DecimalDecimal BinaryBinary OctOct HexHex00 00 00 0011 11 11 1122 1010 22 2233 1111 33 3344 100100 44 4455 101101 55 5566 110110 66 6677 111111 77 7788 10001000 1010 8899 10011001 1111 99

1010 10101010 1212 AA1111 10111011 1313 BB1212 11001100 1414 CC1313 11011101 1515 DD1414 11101110 1616 EE1515 11111111 1717 FF1616 1000010000 2020 10101717 1000110001 2121 1111 1

230)10 =?)2

1100111)2 =?)10

The number 230 can be expressed as: 128 + 64 + 32 + 4 + 2So, the answer is: 11100110

The number 1100111 represents: 64 + 32 + 4 + 2 + 1So, the answer is: 103

(41)10=(?)2

41)10 =?)2

41/2=20 1 (This the least significant bit)20/2=10 010/2=5 05/2=2 12/2=1 0 1/2= 0 1(This the most significant bit) (101001) 2

Note! The Most significant bit is the last bit

4

(41)10=(?)2

41)10 =?)2

41/2=20 1 (This the least significant bit)20/2=10 010/2=5 05/2=2 12/2=1 0 1/2= 0 1(This the most significant bit) (101001) 2

Note! The Most significant bit is the last bit5

(41)10= (101001) 2

PLEASE CHECK YOUR CLASS NOTES FOR THIS LECTURE

6

Integer Numbers (Signed)There are many ways to represent negative integers. The simplest method is called sign-and-magnitude. It means using the leftmost bit as a sign bit (1 for negative, 0 for positive). In a 32-bit system that means that the first bit represents the sign, and the other 31 the absolute value of the number.For example –9 would be:1000 0000 0000 0000 0000 0000 0000 1001This method has one major drawback, it would allow for two zero values –0, and +0. For that reason, it is not used.

+0: 0000 0000 0000 0000 0000 0000 0000 0000

–0: 1000 0000 0000 0000 0000 0000 0000 0000

Integer Numbers (Signed) Without a doubt, your personal computer uses a

method known as two's complement. A little more complicated than sign-and-magnitude but not that much and it has only one zero value.

To have –9 in two's complement, you invert all the bits in the representation of 9 and add 1.

You can achieve the same result by starting from the right; find the first 1 and invert all the bits to the left of that one. +9 = 0000 0000 0000 0000 0000 0000 0000 1001

–9 = 1111 1111 1111 1111 1111 1111 1111 0111

To indicate a signed number, we need to specify the sign to do this we usually set aside the Most Significant bit (MSB) for the sign bit. If that bit is 0, then the rest of the bits giving us the magnitude of a positive number, if MSB is 1, then we have a negative number & we need to interpret the remaining bits based on one of the following methods:

Sign-&-Magnitude 2’s-Complement

Do not forget 2’s-Complement is only for negative values!!!!!!!!

For Positive value do the regular conversion.

9

Word: Depending on the type of hardware being used it can be 8, 16 or 32 bits. The width of a memory cell.

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 1

0 0 0 0 0 1 0 0

0 0 0 0 0 1 0 1

0 0 0 0 0 1 1 0

0 0 0 0 0 1 1 1

10

8 or 16 or 32 or 64 depending on the hardware

Hexadecimal Numbers (HEX)

In computer science, the base-16 numeral system is often used to represent groups of 4 bits. It uses 16 distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen.

Example:+9 = 0000 0000 0000 0000 0000 0000 0000 1001

0 0 0 0 0 0 0 9

-9 = 1111 1111 1111 1111 1111 1111 1111 0111 F F F F F F F 7

Real (Floating Point) Numbers

Real numbers in binary are expressed with a sign, a mantissa (or fraction or significand), and an exponent.

The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely used standard for floating-point computation

The exponent is biased depending on the format (127 for single precision, 1023 for double).

Real (Floating Point) NumbersSuppose +9.0 in binary:

The sign is positive. It will be represented by a sign-and magnitude convention (0) at the leftmost bit.

The exponent will be 3 (the closest power of 2 less or equal than 9 is 8, which is 2 to the power of 3), then biased (3+127=130 for single, 3+1023=1026 for double). That number is then converted to an unsigned binary integer to fit into the 8-bit exponent zone (11-bit zone for double precision).

To compute the mantissa you take the number (9.0) and divide it by 2 to the power of the exponent (in this case 8). You get 1.125. The mantissa is then filled from left to right with 1s and 0s. The first bit represents 0.5 (1 is assumed but not represented), the second bit 0.25, ... For 1.125 we need 1+0.125 which is the third bit at 1, all the others remaining at 0.

Real (Floating Point) Numbers

9.0 in single-precision binary is:0100 0001 0001 0000 0000 0000 0000 0000

A double number is expressed in 64 bits:52 bits for the mantissa, 11 bits for the exponent, and 1 sign bit.

