DOCSLIB.ORG

Number Systems: Binary and Hexadecimal Tom Brennan

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Number Systems: Binary and Hexadecimal Tom Brennan Number Systems: Binary and Hexadecimal Tom Brennan Base 10 The base 10 number system is the common number system. 612 = 2 · 100 + 1 · 101 + 6 · 102 2 + 10 + 600 = 612 4365 = 5 · 100 + 6 · 101 + 3 · 102 + 4 · 103 5 + 60 + 300 + 4000 = 4365 Binary 101010101 = 1 · 20 + 0 · 21 + 1 · 22 + 0 · 23 + 1 · 24 + 0 · 25 + 1 · 26 + 0 · 27 + 1 · 28 = 341 1 + 0 + 4 + 0 + 16 + 0 + 64 + 0 + 256 = 341 The binary number system employs only two digits{0 and 1{and each digit is called a bit. A number can be represented in binary by a string of bits{each bit either a 0 or a 1 multiplied by a power of two. For example, to represent the decimal number 15 in binary you would write 1111. That string stands for 1 · 20 + 1 · 21 + 1 · 22 + 1 · 23 = 15 (1 + 2 + 4 + 8 = 15) History Gottfried Leibniz developed the modern binary system in 1679 and drew his inspiration from the ancient Chinese. Leibniz found a technique in the I Ching{an ancient chinese text{that was similar to the modern binary system. The system formed numbers by arranging a set of hexagrams in a specific order in a manner similar to modern binary. The systems base two nature was derived from the concepts of yin and yang. This system is at least as old as the Zhou Dynasty{it predates 700 BC. While Leibniz based his system of the work of the Chinese, other cultures had their own forms of base two number systems. The Ancient Egyptians employed a type of base two number system called Horus-Eye fractions for their measurements of grain or liquids. The system used sums of binary fractions{which were , , , 1/16, 1/32, and 1/64{to represent fractions of hekats, which were the primary unit of measure in Ancient Egypt and equal to about 4.8 liters. Horus-Eye fractions were in use prior to 1200 BC. Ancient Egyptian multiplication also operated with a system similar to that of the binary number system. The Ancient Indians also had a form of binary. Its purpose differed from those of China and Egypt in that it was meant to describe poetic metre. It was created by the scholar Pingala in about 150 BC, and its duality was derived from describing syllables as either short or long. Applications Binary is used extensively in computer science and telecommunications. It usefulness is derived from the fact that only two digits are required, so a computer can represent the digit 1 with an electrical signal or 0 with no electric signal. The American Standard Code for Information Exchange is used to represent text and symbols within computer memory. It employs seven digit binary strings{which can represent values between 0 and 127(as 26 + 25 + 24 + 23 + 22 + 21 + 20 = 127). Each symbol is represented by a value between 0 and 127. The lower case letter "b", for example,is assigned the number 98 represented by the bit string 1100010 in the ASCII system. Eventually, extended ASCII sets would be released by different computer companies{such as IBM{where data would be processed in 8 bit strings called bytes. The addition of an 8th bit increases the number of possible values excluding 0 from 127 to 255 (because 27 = 128 and 128+127=255). The Braille alphabet also relies on a binary system{because in each position a dot can either be raised from the page or not raised from the page. Information in Braille is organized into cells{rectangular blocks with tiny bumps called raised dots. Each cell has two columns and 3 rows, and there are 63 different cells in braille if you do not count the space. Morse code can be transmitted as a binary code. Telegraph operators send electrical pulses along a telegraph wire, and the presence of a pulse{the on state{represents a one while the absence of a pulse{the off state{ represents a 0. A bit string of 1 represents a short mark, 111 represents a longer mark, 0 represents the gap between the dots and dashes within a character, 000 represents the gap between letters, and 0000000 represents the gap between words. The gray code{also known as the reflected binary code{is a form of binary number system that is employed in color television and bar code scanners. It was first created to address a specific problem with the binary number system that is encountered when using physical switches. In many instances consecutive numbers do not share common digits in their binary representations. The quintessential example of this