An Introduction to Programming through C++

Abhiram G. Ranade

Reading: Ch. 2: A bird’s eye view

A basic question

• How is a computer able to do so many things?– Search for information– Predict weather– Process pictures and say what is in them– Play chess..

• Goal of this lecture sequence: provide high level answers–Most real life problems can be viewed as mathematical problems on

numbers– A computer is good at solving math problems

• High level answers will give good background for later discussion.

Outline

• How to express real life problems as numerical problems.– Picture processing– Predicting the weather– Processing text/language

• Algorithms and Programs– Enable us to tell a computer what operations to perform

• How a computer does the required operations– Digital circuits– How numbers are represented – Parts of a computer–Machine language program, compilation.

“What is in this picture?”

https://en.wikipedia.org/wiki/File:Jackson%27s_Chameleon_2_edit1.jpg

How to represent black and white pictures using numbers

• Suppose picture is 10cm x 10cm.• Break it up into 0.1 mm x 0.1 mm squares• 1000 x 1000 squares. 1 million “pixel”s• If square is mostly white, represent it as 0. • If square is mostly black, represent it as 1.• Picture = 1 million numbers!

Picture, Representation, Reconstruction

0 0 0 1 1 1 0 0 0 00 0 1 0 0 0 1 1 0 00 1 0 0 0 0 0 0 1 01 0 0 0 0 0 0 0 1 01 0 1 0 0 0 1 0 0 11 0 0 0 0 0 0 0 0 11 0 0 1 1 1 0 0 1 00 1 0 0 0 0 0 0 1 00 0 1 0 0 0 1 1 0 00 0 0 1 1 1 0 0 0 0

(a) (b) (c)

Remarks

• Better representation if picture divided into more cells.• Pictures with different “gray levels”: use numbers 0,

0.1, …, 1.0 to represent level of darkness rather than just 0, 1.

• Pictures with colours: picture = 3 sequences– sequence for red component, – sequence for blue component, – sequence for green component

• Add up the colours to get the actual colour.

Computer vision/Image processing

Input: sequence P of 1 million numbers (0 or 1) • Representing a 10cm x 10cm black and white picture, • Given in left to right, top to bottom order.A very simple image processing problem: • Is there a vertical line in the picture?Expressing the problem mathematically: What property does the sequence need to have if it is to contain a vertical line?• All 0s, except for 1s in consecutive rows of some column• Going down a column = move 1000 positions in the sequence• 1s in positions i, i+1000, i+2000, i+3000, i+4000,… for some i• ”Is there a vertical line?” = “Does sequence P satisfy above property?”One way of solving the problem:• Try all values of i..

Does the picture contain a chameleon?

• In principle, same as asking whether the picture contains a single vertical line:– Identify a set of properties that the sequence of numbers

representing the picture must satisfy if the picture contains a chameleon.

– Identify computations that can check if the given number sequence satisfies the required properties.

• In practice requires enormous ingenuity• Main concern of the deep subject “Computer Vision”

Weather prediction

• Divide the surface of the earth into small regions (like pixels).

• Let pi, ti, hi = pressure, temperature, humidity in region i• Laws of physics tell us how to the values will change with

time.• We can measure current pressure, humidity, temperature

values, and calculate what will happen tomorrow!• Smaller the regions, better will be the accuracy. (Smaller

the pixels, better will be the picture representation).

Language/text using numbers

• Define a code for representing letters.• Commonly used code: ASCII – (American Standard Code for Information Interchange)

• Letter ‘a’ = 97 in ASCII, ‘b’ = 98, …• Uppercase letters, symbols, digits also have codes. Code also for space

character.• Words = sequences of ASCII codes of letters in the word.• ‘computer’ = 99, 111,109,112,117,116,101,114.• Sentences/paragraphs = larger sequences.• Does the word “computer” occur in a paragraph?– Does a certain sequence of numbers occur inside another sequence of numbers?

Exercises

• What pattern of 1s and 0s would correspond to a ”+” of any size being present at the center of an otherwise white picture?

• Suppose you are given a sentence in a language you cannot understand. Would you still be able to count the number of words in the sentence? Can you express this as a question on sequences of numbers representing the ASCII codes of different characters?

• How will you represent Chess playing as a question on numbers? Start by representing a chess board with pieces on it using numbers.

Summary of what we discussed

• Questions about pictures, weather, documents can be converted to questions about properties of number sequences.

• Finding answers requires solving interesting math problems.