So 9.0 double-precision binary is: 0100 0000 0010 0010 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000

See a IEEE-754 Floating-Point Conversion calculator at http://babbage.cs.qc.edu/IEEE-754/Decimal.html

Real (Floating Point) Numbers

Now let's do the reverse. Let's convert a binary IEEE-754 floating-point number into a decimal.

1100 0011 0110 0000 0000 0000 0000 0000

1. Determine the sign: That one is easy. 1 is negative, 0 is positive. We have negative number.

2. Determine the exponent: 10000110 is 128+4+2=134. That our biased exponent. Subtract the bias (127) to find the true exponent: 7.Finally, let's calculate 2 to the power of 7, we get 128. This means our number is between 128 and 255.

3. Determine the mantissa: The first bit represents 0.5, the second 0.25, the third 0.125.... Here we have the two first bits at one: 0.5+0.25 = 0.75. Since 1 is always assumed, our mantissa value is 1.75.

4. Do the final computation: Multiply the final value of step #2 by the final value of step #3. 128 times 1.75 is 224. So our number is

–224.0

Check the class examples for computation

CharactersCharacters are expressed using the ASCII code:'A' = 65 = 01000001'g' = 103 = 01100111'$' = 36 = 00100100'+' = 43 = 00101011Note that digits expressed in ASCII are different

than integers or doubles.'9' = 57 = 00111001

See the ASCII code at www.asciitable.com

Programming Languages Many languages exist to program instructions to the computer.1st generation: Machine languages (pure binary) Example: 101011101010101010111010101011 2nd generation: Assembly languages (mnemonic codes) Example: MV R1,R3 3rd generation: High-level languages (C, Fortran, Java)Example: y = x * x + 2; 4th generation: Closer to human languages than typical high-level programming languages (MATLAB, SQL)Example: find all records where name is “Kim”

Solving Problems Here are the steps required to solve a

problem with a computer program:

1. Define the problem2. Analyze the problem.3. Design a solution.4. Implement the solution.5. Test the program.6. Update and maintain the program.

Why MATLAB for CPS118? It is a very popular software and language with students, engineers, and scientists in universities, research institutes and industries all over the world. It is being used by many advanced courses in the science program. It is powerful and easy to use. Programs are platform-independent. Programs written on one platform will run on all the other platforms without change (Windows, Unix, Linux, Macintosh). MATLAB contains an extensive library of predefined functions. MATLAB has advanced capabilities for plotting and graphics making it a visually attractive environment for new programmers.

MATLAB Avantages Its basic data element is the matrix (like 2 two-dimensional grid). A simple integer is considered an matrix of one row and one column. Several mathematical operations that work on arrays or matrices are built-in to the Matlab environment. For example, cross-products, dot-products, determinants, inverse matrices. Vectorized operations. Adding two arrays together needs only one command, instead of a for or while loop. The graphical output is optimized for interaction. You can plot your data very easily, and then change colors, sizes, scales, etc, by using the graphical interactive tools. Matlab’s functionality can be greatly expanded by the addition of toolboxes. These are sets of specific functions that provided more specialized functionality. Ex: Excel link allows data to be written in a format recognized by Excel, Statistics Toolbox allows more specialized statistical manipulation of data (Anova, Basic Fits, etc)

Matlab Drawbacks It uses a large amount of memory and on slow computers it is very hard to use. It sits “on top” of Windows, getting as much CPU time as Windows allows it to have. This makes real-time applications very complicated. MATLAB is an interpreted language. The main disadvantage of interpreted languages is execution speed. When a language is compiled, all of the code is analyzed and processed efficiently, before the programmer distributes the application. With an interpreted language, the computer running the program has to analyze and interpret the code before it can be executed (each and every time), resulting in slower processing performance. For applications where performance is critical, a fast language like C is a much better choice.

End of Lesson

Please check your class lecture notes for more examples (What is written on blackboard)

Computers are electronic systems that can transmit, store and manipulate information (data).

Data can be numeric, character, graphic and sound.

For beginner programmers, the two most important are character and numeric.

To manipulate data, a computer needs a set of instructions called a program.

To write such programs is the object of thiscourse.

25

An algorithm is a series of instructions on how to solve the problem. We need algorithms before we can write programs.

Algorithms have the following characteristics:

The order of execution of operations must be correct.

The operations must be clear and unambiguous.

The operations must be things that are feasible.

They must produce a result.

They must stop in a finite amount of time.

Precision: must give the correct results. Uniqueness: same inputs will give the same results. Finiteness: it must stop eventually. Input: most algorithms need input. Output: the answers are the outputs. Generality: must work with multiple data sets

26

A Simple Algorithm

Baking Bread:1. Add dry ingredients.

