Презентация Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation онлайн

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Слайды и текст к этой презентации:

№1 слайд
Mathematics for Computing -
Содержание слайда: Mathematics for Computing 2016-2017
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Topics - Number
Содержание слайда: Topics 2016-17 Number Representation Logarithms Logic Set Theory Relations & Functions Graph Theory
№3 слайд
Assessment In Class Test
Содержание слайда: Assessment In Class Test (Partway through term, 31/10) (20% of marks) ‘Homework’ (3 parts for 10% of marks) Two hour unseen examination in May/June 2017 (70% of marks)
№4 слайд
Lecture tutorial plans
Содержание слайда: Lecture / tutorial plans Lecture every week 18:00 for 18:10 start. 1 – 2½ hours (with break) Tutorials (problems and answers) one week in two (~1½ hours) Compulsory In-Class Test, October 31st Lecture Notes etc. will appear on Moodle Class split in two rooms
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Provisional Timetable
Содержание слайда: Provisional Timetable
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Course Textbook Schaum s
Содержание слайда: Course Textbook Schaum’s Outlines Series Essential Computer Mathematics Author: Seymour Lipschutz ISBN 0-07-037990-4
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Maths Support http
Содержание слайда: Maths Support http://www.bbk.ac.uk/business/current-students/learning-co-ordinators/eva-szatmari See separate powerpoint file.
№8 слайд
Lecture Rule Communication is
Содержание слайда: Lecture 1 Rule 1 Communication is not easy, How do you tell a computer what to do?
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Welcome Rule We want to get
Содержание слайда: Welcome Rule 1 We want to get the computer to do NEW complicated things We start by learning the basics of its language, Numerical Representation, Logic …
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Memory for numbers We don t
Содержание слайда: Memory for numbers We don’t know how our memory stores numbers We know how a computer does (we designed it) Full glass is 1, empty is 0
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Great, we know how to store
Содержание слайда: Great, we know how to store 1 and 0 in the computer memory Great, we know how to store 1 and 0 in the computer memory How do we store 0,1,2,3? We use two cups!
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If we want extra numbers we
Содержание слайда: If we want extra numbers we add an extra cup! If we want extra numbers we add an extra cup! Each cup we add doubles the number of values we can store
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We don t need the cups now.
Содержание слайда: We don’t need the cups now. We don’t need the cups now. Let’s understand how this works We shall look for repetitive patterns and see what they mean
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Содержание слайда:
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Convert from Binary to
Содержание слайда: Convert from Binary to Decimal When we translate from the binary base (base 2) the decimal base (base 10 – ten fingers) The first binary digit tells us whether to add 1 The second binary digit tells us whether to add 2 The third binary digit tells us whether to add 4 The fourth binary digit tells us whether to add ??
№16 слайд
Convert from Binary to
Содержание слайда: Convert from Binary to Decimal When we translate from the binary base to the decimal base The first binary digit tells us whether to add 1 Every digit afterwards tells us whether to add exactly two times as much a the previous digit Lets try this out
№17 слайд
The binary system computer
Содержание слайда: The binary system (computer) The way the computer stores numbers Base 2 Digits 0 and 1 Example: 110110112   msd lsd (most significant digit) (least significant digit)
№18 слайд
The decimal system ours
Содержание слайда: The decimal system (ours) Probably because we started counting with our fingers Base 10 Digits 0,1,2,3,4,5,6,7,8,9 Example: 7641321910   msd lsd
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Significant Figures
Содержание слайда: Significant Figures Significant Figures: Important in science for precision of measurements. All non-zero digits are significant Leading zeros are not significant e.g.  = 3.14 (to 3 s.f.) = 3.142 (to 4 s.f.) = 3.1416 (to 5 s.f.)
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Some binary numbers!!!
Содержание слайда: Some binary numbers!!!
