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

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Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №1Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №2Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №3Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №4Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №5Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №6Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №7Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №8Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №9Mathematics for Computing 2016-2017. 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Mathematics for Computing 2016-2017
Описание слайда:
Mathematics for Computing 2016-2017

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Topics 2016-17 Number Representation Logarithms Logic Set Theory Relations & Functions Graph Theory
Описание слайда:
Topics 2016-17 Number Representation Logarithms Logic Set Theory Relations & Functions Graph Theory

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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)
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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)

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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
Описание 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
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Provisional Timetable

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Course Textbook Schaum’s Outlines Series Essential Computer Mathematics Author: Seymour Lipschutz ISBN 0-07-037990-4
Описание 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://www.bbk.ac.uk/business/current-students/learning-co-ordinators/eva-szatmari See separate powerpoint file.
Описание слайда:
Maths Support http://www.bbk.ac.uk/business/current-students/learning-co-ordinators/eva-szatmari See separate powerpoint file.

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Lecture 1 Rule 1 Communication is not easy, How do you tell a computer what to do?
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Lecture 1 Rule 1 Communication is not easy, How do you tell a computer what to do?

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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 …
Описание 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 know how our memory stores numbers We know how a computer does (we designed it) Full glass is 1, empty is 0
Описание 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 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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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 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
Описание 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. We don’t need the cups now. Let’s understand how this works We shall look for repetitive patterns and see what they mean
Описание 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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Mathematics for Computing 2016-2017. Lecture 1: Course Introduction and Numerical Representation, слайд №14
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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 ??
Описание 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 ??

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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
Описание 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

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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)
Описание 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)

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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
Описание 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 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.)
Описание 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!!!
Описание слайда:
Some binary numbers!!!

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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)
Описание 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)

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Convert from Binary to Decimal Example of how to use what we learned to convert from binary to decimal
Описание слайда:
Convert from Binary to Decimal Example of how to use what we learned to convert from binary to decimal

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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)
Описание 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
Описание слайда:
Convert from Decimal to Binary

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What Happens when we Convert from Decimal to Binary
Описание слайда:
What Happens when we Convert from Decimal to Binary

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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
Описание 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

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Convert from Decimal to Binary
Описание слайда:
Convert from Decimal to Binary

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Numbers we can already represent Natural numbers: 1, 2, 3, 4, … Alternative versions of the number six Decimal: 6 Alphabetically: six Roman: VI Tallying:
Описание 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:

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What’s still missing Fractional numbers (real numbers) Versions of one and a quarter Mixed number: 1¼, Improper fraction: 5/4, Decimal: 1.25
Описание 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

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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
Описание 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

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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.
Описание 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.

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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
Описание 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 in binary
Описание слайда:
Representing decimal numbers in binary

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Convert fractional part from Decimal to Binary To convert the decimal part:
Описание слайда:
Convert fractional part from Decimal to Binary To convert the decimal part:

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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
Описание 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

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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
Описание 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

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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
Описание 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

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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
Описание 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

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Negative numbers – Two’s Complement (4 bits)
Описание слайда:
Negative numbers – Two’s Complement (4 bits)

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Computer representation Fixed length Integers Real Sign
Описание слайда:
Computer representation Fixed length Integers Real Sign

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Bits, bytes, words Bit: a single binary digit Byte: eight bits Word: Depends!!! Long Word: two words
Описание слайда:
Bits, bytes, words Bit: a single binary digit Byte: eight bits Word: Depends!!! Long Word: two words

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Integers A two byte integer 16 bits 216 possibilities  65536 -32768  n  32767 or 0  n  65535
Описание слайда:
Integers A two byte integer 16 bits 216 possibilities  65536 -32768  n  32767 or 0  n  65535

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Signed integers
Описание слайда:
Signed integers

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Real numbers ‘Human’ form: 4563.2835 Exponential form: 0.45632835 x 104 General form: m x be Normalised binary exponential form: m x 2e
Описание 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

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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
Описание 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

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Real numbers 2 For a 32 bit real number Sign, 1 bit Significand, 23 bits Exponent, 8 bits
Описание слайда:
Real numbers 2 For a 32 bit real number Sign, 1 bit Significand, 23 bits Exponent, 8 bits

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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.
Описание 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.

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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
Описание 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

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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
Описание 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

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Convert from Decimal to Base 7
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Convert from Decimal to Base 7

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Convert from Base 7 to Decimal
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Convert from Base 7 to Decimal

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Convert from Decimal to Base 5 and back
Описание слайда:
Convert from Decimal to Base 5 and back

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Octal Base eight Digits 0,1,2,3,4,5,6,7 Example: 1210 = 148 = 11002 100110111102 Binary
Описание слайда:
Octal Base eight Digits 0,1,2,3,4,5,6,7 Example: 1210 = 148 = 11002 100110111102 Binary

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Convert from Binary to Octal and back
Описание слайда:
Convert from Binary to Octal and back

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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
Описание 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

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Convert from Binary to Hexadecimal and back
Описание слайда:
Convert from Binary to Hexadecimal and back

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Writing down the hexadecimal conversion table
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Writing down the hexadecimal conversion table

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Extra Slides 1 0 1 0 0 1 1 +1 1 1 0 1 1 1
Описание слайда:
Extra Slides 1 0 1 0 0 1 1 +1 1 1 0 1 1 1

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End of Lecture
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End of Lecture

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Extra Slides The following slides present the same information already appearing in other slides, in a different manner.
Описание 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 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
Описание 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

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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
Описание 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

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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.
Описание 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.

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Some hexadecimal (and binary) numbers!!!
Описание слайда:
Some hexadecimal (and binary) numbers!!!

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End
Описание слайда:
End

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