2. Mix.

3. Add water.

4. Knead.

5. Let rise.

6. Bake.

Another Complex AlgorithmSorting mail:

1. Get piece of mail from mail box

2. If piece is personal

2.1 Read it

else

if piece is a magazine

2.1.1 Put in magazine rack

else

if piece is a bill

2.1.1.1 Pay it

else

if piece is junk mail

2.1.1.1.1 Throw it away

About Algorithms 1 Algorithms are a set of instructions for solving a problem.

Once you have created an algorithm, you no longer need to understand the principles on which the algorithm is based.

For example, once you have the directions to Ryerson, you don't need a map any more. The information needed to find the correct route is contained in the algorithm. All you have to do is follow the directions.

About Algorithms 2 Algorithms are a way of sharing expertise with others.

Once you have encoded the necessary intelligence to solve a problem in an algorithm, many people can use your algorithm without needing to become experts in a particular field.

Algorithms are especially important to computers because computers are really machines for solving problems.

For a computer to be useful, we must give it a problem to solve and a technique for solving the problem.

About Algorithms 3 You can make computers "intelligent" by programming them with various algorithms to solve problems.

Because of their speed and accuracy, computers are well suited for solving tedious problems such as searching for a specific gene sequence in a DNA mapping.

Finally, not all problems can be solved with computers because the solutions to some problems can't be stated in an algorithm.

Writing an Algorithm We will now learn how to write an algorithm to solve a simple problem:

Sort the following numbers in ascending order:

7 2 8 3 5

Think about how you would solve this problem for a moment.

Sorting is a common problem handled by computers.

Different types of computer systems:

Mainframes

Workstations

Desktop computers

Palmtop Computers or Personal Digital Assistants (PDA)

Laptop or notebook computers

Tablets

33

A. HardwarePhysical components that together provide the functionality required by a computing machineCPU (Central Processing Unit) - ALU (Arithmetic Logic Unit) - Internal RegistersMemory - RAM (Random Access Memory) aka Read/Write Memory - ROM (Read Only Memory) -CacheI/O Device -Keyboard -Mouse -Display -Printer

34

Components of a Computer

Storage Types CPU registers: Only a few cells on CPU.

Read-Only Memory (ROM): Thousands of cells containing pre-set computer instructions.

Main memory (RAM): Billions of cells on circuits separate from CPU. Memory location picked-up at random and access time is the same.

Secondary storage: Hundreds and thousands of billions of cells on disks (magnetic or optic), flash drives or tapes.

Secondary storage is not volatile (data is kept even when power is off).

Internal Representations Bit: Binary digit (0 or 1).

Byte: A group of 8 bits. One character.

Word: The width of a memory cell.

Nibble: A group of 4 bits. Used to make bit sequences easier to read.

Each byte of main memory has an address.

All numbers and all characters are represented in binary code.

Understand what MATLAB is and why it is widely used in engineering and science

Start the MATLAB program and solve simple problems in the command window

MATLAB is one of a number of commercially available, sophisticated mathematical computation toolsOthers include

MapleMathematicaMathCad

MATLAB is both a powerful computational environment and a programming language that easily handles matrix and complex arithmetic.

It is a large software package that has many advanced features built in, and it has become a standard tool for many working in science or engineering disciplines.

Among other things, it allows easy plotting in both two and three dimensions.

What Is MATLAB?

Numerical calculations◦ Especially:matrices

Graphics MATLAB stands for Matrix Laboratory

Easy to use Built in programming language

MATLAB was originally written in Fortran, then later rewritten in C

MATLAB is updated regularly The Mathworks packages their software in

groups, called releases, New releases are issued twice a year in the spring and in the fallRelease 2013a includes◦ MATLAB 8.1◦ A number of specialized “toolboxes”

MATLAB has two different methods for executing commands: interactive mode and batch mode.

In interactive mode, commands are typed (or cut and pasted) into the command window.

In batch mode, a series of commands is saved in a text file with a .m extension. The batch commands in a file are then executed by typing the name of the file at the MATLAB command prompt.

What Is MATLAB?

MATLAB is an interpreted language. Commands are executed line by line.

It supports two basic types: characters like 'a' (16 bits) and doubles like 8.25 (64 bits).

Most of the time MATLAB works with arrays or matrices of real numbers or characters.

Other types are also supported: complex, symbolic (if installed), 16-bit and 8-bit integers, etc...

The MATLAB Language

• A standardized approach to problem solving commonly found in all science disciplines.

• 1.first you state the problem, • 2.then describe the input and output.• 3. Develop a plan for solving the problem, called an

algorithm. • 4.Then you actually solve the problem, in our case

using MATLAB, and • 5. finally you test your solution for reasonableness.

If you use a consistent problem solving strategy you increase the chance that your result is correct

MATLAB uses a standard windows menu bar

To exit MATLAB use the close icon

Command WindowEnter commands at the prompt

MATLAB WindowsCommand History Window Records all commands issued in the command window – including mistakes

Current Folder WindowLists files stored in the current directory

Workspace Window

The MATLAB DesktopCu

rren

t Fol

der W

indo

w

Command Window

Workspace Window

Command History Window