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Convert from Binary to
Содержание слайда: Convert from Binary to Decimal Lets make this more mathematical, We now use powers of 2 to represent 1,2,4,8,… Note that the power is the index of the digit, when the indices start from 0 (first index is 0) (digit with index 6 corresponds to 26)
№22 слайд
Convert from Binary to
Содержание слайда: Convert from Binary to Decimal Example of how to use what we learned to convert from binary to decimal
№23 слайд
Idea for Converting Decimal
Содержание слайда: Idea for Converting Decimal to Binary Digit at position 0 is easy. It is 1 if the number is even and 0 otherwise Why? In a binary number only the least significant digit (20=1)
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Convert from Decimal to Binary
Содержание слайда: Convert from Decimal to Binary
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What Happens when we Convert
Содержание слайда: What Happens when we Convert from Decimal to Binary
№26 слайд
Decimal to Binary conversion
Содержание слайда: Decimal to Binary conversion Algorithmically: Natural Numbers 1. Input n (natural no.) 2. Repeat 2.1. Output n mod 2 2.2. n  n div 2 until n = 0
№27 слайд
Convert from Decimal to Binary
Содержание слайда: Convert from Decimal to Binary
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Numbers we can already
Содержание слайда: Numbers we can already represent Natural numbers: 1, 2, 3, 4, … Alternative versions of the number six Decimal: 6 Alphabetically: six Roman: VI Tallying:
№29 слайд
What s still missing
Содержание слайда: What’s still missing Fractional numbers (real numbers) Versions of one and a quarter Mixed number: 1¼, Improper fraction: 5/4, Decimal: 1.25
№30 слайд
Decimal numbers base String
Содержание слайда: Decimal numbers (base 10) String of digits - symbol for negative numbers Decimal point A positional number system, with the index giving the ‘value’ of each position. Example: 3583.102 = 3 x 103 + 5 x 102 + 8 x 101 + 3 x 100 + 1 x 10-1 + 0 x 10-2 + 2 x 10-3
№31 слайд
Representing Decimal numbers
Содержание слайда: Representing Decimal numbers in Binary We can use two binary numbers to represent a fraction by letting the first number be the enumerator and the other be denominator Problem: we want operation such as addition and subtraction to execute fast. This representation is not optimal.
№32 слайд
Representing Fractions in
Содержание слайда: Representing Fractions in Binary Use a decimal point like in decimal numbers There are two binary numbers the first is the number before the (radix) point and the other after the point
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Representing decimal numbers
Содержание слайда: Representing decimal numbers in binary
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Convert fractional part from
Содержание слайда: Convert fractional part from Decimal to Binary To convert the decimal part:
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Negative numbers First bit
Содержание слайда: Negative numbers First bit (MSB) is the sign bit If it is 0 the number is positive If it is 1 the number is negative Goal when definition was chosen: Maximize use of memory Make computation easy
№36 слайд
Negative Numbers Calculate
Содержание слайда: Negative Numbers – Calculate two’s Complement The generate two’s complement Write out the positive version of number, Write complement of each bit (0 becomes 1 and 1 becomes 0) Add 1 The result is the two’s complement and the negative version of the number
№37 слайд
Negative Numbers Two s
Содержание слайда: Negative Numbers – Two’s Complement (examples) 3bit 8bit 011 310 00011101 2910 number 100 11100010 complement + 001 00000001 +1 === ======== 101 -310 11100011 -2910 2’s complement
№38 слайд
Negative numbers Two s
Содержание слайда: Negative numbers – Two’s Complement(3 bits) First bit (MSB) is the sign bit If it is 0 the number is positive If it is 1 the number is negative Goal when definition was chosen: Maximize use of memory Make computation easy
№39 слайд
Negative numbers Two s
Содержание слайда: Negative numbers – Two’s Complement (4 bits)
№40 слайд
Computer representation Fixed
Содержание слайда: Computer representation Fixed length Integers Real Sign
№41 слайд
Bits, bytes, words Bit a
Содержание слайда: Bits, bytes, words Bit: a single binary digit Byte: eight bits Word: Depends!!! Long Word: two words
№42 слайд
Integers A two byte integer
Содержание слайда: Integers A two byte integer 16 bits 216 possibilities  65536 -32768  n  32767 or 0  n  65535
№43 слайд
Signed integers
Содержание слайда: Signed integers
№44 слайд
Real numbers Human form .