occurs with the numbers three and four. Three is represented by 011 and four is represented by 100. To switch from three to four all three switches must change state, and it is very unlikely that they all three changes will occur at the exact same time. This can lead to some ambiguity when transmitting codes, as it may be difficult to tell when switches are displaying a position or an intermediate between two positions. Gray code differs from the traditional binary systems by only changing one switch at a time. Gray codes have been extremely useful in the field of telegraphy{Emile Baudot received the French Legion of Honor medal for his work in telegraphy with Gray codes in 1878. However, they were made famous and named for Frank Gray, who figured out how to use a machine called the "PCM tube" to convert analog signals to gray codes in 1953{this method was key in the development of color television. Binary Addition Binary addition functions in a manner very similar to decimal addition. However, instead of carrying a one every time two digits add to 10 or a value greater than 10, a one is carried every time two digits add up to 2 or greater(this could only occur if both binary digits were one and a one is carried from the addition of the previous digits. 1010+1111 First Column: 0+1 =1 2 Second Column: 1+1=0 (one will be carried) Third Column: 0+1+1 (the carried one)=0 (another one will be carried) Fourth Column: 1+1+1(carried one)=1 (one will be carried) Fifth Column: 0+0+1(the carried one)=1 So the sum 1010+1111 is 11001. An easy way to verify this result is to convert the three binary numbers to decimal numbers. The binary number 1010 converts to 10 (1 · 23 + 1 · 21 = 10) The binary number 1111 converts to 15 (1 · 23 + 1 · 22 + 1 · 21 + 1 · 20 = 15) The binary number 11001 converts to 25 (1 · 24 + 1 · 23 + 1 · 20 = 25) It is easy to verify that 10 + 15 = 25 Binary Subtraction Binary subtraction is more complicated than binary addition due to the fact that borrowing is often necessary in binary subtraction. Borrowing in binary is similar to borrowing in decimal, but instead of borrowing a 1 · 101 you borrow a 1 · 21. This can get complicated when you have to borrow from a 0 or multiple 0s, like when subtracting 111 from 1000. In situations like these it may be simpler to convert to decimal and then subtract. For that example the difference is one (8-7=1). Pattern recognition can also be useful. 100-11=1, 10000-1111=1, and 100000-11111=1, or in more general terms, the difference between a string with a one followed by n zeroes and a string of n ones is 1. 10101-1111 First Column: 1-1=0 Second Column: 0-1 (in this instance it is necessary to borrow) 10-1=1 After borrowing in the second column the remaining three digits of the larger number are 100. 100-11=1. So difference between 10101-1111= 110. We can verify this result through decimal subtraction. 21 (24+22+20 = 21) - 15 (23 + 22 + 21 + 20) = 6 (22 + 21 = 6) Most computers use a method called the complement method to subtract binary numbers. The method has five steps. The first is to add zeros to the end of the smaller binary string if necessary to ensure that the two strings have the same number of bits. The second step is to switch all digits in the second term{all zeros should be switched to ones and all ones switched to zeros. The third step is to add one to the new second term. The fourth step is to add the the two terms together. And the fifth and final step is to discard the first term{as there will be an extra digit. 10101-1111 10101 01111 { 3 complement of second string: 10000 add one to second string: 10001 addition of the two strings 10101+10001=100110 discard the first digit 00110 Binary Multiplication Like binary addition, binary multiplication is very similar to decimal multiplication. The main difference between the two is that when adding up the results of the multiplication{in the same manner as decimal multiplication{a one is carried when the digits add to two or greater instead of ten or greater. 1010 1111 x |1010 {1010 -1010 1010 + 10010110 This in decimal converts to 150 (27 + 24 + 22 + 21 = 150) We can verify that this multiplication is correct easily. 1010 in decimal is 10 and 1111 in decimal is 15. 15 ∗ 10 = 150 Hexadecimal The Hexadecimal number system employs sixteen digits: 0-9 represent themselves and then 10-15 are repre- sented by a,b,c,d,e, and f. To differentiate hexadecimal numbers from decimal numbers the can be written with subscripts.