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A historical remark

• Computers are used to solve problems, but problem solving is not new, humans have been solving numerical problems for millenia – Astronomical calculations for navigation– Land records/geometry– Poetry raised some interesting computational problems and some were

solved very elegantly by ancient Indian mathematicians– Last 500 years: development of differential equations to analyze structures,

weather, ...

• “Solving problems” = deciding what operations to perform to calculate the required answer

• Algorithms = A precise description of the operations needed to solve a problem.

You already know many algorithms!

• The procedures you have learnt in primary school for arithmetic on numbers with many digits are really algorithms.

• Primary school algorithms contain all ingredients that are present in advanced algorithms– Arithmetic operations: “Add the least significant digit of the first number to the

least significant digit of the second number”– Conditional operation: “Set carry = 1 if the previous sum was greater than 9”– Repetition: “Repeat as many times as there are digits”

• Other algorithms you know– determine whether a number is prime, – finding the greatest common divisor– Drawing a polygon on the screen

• All these algorithms will be useful on computers!

Programs

• Algorithms written in precise syntax/language.• C++ is one such language.• Other languages: Fortran, Basic, Lisp, …• All these languages can be used to specify

arithmetic operations, conditional execution, and repetition.– You have already seen how to specify repetition – use

the repeat statement!

Exercise

• Write down the algorithm for multiplying an n digit number by another n digit number that you learned in primary school.– Break up the description into parts, e.g. in the first part we

do multiplications and in the second we do additions.– To describe what operations you perform, it might be useful

to give names e.g. Let Qi, Ri denote the ith least significant digits of the multiplicand and the multiplier.

– Are there phases in the algorithm in which you do similar operations? If yes can you say what happens in “ith phase”?

What we discussed

• Notion of problem solving, algorithms is old• An algorithm specifies a sequence of operations–May contain arithmetic operations–May contain operations to be executed conditionally–May ask for sequence of operations to be repeated

• You already know many nice algorithms– Arithmetic on numbers, matrices, root finding...– Very useful for programming

• Programs are formal descriptions of algorithms, in specific languages.

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How a computer solves (numerical) problems

Outline• Digital circuits• How numbers are represented • Parts of a computer• Machine language program, compilation.

Digital circuits: building blocks of computers

• Digital circuits: interpret electrical potential/voltage as numbers.• Simplest convention– Voltage above 1 volt = number 1, Voltage between 0 and 0.2 volt = number 0– Circuit designed so that voltage will never be between 0.2 and 1 volt, hence no ambiguity.

• Current may also be used, e.g. current < some value represents 0…• Charge stored on a capacitor may also denote numbers– Capacitor has low charge = number 0, High charge = number 1– Once charge is stored on a capacitor, it persists. “Memory”

• Once you can represent 0 and 1, you can represent anything: NEXT• We can design circuits which perform arithmetic: – Circuit inputs: sets of voltages representing two numbers– Circuit outputs: set of voltages representing their sum! – Or product, or quotient …

Standard representation for non-negative numbers

• Use many capacitors to store a single number– Typical number of capacitors used : 8, 16, 32, 64

• Example: To store 25 using 32 bits:– Represent 25 in binary.

25 Decimal = 00000000000000000000000000011001

• So store the following charge pattern (H=High, L=Low) LLLLLLLLLLLLLLLLLLLLLLLLLLLHHLLH• Range stored: 0 to 11111111111111111111111111111111 = 232 –

1. • If your numbers are likely to be larger, then use 64 bits.• Transmitting numbers: send high or low voltages on as many wires.

Binary representation revision

• Binary number an-1an-2...a1a0 . a-1a-2…a-k

– Example: 101.11

• Decimal value v = ∑i ai2i – 1*22 + 0*21 + 1*20 + 1*2-1 + 1*2-2 = 5.75

• Converting a decimal integer v to binary– Divide v by 2, remainder gives a0

– Repeat previous step with the quotient to get a1, a2, …

• Converting fraction f to binary– If f > 0.5, a-1 = 1– Similarly other bits…

Representing integers that can be positive or negative

• One of the bits is used to indicate sign• Sign bit = 0 (low charge/voltage) means positive number, = 1 means

negative number.• To store -25 use10000000000000000000000000011001• Leftmost bit = sign bit• Max positive number: 231 - 101111111111111111111111111111111• Range stored: -231 – 1 to 231 – 1.• Actual representation used: –more complex. “Two’s complement”.

int nsides;

• C++ will typically designate some 32 capacitors in your computer for the variable nsides.

