ITM207 - Notes Week 1 Natrual numbers Zero and any numbers obtained by adding 1 to it 0, 3, 100, 100000 Negative number A number with a - sign before it -1, -4, -1000, -10100 Integers A natural number, a negative number Ex: 249, 0, -324232, -34 Rational numbers An integer or the quotient of two integers Ex: -249, -1, 0, 3/7, - ⅖ Decimal base Numbers from 0-9 “A number system which uses digits from 0 to 9 to represent a number with base 10 is the decimal system number. The number is expressed in base-10 where each value is denoted by 0 or first nine positive integers. Each value in this number system has the place value of power 10.” Positional Notation Ex: 600 + 40 + 2 600 ones + 40 ones + 2 ones 642 is 600 + 40 + 2 in BASE 10 The base of a number determines the number of different digit symbols (numerals) an the values of digit positions BECAUSE 6 x 10^2 = 6 x 100 = 600 4 x 10^1 = 4 x 10 = 40 2 x 10^0 = 2 x 1 = 2 (2 is like this because it is a base number 0-9) The Base Equation would be dn * Rn - 1 + dn - 1 * Rn - 2 + ... + d2 * R1 + d1 * R0 >>> D is the digit in the ith position, n is the number of digits in the number Equation of 642 would be: 642 is 6 x 10^2 + 4 x 10 + 2 x 1 What if 642 has a base of 13? + 6 x 13^2 = 1014 + 4 x 13^1 = 52 + 2 x 13^0 = 2 = 1068 in base 10 642 in base 13 is 1068 in base 10 Binary Decimal in base 10 an has 10 digit symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Binary is base 2 an has 2 digit symbols 0, 1 For a number to exist in a given base it can only contain digits in that base from 0 up to but not including the base Bases higher than 10 How are digits in bases higher than the 10 represented With distinct symbols for 10 an above Base 16 has 16 digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F What bases can these numbers be in? 122, 198, 178, G1A4 Converting hexadecimal to decimal D x 16^2 = 13 x 256 = 3328 + E x 16^1 = 14 x 16 = 224 + F x 16^ º = 15 x 1 = 15 = 3567 in base 10 Check slides for Octal stuff… All dividing stuff is in Lecture slides Binary Numbers and computers Low voltage = 0 High Voltage = 1 All bits have 0 or 1 Or other way around but we dont have to worry about that Bytes 8 bits The number of bits in a word determines the word length of the computer, which is usually a multiple of 8. • 32-bit machines • 64-bit machines • Etc. Binary and Computers Practice equation Tips: Start from left of number always (11011101) = (?)base10 1x2^0 = 1 0x2^1= 0 1x2^2= 4 1x2^3 8 1x2^4 16 0x2^5 0 1x2^6 64 1x2^7 128 It all Equals 221 Week 2 Analog and Digital Information Information can be represented in One of Two Ways: Analog or Digital Analog Data A continuous representation, analogous to the actual info it represents Digital Data A discrete representation breaking the information up into separate elements Computers cannot work well with Analog data, so we digitize the data Digitize Break data into pieces and represent those pieces separately - Those discrete elements are individually represented using binary digits Why do we use Binary to represent digitised data? Electronic Signals Important facts about electronic signals - An Analog Signal continually fluctuates in voltage up and down - A digital signal has only a high or low state corresponding to the two binary digits - All electronic signals (Both Analog and Digital) Degrade as they move down a line - The voltage of the signal fluctuates due to environmental effects Digital signal: Jump sharply b/w two extremes referred as pulse-code modulation (PCM) Periodically a Digital signal is reclocked to regain its original shape as long as it is reclocked before to much degradation occurs no info is lost Binary Representations - Each bit can be either 0 or 1, so it can represent a choice between two possibilities (or “two things) - Two bits can represent four things (possible gears: Park, Drive, Reverse, Neutral // Park represented by 00, drive by 01, reverse by 10, neutral by 11) How many things can 3 bits represent? Would be 8 because 2^3 How many things can 4 bits represent? Would be 16 because 2^4 How many things can 8 bits represent? Would be 256 because 2^8 How many bits are needed to represent 32 things? 6 Bits. One Hundred Things? 7. How many things can n bits represent? 2^n Different things What happens every time you increase the number of bits by one? Representing Natural Numbers - Convert to Binary - Computers store data in fixed size chunks so we have leading zeros - What do integers include that the natural numbers do not? Representing Negative Values Fixed Sized Numbers Half the natural numbers will represent themselves the other half will represent negative integers Using two decimal digits, Let 1 through 49 represent 1 through 49 Let 50 through 99 represent -50 through -1 To perform addition, add the numbers and discard any carry to the hundreds digit 5 + 94 = 99 , 5 + -6 = -1 -4 + 6 = 2 , 96 + 6 = 2 -2 + -4 = -6 , 98 + 96 = 94 Now I try it 48 - 1 = 47 (Signed Magnitude) 48 - 99 = 47 To perform subtraction, use A - B = A + (-B) Add the negative off the second to the first -5 - (-3) = -8 New scheme 95 - 3 = -3= 97, 95 + 97 = 92 4 - 3 , 4 - (-3) , -1 - 2 —-----Called Tens Complement representation because we can use this formula to compute the representation of a negative number Negative (I) = 10^k -1, where k is the number of digits For example -3 is negative(3) so using two digits its representation is Negative(3) = 