Foundations of Computer Science Computing …it is all about Data Representation, Storage, Processing, and Communication of Data 10/4/20151CS 112 – Foundations of Computer Science

A Computational Machine  What capabilities should a computing machine have ?  Well..any computational problem involves two primary elements: Data (numbers, text strings, or images,...) used for input, manipulation and output Instructions (that describes the operations (e.g. arithmetic or comparison) that must be performed on data pertinent to the problem being solved Von Neumann – provided the insight that both data and instructions could be stored together in the computer 10/4/20152CS 112 – Foundations of Computer Science

A Computational Machine  Thus, the machine must have the following capabilities: Representation & Storage (of both Data, and the Instructions that defines operations on data, why?) Data Processing (interpret Instructions and carry out operations defined by these instructions on Data) Data input and Data output (I/O) (why?) Data Communication (optional, if the machine is connected to other similar machines – via a network) 10/4/20153CS 112 – Foundations of Computer Science

Computer Capabilities  A transistor can be set to either on (1) or off (0) by controlling the direction of the electric current passing through it  Hundreds of Millions of transistors can be packed into a small ‘compact’ unit called the integrated circuit (IC) (a.k.a chips) Transistor is said to be storing a 1 here 5 Volts = 1 is stored Power source resistance Electric current transistor Transistor is said to be storing a 0 here 0 Volts = 0 is stored Power source No resistance Electric current transistor 10/4/20154CS 112 – Foundations of Computer Science

Computer Capabilities “Information in the Computer is bits + context”  Both data and instructions are encoded in the computer using Bits (Binary digits)  Think of a Bit as a code that can be either equal to 0 or 1  Encoding means that for each data value or algorithm instruction you use a code consisting of one or more bits to represent it.  For example, Street Traffic lights is a form of encoding for instructions to motorists (red = ‘stop’, yellow = ‘slow down’, green = ‘go’)  In computers bits are grouped together in 8-bit chunks called Bytes (e.g , , and ) 10/4/20155CS 112 – Foundations of Computer Science

Computer Capabilities Representation the encoding data using bits, and the encoding instructions using bits Storage storing bits used to encode data and instructions Data Processing for each instruction: 1) Interpreting the bits of the instruction, and 2) doing the operation specified in the instruction on the bits of the data specified in the instruction 10/4/20156CS 112 – Foundations of Computer Science

Computer Capabilities Data Input converting data transferred into the machine (say through a keyboard or mouse) into bits (and storing these bits) Data Output converting bits of data into appropriate format for transfer outside the machine (e.g. to the screen) Data Communication Transfer bits of data from one computer to another 10/4/20157CS 112 – Foundations of Computer Science

Computer Capabilities  What are some of the different types of data that may be encountered? Text (sequences of characters used to represent names, descriptions, etc.) Numeric (to represent different quantities, e.g. age, length, height, amount of money, growth rates) Image (to represent photographic or synthesized pictures) Audio (to represent audio signals of humans and otherwise) Video (to represent video signals) 10/4/20158CS 112 – Foundations of Computer Science

Computer Capabilities  What are some of the different types of instructions that may be used on this data? Text (extracting characters, Concatenating characters together, comparing characters) Numeric (arithmetic +, -, *, , comparison among other operations) Audio (manipulate pitch, volume, high or low frequencies in an audio signal) Image (manipulate pixels properties, e.g. color) 10/4/20159CS 112 – Foundations of Computer Science

Data Representation & Storage Representation by encoding works as follows:  First you have a value set; a fixed set of values you want to encode (data values likes numbers, characters, instructions to motorists,etc.).  You have a symbol set; a fixed set of symbols to use in constructing codes for these values (for computers this is generally the symbols {0,1}).  Using combinations of the symbols found in the symbol set you construct a unique code corresponding to each value in the value set. 10/4/201510CS 112 – Foundations of Computer Science

Data Representation & Storage Exercise I To save Money a new traffic light device will be deployed where only two colors are available red & green; propose an encoding scheme that allows the stop, slow down, and go instructions to be given to motorists using this new traffic light device Answer  What’s the value set ?  What’s the symbol set?  What’s the encoding scheme? 10/4/201511CS 112 – Foundations of Computer Science

Data Representation & Storage Exercise I To save Money a new traffic light device will be deployed where only two colors are available red & green; propose an encoding scheme that allows the stop, slow down, and go instructions to be given to motorists using this new traffic light device Answer  What’s the value set ? stop, slow down & go  What’s the symbol set? red & green  What’s the encoding scheme? stop = red, go = green & slow down = red & green 10/4/201512CS 112 – Foundations of Computer Science

Numeric Representation  What does 943 mean?  As a decimal number 10/4/201513CS 112 – Foundations of Computer Science

Numeric Representation  What does 943 mean?  As a decimal number 9 * 10 2 = 9 * 100= * 10 1 = 4 * 10= * 10 0 = 3 * 1= called positional notation here represented in base 10 10/4/201514CS 112 – Foundations of Computer Science

Binary Representation  Generally speaking binary representation (base 2) of integers will use the symbols 0 & 1 in a positional representation where positions represent powers of or thus = = 25 10/4/201515CS 112 – Foundations of Computer Science

Binary Representation  Represent the decimal number 125 in binary  The standard algorithm (1) Start with your number, here 125, in base 10 (2) Divide the number (125) by 2 and record the remainder 125 / 2 has a quotient of 62 and a remainder of 1 (3) If the quotient = 0 stop, else Go to step 2 and repeat using the quotient as the number (4) Record the remainders recorded from right to left 10/4/201516CS 112 – Foundations of Computer Science