Load more

Recommended publications
  • 13054-Duodecimal.Pdf
    Universal Multiple-Octet Coded Character Set International Organization for Standardization Organisation Internationale de Normalisation Международная организация по стандартизации Doc Type: Working Group Document Title: Proposal to encode Duodecimal Digit Forms in the UCS Author: Karl Pentzlin Status: Individual Contribution Action: For consideration by JTC1/SC2/WG2 and UTC Date: 2013-03-30 1. Introduction The duodecimal system (also called dozenal) is a positional numbering system using 12 as its base, similar to the well-known decimal (base 10) and hexadecimal (base 16) systems. Thus, it needs 12 digits, instead of ten digits like the decimal system. It is used by teachers to explain the decimal system by comparing it to an alternative, by hobbyists (see e.g. fig. 1), and by propagators who claim it being superior to the decimal system (mostly because thirds can be expressed by a finite number of digits in a "duodecimal point" presentation). • Besides mathematical and hobbyist publications, the duodecimal system has appeared as subject in the press (see e.g. [Bellos 2012] in the English newspaper "The Guardian" from 2012-12-12, where the lack of types to represent these digits correctly is explicitly stated). Such examples emphasize the need of the encoding of the digit forms proposed here. While it is common practice to represent the extra six digits needed for the hexadecimal system by the uppercase Latin capital letters A,B.C,D,E,F, there is no such established convention regarding the duodecimal system. Some proponents use the Latin letters T and E as the first letters of the English names of "ten" and "eleven" (which obviously is directly perceivable only for English speakers).
    [Show full text]
  • Maths Week 2021
    Maths Week 2021 Survivor Series/Kia Mōrehurehu Monday Level 5 Questions What to do for students 1 You can work with one or two others. Teams can be different each day. 2 Do the tasks and write any working you did, along with your answers, in the spaces provided (or where your teacher says). 3 Your teacher will tell you how you can get the answers to the questions and/or have your work checked. 4 When you have finished each day, your teacher will give you a word or words from a proverb. 5 At the end of the week, put the words together in the right order and you will be able to find the complete proverb! Your teacher may ask you to explain what the proverb means. 6 Good luck. Task 1 – numbers in te reo Māori The following chart gives numbers in te reo Māori. Look at the chart carefully and note the patterns in the way the names are built up from 10 onwards. Work out what each of the numbers in the following calculations is, do each calculation, and write the answer in te reo Māori. Question Answer (a) whitu + toru (b) whā x wa (c) tekau mā waru – rua (d) ono tekau ma whā + rua tekau ma iwa (e) toru tekau ma rua + waru x tekau mā ono Task 2 - Roman numerals The picture shows the Roman Emperor, Julius Caesar, who was born in the year 100 BC. (a) How many years ago was 100 BC? You may have seen places where numbers have been written in Roman numerals.
    [Show full text]
  • The Hexadecimal Number System and Memory Addressing
    C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte.