100 -3 = 97 Three digits rep = Negative(3) = 10^3 -3 = 997 Twos Complement (The binary number line is easier to read when written vertically) Would the tens complement formula work with the 10 replaced by 2? Do you notice something interesting about the left most bit 1^2 Number Overflow If each value is stored using 8 bits, then 127 + 3 overflow 01111111 00000011 = 10000010 Apparently 127 + 3 is not -126. If we were not representing negative numbers the result would be correct Remember when we said we would always fail in our attempt to map an infinite world onto a finite machine Representing Real Numbers Real Numbers are numbers with a whole part and a fractional part (either of which may be zero) 104.32 0.9999999 357.0 3.14159 In decimal, positions to the right of the decimal point are the tenth hundreths thousandths etc, 10^-1 , 10^-4, 10^-5 Week 3 Gate A device that performs a basic operation on electrical signals Circuit Is a electrical circuit Boolean Expressions Uses Boolean algebra a mathematical notation for expressing two valued logic Logic Diagram A graphical representation of a circuit each gate has its own symbol Truth tables A table showing all possible input values and associated output values NOT GATE A NOT Gate accepts one input signal (0 or 1) and returns the complementary (opposite) signal as output AND GATE An AND Gate accepts two input signals If both are 1 the output is 1 otherwise the output is 0 AND operation a single dot . or asterisk ‘ OR Gate An OR Gate accepts two input signals If both are 0 the output is 0 otherwise the output is 1 OR operation plus sign + XOR Gate An XOR gate accepts two input signals If both are the same the output is 0 otherwise the output is 1 NAND Gate The NAND NOT AND AND Gate accepts two input signals If both are 1 the output is 0 otherwise output is 1 OR gate Opposite of OR Gate If both r 0 output is 1 otherwise output is 0 Gates with more Inputs Some gates can be generalized to accept three or more input values A three input AND gate for ex produced an output of 1 only if all input values r 1 Circuits Combinational Citcuirt All the input values explicitly determine the output Sequential circuit The output is a function of the input values and the existing state of the circuit We describe circuit operations using Boolean expressions Logic diagrams And truth tables Combination Circuits Gates are combined into circuits by using the output of one gate as the input for another Week 4 —__–__–__–__–__—__— In Slides Week 5 Raptor Tips & Rules & Etc.. Context -> Program Source Code -> Executables (Binaries) - A matter or situation regarded as unwelcome or harmful and needing to be dealt with an overcome A thing that is difficult to achieve An inquiry starting from given conditions to investigate or demonstrate a fact result or law A proposition in which something has to be constructed General Problem Solving How to solve it - A new aspect of mathematical method by George Polya 1. Understand the problem - What do I know - What info I have to process - What does the solution look like - What sort of special cases exist - How will I recognize that I have found the solution 2. Devise a Plan (Choose A Strategy) - Divide & conquer - Look For a pattern - Mathematical formulation - Solve an Equation - Daw a diagram/pic - Guess n check - Make a list - Work backwards - etc 3. Carry out the plan 4. Look Back (Reflect) Problem Solving Process -> Understand The Problem -> Solution Design (Raptor) + Implement Test Modify and Maintain + Multiple Iterations + Incremental Refinements Problem Solving Strategies - How TO? - Break down the problem into smaller subproblems divide and conquer - Separation of Concerns Identify Potential patterns of similarity with previously encountered and solved problems - reuse a known solution Reformulate the problem or sub problem to explore potential solutions that would not otherwise be apparent Use abstraction to ignore unnecessary details that would otherwise constitute “Noise” preventing you from seeing a potential solution Use an (Iterative) incremental refinement approach - What the problem an the solution spaces look like after each iteration changes, providing new problem understanding and solution opportunities. What is an Algorithim?\ EX — What is the highest mark? No Mark . …… 1st Mark —--> Person —> Highest Mark Highest Mark Algo is 1. Start 2. Read first mark 3. Designate the first mark to be the highest 4. Read the next mark 5. If new mark is higher make it the new highest mark 6. IF all marks read proceed to step 7 7. Write highest mark 8. End Bold is Processing 1. Finite duration. End occurs within a finite time from start 2. Three core elements - Input - Processing - Output 3. Unambiguous statement of the solution -> Encoding into a computer program is straight forward 4. Deterministic (Most algorithms) - For the same input the algo generates the same output Use random numbers for adding each other and the same random number to division E1 E1’ = E1 E2 E2’ = E2 + Divide “/” Algorithm Refinement - Add an diitional step to account for the trivial case where only a single mark is given as input - Use of variables and relational operators - Use of shorthand symbols such as the assignment operator ←—-- ex of shorthand symbol Flowchart Specification of Algorithms Each shape has a different use - Check Slides When enter text in Prompt box use Quotations