Binary Representation QR 125 / / / / / / / = /4/201517CS 112 – Foundations of Computer Science

Binary Addition How did we learn to add? ?? Do you remember the tables ?? = = = = 10 carry (hmmm… why didn’t they give us this in grade school?) 10/4/201518CS 112 – Foundations of Computer Science

Binary Addition  Example carry /4/201519CS 112 – Foundations of Computer Science

Negative Numbers Sign and Magnitude It's what we're used to: Fortunately there are only two choices for the sign: + and - So why no add a bit, and then use 0 for + and 1 for - Thus > - 38 sign bit 10/4/201520CS 112 – Foundations of Computer Science magnitude

Negative Numbers Among other things this leads to two zeros = +0 and = -0 that aren't really different numbers. Try instead something called 2’s complement 10/4/201521CS 112 – Foundations of Computer Science

Negative Numbers Might think of as clock arithmetic /4/201522CS 112 – Foundations of Computer Science

Addition and Subtraction = 5 or = /4/201523CS 112 – Foundations of Computer Science

Addition and Subtraction = 3 or = /4/201524CS 112 – Foundations of Computer Science

Addition and Subtraction = -3 or = /4/201525CS 112 – Foundations of Computer Science

Negative Numbers 2’s Complement Take the 1’s complement and add 1 5 -> 01011’s comp -> 1010 add > -5 Now go the other way -5 -> 10111’s comp -> 0100 add > 5 Thus the 2’s complement of a 2’s complement is the original number (same is true for the negative) 10/4/201526CS 112 – Foundations of Computer Science

Negative Numbers Find -56 using an eight bit representation 56 -> 's comp add > -56 What does represent? 2's complement it > 71 so the original number must have been /4/201527CS 112 – Foundations of Computer Science

Negative Numbers Find -83 using an eight bit representation What does represent? What does represent? 10/4/201528CS 112 – Foundations of Computer Science

Need More Bits Positive Numbers – Fill new positions to the left with 0 7 in four bits= in eight bits= in sixteen bits= Negative Numbers – Fill new positions to the left with 1 -7 in four bits= in eight bits= in sixteen bits= /4/201529CS 112 – Foundations of Computer Science

Addition and Subtraction Addition – follow the usual rules = = carry out – ignored, it means you passed /4/201530CS 112 – Foundations of Computer Science

Addition and Subtraction Subtraction can be represented as the addition of the negative > 7 + (-5) >-4 + (-3) /4/201531CS 112 – Foundations of Computer Science

Addition and Subtraction So is there any concern? Yes – no matter how many bits you use there is some maximum and some minimum number that can be represented. For four bits those are +7 and -8 respectively = 11 and -6 + (-4) = -10 are beyond a 4-bit rep /4/201532CS 112 – Foundations of Computer Science

Addition and Subtraction So is there any concern? Yes – no matter how many bits you use there is some maximum and some minimum number that can be represented. For four bits those are +7 and -8 respectively = 11 and -6 + (-4) = -10 are beyond a 4-bit rep two positives added to a negative 10/4/201533CS 112 – Foundations of Computer Science

Addition and Subtraction So is there any concern? Yes – no matter how many bits you use there is some maximum and some minimum number that can be represented. For four bits those are +7 and -8 respectively = 11 and -6 + (-4) = -10 are beyond a 4-bit rep two negatives added to a positive 10/4/201534CS 112 – Foundations of Computer Science

Addition and Subtraction So is there any concern? Yes – no matter how many bits you use there is some maximum and some minimum number that can be represented. For four bits those are +7 and -8 respectively = 11 and -6 + (-4) = -10 are beyond a 4-bit rep Both cases are called overflow. 10/4/201535CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement /4/201536CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) (5) (-2) /4/201537CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) (5) 0110(6) (-2) 1001(X) /4/201538CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) (5) 0110(6) 0011(3) (-2) 1001(X) 1010(X) /4/201539CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) (5) 0110(6) 0011(3) 1000(-8) (-2) 1001(X) 1010(X)10111(X) /4/201540CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) /4/201541CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) 1001(-7) (-5) (X) 10/4/201542CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) 1001(-7) 0011(3) (-5) 1010(-6) (X) 1101(-3) 10/4/201543CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) 1001(-7) 0011(3) 0111(7) (-5) 1010(-6) 1101(-3) (X) 1101(-3)10100(4) 10/4/201544CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) 1001(-7) 0011(3) 0111(7) 1111(-1) (-5) 1010(-6) 1101(-3) 0111(7) (X) 1101(-3)10100(4)10110(6) 10/4/201545CS 112 – Foundations of Computer Science

Addition and Subtraction Carry out the following – if there is overflow indicate that – if there is no overflow interpret your answer in decimal form. Numbers are listed in 2's complement. 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 0101(5) 0110(6) 0011(3) 1000(-8) 0011(3) 1110(-2) 1001(X) 1010(X)10111(X) 1101(-3) 1001(-7) 0011(3) 0111(7) 1111(-1) 1010(-6) 1011(-5) 1010(-6) 1101(-3) 0111(7) 1101(-3) 10011(X) 1101(-3)10100(4)10110(6)10111(X) 10/4/201546CS 112 – Foundations of Computer Science