    [Show full text]
  • Bits, Data Types, and Operations
    Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Agenda 1. Data Types 2. Unsigned & Signed Integers 3. Arithmetic Operations Chapter 2 4. Logical Operations 5. Shifting Bits, Data Types, 6. Hexadecimal & Octal Notation and Operations 7. Other Data Types COMPSCI210 S1C 2009 2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. Data Types Computer is a binary digital system. How do we represent data in a computer? Digital system: Binary (base two) system: At the lowest level, a computer is an electronic machine. • finite number of symbols • has two states: 0 and 1 • works by controlling the flow of electrons Easy to recognize two conditions: 1. presence of a voltage – we’ll call this state “1” 2. absence of a voltage – we’ll call this state “0” Basic unit of information is the binary digit, or bit. Values with more than two states require multiple bits. • A collection of two bits has four possible states: Could base state on value of voltage, 00, 01, 10, 11 but control and detection circuits more complex. • A collection of three bits has eight possible states: • compare turning on a light switch to 000, 001, 010, 011, 100, 101, 110, 111 measuring or regulating voltage • A collection of n bits has 2n possible states. COMPSCI210 S1C 2009 3 COMPSCI210 S1C 2009 4 1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
    [Show full text]
  • Unicode Request for Kaktovik Numerals Design
    Unicode request for Kaktovik numerals L2/21-058R Eduardo Marín Silva, [email protected] Kirk Miller, [email protected] Catherine Strand, [email protected] 2021 April 29 This document supersedes L2/20-070 by Eduardo Marín Silva. The Kaktovik numerals are a set of base-20 digits with a sub-base of 5—that is, a penta-vigesimal system. Graphically, the sub-base forms the upper part of the digit and the remaining units the lower part, an iconic design that lends itself to graphical manipulation for arithmetic. Kaktovik numerals are part of the curriculum in the North Slope Borough School District of Alaska. Though designed by speakers of Iñupiaq Eskimo (ISO code [esi]), they are equally suited to the penta-vigesimal systems of other Inuit and Yupik languages of Alaska, Canada and Russia, and they have the support of the Inuit Circumpolar Council. Thanks to Deborah Anderson of the Universal Scripts Project for her assistance. Design Kaktovik numerals were made intentionally distinct from decimal Hindu-Arabic digits so that there could be no confusion between them. In speech as well, Kaktovik digits have been named in Iñupiaq and Hindu-Arabic digits in English in order to keep them distinct. There are 19 counting digits, composed of straight strokes joined at sharp angles, and a graphically distinct zero . The counting digits occupy the space of an upright golden rectangle, with a unit square at bottom and a smaller golden rectangle at top. The top rectangle is occupied by up to three horizontal strokes that tally the quinary sub-base (null, 틅, 틊, 틏).
    [Show full text]
  • Abstract of Counting Systems of Papua New Guinea and Oceania
    Abstract of http://www.uog.ac.pg/glec/thesis/ch1web/ABSTRACT.htm Abstract of Counting Systems of Papua New Guinea and Oceania by Glendon A. Lean In modern technological societies we take the existence of numbers and the act of counting for granted: they occur in most everyday activities. They are regarded as being sufficiently important to warrant their occupying a substantial part of the primary school curriculum. Most of us, however, would find it difficult to answer with any authority several basic questions about number and counting. For example, how and when did numbers arise in human cultures: are they relatively recent inventions or are they an ancient feature of language? Is counting an important part of all cultures or only of some? Do all cultures count in essentially the same ways? In English, for example, we use what is known as a base 10 counting system and this is true of other European languages. Indeed our view of counting and number tends to be very much a Eurocentric one and yet the large majority the languages spoken in the world - about 4500 - are not European in nature but are the languages of the indigenous peoples of the Pacific, Africa, and the Americas. If we take these into account we obtain a quite different picture of counting systems from that of the Eurocentric view. This study, which attempts to answer these questions, is the culmination of more than twenty years on the counting systems of the indigenous and largely unwritten languages of the Pacific region and it involved extensive fieldwork as well as the consultation of published and rare unpublished sources.
    [Show full text]
  • 2 1 2 = 30 60 and 1
    Math 153 Spring 2010 R. Schultz SOLUTIONS TO EXERCISES FROM math153exercises01.pdf As usual, \Burton" refers to the Seventh Edition of the course text by Burton (the page numbers for the Sixth Edition may be off slightly). Problems from Burton, p. 28 3. The fraction 1=6 is equal to 10=60 and therefore the sexagesimal expression is 0;10. To find the expansion for 1=9 we need to solve 1=9 = x=60. By elementary algebra this means 2 9x = 60 or x = 6 3 . Thus 6 2 1 6 40 1 x = + = + 60 3 · 60 60 60 · 60 which yields the sexagsimal expression 0; 10; 40 for 1/9. Finding the expression for 1/5 just amounts to writing this as 12/60, so the form here is 0;12. 1 1 30 To find 1=24 we again write 1=24 = x=60 and solve for x to get x = 2 2 . Now 2 = 60 and therefore we can proceed as in the second example to conclude that the sexagesimal form for 1/24 is 0;2,30. 1 One proceeds similarly for 1/40, solving 1=40 = x=60 to get x = 1 2 . Much as in the preceding discussion this yields the form 0;1,30. Finally, the same method leads to the equation 5=12 = x=60, which implies that 5/12 has the sexagesimal form 0;25. 4. We shall only rewrite these in standard base 10 fractional notation. The answers are in the back of Burton. (a) The sexagesimal number 1,23,45 is equal to 1 3600 + 23 60 + 45.
