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Mathematics for Computing 2016-2017

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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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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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Course Textbook Schaum’s Outlines Series Essential Computer Mathematics Author: Seymour Lipschutz ISBN 0-07-037990-4

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Maths Support 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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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

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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

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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

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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 ??

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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

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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)

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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

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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.)

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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)

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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)

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Convert from Decimal to Binary

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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

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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:

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

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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

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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

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$Representing Decimal numbers in Binary We can use two binary numbers to represent a fraction by letting the first number be the enumerator and the...$

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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.

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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

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Representing decimal numbers in binary

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

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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

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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

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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

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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

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Negative numbers – Two’s Complement (4 bits)

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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

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

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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

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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

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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.

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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

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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

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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 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

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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

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Convert from Binary to Hexadecimal and back

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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

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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.

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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

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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

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$Decimal to Binary conversion 3: Fractional Numbers 1. Input n 2. Repeat 2.1. m  2n 2.2. Output m  2.3. n  frac(m) until n = 0 m  is the integer...$

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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.