Internal Information Representation and Processing

CSCE 106 - Fundamentals of Computer Science

Dr. Awad Khalil

Computer Science & Engineering Department

The American University in Cairo

Internal Information Representation - slide 2 Fundamentals of Computer Science – Awad Khalil

Decimal Number System

We are used to the decimal number system which is a positional number system The decimal number 4386 represents the value: 41000 + 3 100 + 8 10 + 6 1 In general, the decimal number dn dn-1 . . . d1 d0 represents the value:

The digit at the left is called the most significant digit The most significant digit has the largest power of ten The digit at the right is called the least significant digit The concept of zero is a cornerstone in the decimal number system. It enables us to

multiply by 10 by simply adding a zero as a least significant digit. Similarly, we can divide by zero by remove the least significant zero or inserting a

decimal point.

n

i

ii

nn

nn ddddd

0

00

11

11 101010...1010

1234 10 = 12340 1234 / 10 = 123.4

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Other Radices

The positional number system is called also the Arabic number system (the Arabs have discovered the 0)

Other bases (called also radices) are possible A digit in base b is between 0 and b-1

Base 10 digits: 0, 1, …, 9 (called also decimal digits) Base 2 digits: 0 and 1 (called also binary digits or bits) Base 3 digits: 0, 1, 2 Base 8 digits: 0, 1, …, 7 (called also octal digits) Base 16 digits: 0, 1, …, 9, A, B, C, D, E, F (called also hexadecimal digits)

The number (dn dn-1 . . . d1 d0)b represents the value:

Examples:

n

i

ii

nn

nn bdbdbdbdbd

0

00

11

11 ...

(207)8 = 2 82 + 0 81 + 7 80 = 128 + 0 + 7 = 135

(1011)2 = 1 23 + 0 22 + 1 21 + 1 20 = 8 + 0 + 2 + 1 = 11

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Binary, Octal, and Hexadecimal Number Systems

Computers use the binary number system Two binary digits: 0 and 1 called bits

In the hardware circuitry, 0 represents LOW voltage, while 1 represents HIGH voltage

Information inside the computer is represented by 0’s and 1’s

The integer values 0 to 16, represented in four number systems are shown below:

0 0 0 01 1 1 12 10 2 23 11 3 34 100 4 45 101 5 56 110 6 67 111 7 78 1000 10 89 1001 11 9

10 1010 12 A11 1011 13 B12 1100 14 C13 1101 15 D14 1110 16 E15 1111 17 F16 10000 20 10

Decimal Binary Octal Hexadecimal

Advantages of the octal and hexadecimal numbers: Reduce the length of numbers

Are more readable than binary numbers

Can be easily converted to/from binary

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Conversions between Number Systems

A number in base b can be easily converted to base 10 using the following equation:

Examples:

We can also convert a decimal number N to obtain the number (dn dn-1 . . . d1 d0 )b in base b using the following algorithm:

n

i

ii

nn

nn bdbdbdbdbd

0

00

11

11 ...

(2012)3 = 2 33 + 0 32 + 1 31 + 2 30 = 54 + 0 + 3 + 2 = 59

(A0F)16 = 10 162 + 0 161 + 15 160 = 2560 + 0 + 15 = 2575

i := 0;q := N;repeat di := q mod b; q := q div b; i := i + 1;until q = 0;

Question: Convert 50 to base 2Answer:d0 = 50 mod 2 = 0, q = 50 div 2 = 25d1 = 25 mod 2 = 1, q = 25 div 2 = 12d2 = 12 mod 2 = 0, q = 12 div 2 = 6d3 = 6 mod 2 = 0, q = 6 div 2 = 3d4 = 3 mod 2 = 1, q = 3 div 2 = 1d5 = 1 mod 2 = 1, q = 1 div 2 = 0

Thus, 50 = (110010)2

Question: Convert 50 to base 3Answer:d0 = 50 mod 3 = 2, q = 50 div 3 = 16d1 = 16 mod 3 = 1, q = 16 div 3 = 5d2 = 5 mod 3 = 2, q = 5 div 3 = 1d3 = 1 mod 3 = 1, q = 1 div 3 = 0

Thus, 50 = (1212)3

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More Conversions

Suppose we want to convert a number from base 3 to base 2, we can convert it first to base 10 and then to base 2

Example: Convert (1012)3 to base 2

Answer:(1012)3 = 1 33 + 0 32 + 1 31 + 2 30 = 27 + 3 + 2 = 32 d0 = 32 mod 2 = 0, q = 32 div 2 = 16d1 = 16 mod 2 = 0, q = 16 div 2 = 8d2 = 8 mod 2 = 0, q = 8 div 2 = 4d3 = 4 mod 2 = 0, q = 4 div 2 = 2d4 = 2 mod 2 = 0, q = 2 div 2 = 1d5 = 1 mod 2 = 1, q = 1 div 2 = 0

Thus, (1012)3 = 32 = (100000)2

It is easy to convert numbers between the binary, octal, and hexadecimal systems Every 3 binary digits can be converted into an octal digit starting at the least significant bit

Every 4 binary digits can be converted into a hexadecimal digit

Example: (10100110)2 = (246)8 = (A6)16

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Computer Representation of Information

Basic unit of information is the Bit or Binary digit.

With a single bit, we can represent two distinct values 0 and 1.

With two bits, we can represent four distinct values: 00, 01, 10, and 11.

In general, with m bits, we can represent 2m distinct values.

A byte is a grouping of 8 bits.

A word is a grouping of either 16, 32, or 64 bits, depending on the computer system.

A word is typically 32 bits on most systems, a half word is 16 bits, and a double word is 64 bits.

Bit = 0 or 1

Byte = 8 bits

Half Word = 2 bytes = 16 bits

Word = 4 bytes = 32 bits

Long Word = 8 bytes = 64 bits

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Binary Addition

The addition of 3 bits a + b + c produces a sum bit and a carry bit.

Signed Integers are represented in 2’s complement notation.

Addition of integers in 2’s complement notation results in a signed integer.

Although a carry out is produced in the 2nd and 4th computations, it is ignored.

a b c carry sum

0 0 0 0 00 0 1 0 10 1 0 0 10 1 1 1 01 0 0 0 11 0 1 1 01 1 0 1 01 1 1 1 1

a + b + c

Examples:

carry 11110 carry 111011000

01001101 +77 11101101 –19+ 00101011 +43 + 10101110 –82

sum 01111000 +120 sum 10011011 –101

carry 11110 carry 111011000

11001101 –51 01101101 +109+ 00101011 +43 + 10101110 –82

sum 11111000 –8 sum 00011011 +27

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The Discipline of Computing

Computing is a relatively young academic discipline. We distinguish between applications of the computer within other disciplines such as

business, engineering, and sciences, and the nature of computing itself. Peter Denning gave the following broad definition of the field of Computer Science:

Computer Science is the body of knowledge dealing with the design, analysis, implementation, efficiency, and application of processes that transform information. The fundamental question underlying all of computer science is "what can be automated?"

Nine general subject areas combine to make up the discipline of Computing. These are shown below:

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The Discipline of Computing

Artificial

Intelligence

Algorithms and Data Structures

Programming Languages

Computer Architecture

Software Engineering

Numeric and Symbolic Computation

Operating Systems

Database Systems

Human-Computer Communication