    [Show full text]
  • 1. Understanding Decimal, Binary, Octal and Hexadecimal Numbers
    Objectives: 1. Understanding decimal, binary, octal and hexadecimal numbers. 2. Counting in decimal, binary, octal and hexadecimal systems. 3. Convert a number from one number system to another system. 4. Advantage of octal and hexadecimal systems. 1. Understanding decimal, binary, octal and hexadecimal numbers Decimal number systems: Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9 --------10-base system) The decimal system is a "positional-value system" in which the value of a digit depends on its position. Examples: 453→4 hundreds, 5 tens and 3 units. 4 is the most weight called "most significant digit" MSD. 3 carries the last weight called "least significant digit" LSD. number of items that a decimal number represent: 9261= (9× )+(2× )+(6× )+(1× ) The decimal fractions: 3267.317= (3× )+(2× )+(6× )+(7× )+ (3× ) + (6× ) + (1× ) Decimal point used to separate the integer and fractional part of the number. Formal notation→ . Decimal position values of powers of (10). Positional values "weights" 2 7 7 8 3 . 2 3 4 5 MSD LSD Binary numbers: . Base-2 system (0 or 1). We can represent any quantity that can be represented in decimal or other number systems using binary numbers. Binary number is also positional–value system (power of 2). Example: 1101.011 1 1 0 1 . 0 1 1 MSD LSD Notes: . To find the equivalent of binary numbers in decimal system , we simply take the sum of products of each digit value (0,1)and its positional value: Example: = (1× ) + (0× ) + (1× ) + (1× )+ (1× )+ (0× ) +(1× ) = 8 + 0 + 2 + 1 + + 0 + = In general, any number (decimal, binary, octal and hexadecimal) is simply the sum of products of each digit value and its positional value.
    [Show full text]
  • Bit, Byte, and Binary
    Bit, Byte, and Binary Number of Number of values 2 raised to the power Number of bytes Unit bits 1 2 1 Bit 0 / 1 2 4 2 3 8 3 4 16 4 Nibble Hexadecimal unit 5 32 5 6 64 6 7 128 7 8 256 8 1 Byte One character 9 512 9 10 1024 10 16 65,536 16 2 Number of bytes 2 raised to the power Unit 1 Byte One character 1024 10 KiloByte (Kb) Small text 1,048,576 20 MegaByte (Mb) A book 1,073,741,824 30 GigaByte (Gb) An large encyclopedia 1,099,511,627,776 40 TeraByte bit: Short for binary digit, the smallest unit of information on a machine. John Tukey, a leading statistician and adviser to five presidents first used the term in 1946. A single bit can hold only one of two values: 0 or 1. More meaningful information is obtained by combining consecutive bits into larger units. For example, a byte is composed of 8 consecutive bits. Computers are sometimes classified by the number of bits they can process at one time or by the number of bits they use to represent addresses. These two values are not always the same, which leads to confusion. For example, classifying a computer as a 32-bit machine might mean that its data registers are 32 bits wide or that it uses 32 bits to identify each address in memory. Whereas larger registers make a computer faster, using more bits for addresses enables a machine to support larger programs.