• The bit pattern stored in it will be interpreted as having a sign and a magnitude.

unsigned int nsides;• 32 capacitors will be given, but the bit pattern in it will be

interpreted as 32 bit binary number.• 10000000000000000000000000011001 will mean 231+25

Bits, bytes, half-words, words

• Bit = 1 binary “digit”, (one number = 0 or 1)• byte = 8 bits• half-word = 16 bits• word = 32 bits • double word = 64 bits

“one byte of memory” = memory capable of storing 8 bits = 8 capacitors.

Representing Real numbers

• Use analogue of “scientific notation”:significand * 10exponent

e.g. 6.022 * 1023

• Same idea, but significand, exponent are in binary, thus number is:significand * 2exponent

• “Single precision”: store significand in 24 bits, exponent in 8 bits. – Fits in one word! – 24 bits of significand = 7-8 decimal digits

• “Double precision”: store significand in 53 bits, exponent in 11 bits. – Fits in a double word!– 53 bits of significand = 16-17 decimal digits

• Actual representation: more complex. “IEEE Floating Point Standard”.

Indicative example

Let us represent Avogadro’s number 6.022 x 1023

• Convert to binary: 1.11111110001010111 x 21001110

• Use 23 bits for magnitude of fraction, 1 bit for sign of fraction.• Use 7 bits for magnitude of exponent, 1 bit for sign of exponent• 0111111110001010111000001001110• Decimal point is assumed after 2nd bit.• Inherently imprecise: fraction represented only to certain finite

number of bits.• IEEE Floating Point Representation: more complex.

Exercise

• Suppose I want to represent 8 digit telephone numbers. I should use ___ (Fill in 8,16,32,64) bit _____ (Fill in signed or unsigned) representation.

• Which is bigger, byte or half-word?• What is roughly the largest number that can be

represented using 64 bits?– Hint: 210 = 1024 ≈ 103

What we discussed

• Numbers are represented by sequence of 0s and 1s• The same sequence may mean one number as an unsigned integer, a signed

integer, or a floating point number• The capacitors only store high or low charge, they are not “aware” that the charge

represents numbers.• So long as we remember what type of number we are storing, there will be no

problem.• As a user, you don’t need type/read binary numbers.– C++ will convert binary numbers to decimal system while printing– C++ will accept numbers typed in decimal by you and itself convert it to binary for use on

the computer.– But you should know (roughly) what range of numbers can be stored in k bit unsigned/signed/

floating formats

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How a computer works

Outline• Overall computer organization• How the parts work

Organization of a computer

Memory

• Organized into bytes (groups of 8 capacitors)• Memories of present day computers contain few

Gigabytes, Giga = 230 ≈ 109 , billion.• Each byte in the memory is assigned a distinct number, or

an address. Much like houses on a street!• In a memory with N bytes, the addresses go from 0 to N-1• The memory circuit communicates with the rest of the

world through 3 sets of wires or “ports”: address port, data port, control port.

Storing/writing data into memory

Suppose we want to store a number D in the byte with address A.• Place the number A on the address port.– Address port consists of several wires, place bits of A on respective

wires.– “Place a number” = “Set the voltage on the wire appropriately”

• Place the number D on the data port.• Place the number 1 on the control port.• Hold the values on the ports steady for 1 “clock cycle”.⇒ Capacitors at address A will get data D!• Clock cycle = how long? memory designer will tell you..

Reading what is stored

Suppose we wish to know what is stored in address A of the memory• Place A on address port• Place 0 on control port.• After 1 clock cycle, the values in the capacitors at address A

will appear on the data port.• Data port connects to the rest of the world by wires.– Through them the rest of the world will know what was in address

A.

• Reading the value at an address does not destroy it.

Remarks

• You are not expected to understand what circuitry is present in a memory.

• Instead of reading or writing a byte at a time, entire word starting at given address A may be read or written into. (“word oriented” memory)– “Word starting at address A”: data stored in bytes having

address A, A+1, A+2, A+3.– The data port will have 32 wires to accept 32 bits = 1 word

• Similarly we can have “double word oriented memory”…

Arithmetic Unit

Ports: Input1, Input2, Output, ControlInput1, Input2, Output will consist of w wires, w = size of memory data portControl = several wires. Number appearing on the control wires will say what operation should be performed.• 1 cycle after values are placed on Control, the Output

will take the commanded value: sum, product, …

Peripherals: keyboard, screen, disk…

• Also have control port and data port like organization.• Value placed on control port tells the peripheral what to

do with the value on the data port, or itself places values on the data port.– Screen: data port value will be suitably interpreted and

shown on the screen.– Keyboard: some value corresponding to what user types will

be placed on the data port by the keyboard circuits.– Value placed on the data port can be sensed by the rest of the

computer

Control Unit

• Tells other parts of the computer what to do.– Sends numbers on control wires of each unit

• The control unit decides what to tell other units by reading a “machine language program” stored in the memory.