Содержание слайда: Real numbers ‘Human’ form: 4563.2835 Exponential form: 0.45632835 x 104 General form: m x be Normalised binary exponential form: m x 2e
№45 слайд
Real numbers Conversion from
Содержание слайда: Real numbers Conversion from Human to Exponential and back 655.54 = 0. 65554 * 103 0.000545346 = 0. 545346 *10-3 0.523432 * 105 = 52343.2 0.7983476 * 10-4 = 0.00007983476
№46 слайд
Real numbers For a bit real
Содержание слайда: Real numbers 2 For a 32 bit real number Sign, 1 bit Significand, 23 bits Exponent, 8 bits
№47 слайд
Types of numbers Integers , -
Содержание слайда: Types of numbers Integers: …, -3, -2, -1, 0, 1, 2, 3, … Rational numbers: m/n, where m and n are integers and n  0. Examples: ½, 5/3, ¼ = 0.25 1/3 = 0.3333… Irrational numbers, examples: 2  1.414,   22/7  3.14159 e  2.718.
№48 слайд
Other representations Base
Содержание слайда: Other representations Base Index form Number = baseindex e.g. 100 = 102 Percentage form Percentage = number/100 e.g. 45% = 45/100 = 0.45 20% = 20/100 = 0.2 110% = 110/100 = 1.1
№49 слайд
Other number systems Bases
Содержание слайда: Other number systems Bases can be any natural number except 1. Common examples are : Binary (base 2) Octal (base 8) Hexadecimal (base 16) We’ll show what to do with base 5 and 7 and then deal with the octal and hexadecimal bases
№50 слайд
Convert from Decimal to Base
Содержание слайда: Convert from Decimal to Base 7
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Convert from Base to Decimal
Содержание слайда: Convert from Base 7 to Decimal
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Convert from Decimal to Base
Содержание слайда: Convert from Decimal to Base 5 and back
№53 слайд
Octal Base eight Digits , , ,
Содержание слайда: Octal Base eight Digits 0,1,2,3,4,5,6,7 Example: 1210 = 148 = 11002 100110111102 Binary
№54 слайд
Convert from Binary to Octal
Содержание слайда: Convert from Binary to Octal and back
№55 слайд
Hexadecimal Base sixteen
Содержание слайда: Hexadecimal Base sixteen Digits 0,1,2,3,4,5,6,7,8,9,A(10), B(11), C(12),D(13),E(14),F(15). Example B316 = 17910 = 101100112 110101012 Binary
№56 слайд
Convert from Binary to
Содержание слайда: Convert from Binary to Hexadecimal and back
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Writing down the hexadecimal
Содержание слайда: Writing down the hexadecimal conversion table
№58 слайд
Extra Slides
Содержание слайда: Extra Slides 1 0 1 0 0 1 1 +1 1 1 0 1 1 1
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End of Lecture
Содержание слайда: End of Lecture
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Extra Slides The following
Содержание слайда: Extra Slides The following slides present the same information already appearing in other slides, in a different manner.
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Decimal to Binary conversion
Содержание слайда: Decimal to Binary conversion 1: Mathematical Operations n div 2 is the quotient. n mod 2 is the remainder. For example: 14 div 2 = 7, 14 mod 2 = 0 17 div 2 = 8, 17 mod 2 = 1
№62 слайд
Decimal to Binary conversion
Содержание слайда: Decimal to Binary conversion 2: Natural Numbers 1. Input n (natural no.) 2. Repeat 2.1. Output n mod 2 2.2. n  n div 2 until n = 0
№63 слайд
Decimal to Binary conversion
Содержание слайда: Decimal to Binary conversion 3: Fractional Numbers 1. Input n 2. Repeat 2.1. m  2n 2.2. Output m  2.3. n  frac(m) until n = 0 m  is the integer part of m frac(m) is the fraction part.
№64 слайд
Some hexadecimal and binary
Содержание слайда: Some hexadecimal (and binary) numbers!!!
№65 слайд
End
Содержание слайда: End
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