    [Show full text]
  • Chapter 6 MISCELLANEOUS NUMBER BASES. the QUINARY
    The Number Concept: Its Origin and Development By Levi Leonard Conant Ph. D. Chapter 6 MISCELLANEOUS NUMBER BASES. THE QUINARY SYSTEM. The origin of the quinary mode of counting has been discussed with some fulness in a preceding chapter, and upon that question but little more need be said. It is the first of the natural systems. When the savage has finished his count of the fingers of a single hand, he has reached this natural number base. At this point he ceases to use simple numbers, and begins the process of compounding. By some one of the numerous methods illustrated in earlier chapters, he passes from 5 to 10, using here the fingers of his second hand. He now has two fives; and, just as we say “twenty,” i.e. two tens, he says “two hands,” “the second hand finished,” “all the fingers,” “the fingers of both hands,” “all the fingers come to an end,” or, much more rarely, “one man.” That is, he is, in one of the many ways at his command, saying “two fives.” At 15 he has “three hands” or “one foot”; and at 20 he pauses with “four hands,” “hands and feet,” “both feet,” “all the fingers of hands and feet,” “hands and feet finished,” or, more probably, “one man.” All these modes of expression are strictly natural, and all have been found in the number scales which were, and in many cases still are, in daily use among the uncivilized races of mankind. In its structure the quinary is the simplest, the most primitive, of the natural systems.
    [Show full text]
  • Number Systems and Number Representation Aarti Gupta
    Number Systems and Number Representation Aarti Gupta 1 For Your Amusement Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct -- http://www.electronicsweekly.com 2 Goals of this Lecture Help you learn (or refresh your memory) about: • The binary, hexadecimal, and octal number systems • Finite representation of unsigned integers • Finite representation of signed integers • Finite representation of rational numbers (if time) Why? • A power programmer must know number systems and data representation to fully understand C’s primitive data types Primitive values and the operations on them 3 Agenda Number Systems Finite representation of unsigned integers Finite representation of signed integers Finite representation of rational numbers (if time) 4 The Decimal Number System Name • “decem” (Latin) => ten Characteristics • Ten symbols • 0 1 2 3 4 5 6 7 8 9 • Positional • 2945 ≠ 2495 • 2945 = (2*103) + (9*102) + (4*101) + (5*100) (Most) people use the decimal number system Why? 5 The Binary Number System Name • “binarius” (Latin) => two Characteristics • Two symbols • 0 1 • Positional • 1010B ≠ 1100B Most (digital) computers use the binary number system Why? Terminology • Bit: a binary digit • Byte: (typically) 8 bits 6 Decimal-Binary Equivalence Decimal Binary Decimal Binary 0 0 16 10000 1 1 17 10001 2 10 18 10010 3 11 19 10011 4 100 20 10100 5 101 21 10101 6 110 22 10110 7 111 23 10111 8 1000 24 11000 9 1001 25 11001 10 1010 26 11010 11 1011 27 11011 12 1100 28 11100 13 1101 29 11101 14 1110 30 11110 15 1111 31 11111 ..
    [Show full text]
  • 1 Evolution and Base Conversion
    Indian Institute of Information Technology Design and Manufacturing, Kancheepuram logo.png Chennai { 600 127, India Instructor An Autonomous Institute under MHRD, Govt of India N.Sadagopan http://www.iiitdm.ac.in Computational Engineering Objectives: • To learn to work with machines (computers, ATMs, Coffee vending machines). • To understand how machines think and work. • To design instructions which machines can understand so that a computational task can be performed. • To learn a language which machines can understand so that human-machine interaction can take place. • To understand the limitations of machines: what can be computed and what can not be computed using machines. Outcome: To program a computer for a given computational task involving one or more of arithmetic, algebraic, logical, relational operations. 1 Evolution and Base Conversion Why Machines ? We shall now discuss the importance of automation; human-centric approach (manual) versus machines. Although machines are designed by humans, for many practical reasons, machines are superior to humans as far as problem solving is concerned. We shall highlight below commonly observed features that make machines powerful. • Reliable, accurate and efficient. • Can do parallel tasks if trained. • Efficient while peforming computations with large numbers. If designed correctly, then there is no scope for error. • Good at micro-level analysis with precision. • Consistent It is important to highlight that not all problems can be solved using machines (computers). For example, (i) whether a person is happy (ii) whether a person is lying (not speaking the truth) (iii) reciting the natural number set (iv) singing a song in a particular raga. Computer: A computational machine In this lecture, we shall discuss a computational machine called computer.
    [Show full text]