• Machine language program = sequence of numbers representing machine language instructions

• Machine language instruction examples:– “Make the ALU add the numbers in addresses x,y and store the

result in address z”– Same but multiply instead of add.

What we discussed

• Memory, ALU, Peripherals communicate with the rest of the world through– ”Data port”, “Address port”, “Control port”

• Control unit places values on control ports of other devices and tells them what to do.

• Control unit arranges movement of data between other parts of the computer.

• Control unit know what to tell others by reading a ”machine language program”

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Machine language instruction

Machine language instruction = sequence of numbers• Possible structure of machine language instruction:– First number = says what operation to perform (“operation code”) – Second and third numbers : addresses in memory from where the

operands are to be taken– Fourth number: address in memory where the result is to be stored.

• Machine language instructions are designed by the computer designer.– A machine language instruction will be designed for every operation

the computer can perform

(Fictitious) Examples of machine language instructions

• Hypothetical instruction: 57, 100, 200, 300• Operation code 57 might mean “multiply”• On reading the above instruction control unit does the following:– Tells the memory to read the words at the first two addresses and send them to the

Arithmetic unit.– Tells the arithmetic unit to perform multiplication by sending appropriate number on

its control wires.–Moves the result from the arithmetic unit to the memory– Tells memory to store the received word into the word at the third address

• This instruction causes the product of the numbers stored in addresses 100, 200 to be stored in the address 300.

• 58 might mean the same thing as above, except perhaps the numbers would be added.

Machine language program (hypothetical) example

• Example: 57, 100, 100, 100, 57, 100, 100, 100• This contains two instructions.• Both instructions cause the word at address 100 to be

multiplied by itself and the result stored back in address 100.• After executing the first instruction, address 100 would

contain the square of the number that was present before.• The second operation would repeat the squaring operation.• Thus this is a machine language program to compute the

fourth power of a number.

Exercise

• Give a machine language program which computes the cube of the number stored in address 100 and stores in address 200. Hint: Modify the fourth power program slightly.

Control unit operation

• Control unit must be told where the machine language program is in memory.

• The control unit then fetches the instructions constituting the program, interprets the codes, and performs the required operation.

• After one instruction is fetched and executed, it fetches the next instruction and repeats the process.

• Yes, the circuitry in the control unit is very tricky, but it can be built!

Remarks

• We said 57 is the code for multiplication.. – The actual code used on different machines will be different.– You do not need to remember that 57 = multiplication.

• Actual machine languages are more complex, will have many more instructions.

• On early computers, you would have to write machine language programs.– Decide what operation you want to perform.– Look up the manual and find its code.– Enter the code into the memory of the computer.– Enter the addresses of operands, result.– Repeat.Process is laborious, error-prone.

Machine language programs and C++ programs

• On a modern computer you write a C++ program.• A prewritten program, “compiler”, translates your C++

program to a machine language program.–When you type s++ square.cpp the compiler is called upon to

compile your file square.cpp. – It creates the “machine language program” which by default is

called a.out on unix.

• When you type ./a.out :– a.out gets loaded into memory by the “loader” program – Then a.out executes.

Concluding Remarks

• In order to solve problems on a computer, they must be formulated as problems on numbers.

• Numerical codes can represent non numerical entities– Letters and other symbols: ASCII code

• Algorithm: precise sequence of calculations needed to solve a certain problem– Human beings have been using algorithms well before computers were invented, for

pencil paper calculations.– You yourself know many algorithms.– Computer algorithms are very similar to paper pencil algorithms

• Very natural strategy for designing algorithms: – Think about how you would solve the problem using pencil/paper, given enough time.– Understand the structure of the required calculations.– Express those in a programming language. (Rest of the course!)

Concluding Remarks

• Current/charge/voltage values in the computer circuits represent bits (0 or 1). – Larger numbers can be represented using sequence of bits.– In a fixed number of bits you can represent a fixed number of

numbers.

• Circuits can be designed which take as input voltages representing numbers and produce voltages representing their product/sum/…

• Memory in a computer is organized as a sequence of bytes, each byte can be identified by its address.

Concluding Remarks

• Machine language program : sequence of machine language instructions–Must be present in the memory

• Control unit reads machine language instructions and interprets them– Decides what needs to be done– Sends control signals to other units and makes them do the needful

• Users write program written in high level language e.g. C++– User program compiled into machine language by compiler– Loader loads compiled program into memory and then it executes.