3 Data Representation

8628-28 r4 vs.fm Page 9 Thursday, January 2, 2 2:4 PM.3 Data Representation 9 appears at Level 3, uses short mnemonics such as ADD, SUB, and MOV, which are easily translated to the ISA level. Assembly

### Number Systems and Data Representation CS221

Number Systems and Data Representation CS221 Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk, or as a state in a transistor, core,

### Lesson Plan. Preparation

Math Computer Technician Practicum Lesson Plan Performance Objective Upon completion of this lesson, each student will be able to convert between different numbering systems and correctly write mathematical

### Digital Number Representation

Review Questions: Digital Number Representation 1. What is the significance of the use of the word digit to represent a numerical quantity? 2. Which radices are natural (i.e., they correspond to some aspect

### 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

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers. Factors 2. Multiples 3. Prime and Composite Numbers 4. Modular Arithmetic 5. Boolean Algebra 6. Modulo 2 Matrix Arithmetic 7. Number Systems

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement

Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement Topic 1: Binary, Octal, and Hexadecimal Numbers The number system we generally use in our everyday lives is a decimal

### 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

### EE 3170 Microcontroller Applications

EE 37 Microcontroller Applications Lecture 3 : Digital Computer Fundamentals - Number Representation (.) Based on slides for ECE37 by Profs. Sloan, Davis, Kieckhafer, Tan, and Cischke Number Representation

### Section Summary. Integer Representations. Base Conversion Algorithm Algorithms for Integer Operations

Section 4.2 Section Summary Integer Representations Base b Expansions Binary Expansions Octal Expansions Hexadecimal Expansions Base Conversion Algorithm Algorithms for Integer Operations Representations

### 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

### MT1 Number Systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number:

MT1 Number Systems MT1.1 Introduction A number system is a well defined structured way of representing or expressing numbers as a combination of the elements of a finite set of mathematical symbols (i.e.,

### Decimal Numbers: Base 10 Integer Numbers & Arithmetic

Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number