Information Science 1

‐Representa*on of Data 

in Memory‐ 

College of Information Science and Engineering

Ritsumeikan University

Week 03 

Topics covered

  Basic terms and concepts of “The Structure of a Computer”

  Positional numbering systems – decimal

– binary

– octal

– hexadecimal

  Conversion among different bases -  to decimal and from decimal

- other conversions

  Test

Recall Week 02

  Digital system

  Data, Binary

  Memory, RAM

  Bit, Byte, Computer Word, Address

  CPU

  CU, ALU, MAR, MDR, IR, GPR, PC, PSW

  Machine Cycle, Fetch, Execute,

Automatic Sequence Control

  Input, Output

  Computer Bus

Objectives of this class

  To understand the fundamentals of

numerical data representation and

manipulation in computers

  To master the skill of representing decimal

numbers in the binary, octal, and

hexadecimal systems

  To be able to speak aloud a number in any

of the four bases

  To be able to convert from decimal to the

above three numbering systems

Recall: Computer as a digital system

  A bit is the most basic (and, hence, the

smallest) unit of information in a computer

-  It is a state of “on” or “off” in a digital circuit

-  Sometimes these states are also called as

“high” or “low” (voltage)

  A byte is a group of eight bits

-  A byte is the smallest (in principle)

addressable unit of computer storage

-  “Addressable” means that a particular byte

can be retrieved according to its location

Computer words and nibbles

  A word is a contiguous group of bytes

- Words can be any number of bytes or bits (word

sizes of 32 or 64 bits are now most common)

-  In a word-addressable system, a word is the

smallest addressable unit of memory storage

  A group of four bits is called a nibble (or

nybble)

-  A byte, therefore, consists of two nibbles:

a high-order nibble, and a low-order nibble

00111100010110001010011000110111101

Positional numbering system

  A positional numbering system (or positional notation system) is a numeral system in which each digit is related to the next by a constant multiplier called the base or radix of that system

– The value of each digit position is, therefore, the value of its digit multiplied by a power of the base, where the power is determined by the digit's position counted from the separator (which is usually a dot “.” or comma “,”)

– The value of a number is then calculated as the sum of the values of all positions

Decimal system

  Decimal numbers have radix (base) = 10 (in Latin, decima means a tenth part )

  Symbols used: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

  Position weights:

… 103 102 101 100 . 10-1 10-2 10-3 … 1000 100 10 1 1/10 1/100 1/1000

— For example, the decimal number 36.250: 36.250 = 3*10 + 6*1+ 2*1/10 + 5* 1/100 +0* 1/1000

Binary system

  Binary numbers have radix = 2 (in Latin, bini means two together )

  Symbols used: 0 and 1

  The radix (≠10) is denoted by a subscript (2

  Position weights:

… 23 22 21 20 . 2-1 2-2 2-3 … 8 4 2 1 1/2 1/4 1/8

— For example, the binary equivalent of 36.250: 36.250 = 1*32 + 1*4 + 1/4 = 100100.01(2

Octal system

  Octal numbers have radix = 8 (in Latin, octo means eight)

  Symbols used: 0, 1, 2, 3, 4, 5, 6, and 7

  Convenient when bits are grouped in triplets   Position weights:

… 83 82 81 80 . 8-1 8-2 2-3 … 512 64 8 1 1/8 1/64 1/512

— For example, the octal equivalent of 36.250: 36.250 = 4*8 + 4*1 + 2*1/8 = 44.2(8

Hexadecimal system

  Hexadecimal numbers have radix = 16 (in Greek, hexa means six)

  Symbols used: 0, …, 9, A, B, C, D, E, and F

  Convenient to represent nibbles (hextets)

  Position weights:

… 163 162 161 160 . 16-1 16-2 16-3 … 4096 256 16 1 1/16 1/256 1/4096

— For example, the hexadecimal equivalent of 36.250: 36.250 = 2*16 + 4*1 + 4*1/16 = 24.4(16

Radix r numbers

  For any radix r number represented with n

+m+1 digits as

Number(r → an...a1 a0 . a-1...a-m ,

its decimal equivalent is calculated as

follows:

Number(10 =

an×r n +...+a1×r 1 +a0 +a-1×r -1 +...+a-m×r –m

= ∑∑−−==

nn

mmii

ii

iirraa∑∑

−−==

nn

mmii

ii

iirraa

Examples: ASCII (main code)

Examples: RGB color codes

Conversion among different systems

  We already learned these 3 conversions:

Hexadecimal

Decimal Octal

Binary

  There still remain 9 other possible

conversions

Converting decimal to another base

  The rule

— Left of the separator (i.e. of the decimal

point): repeatedly divide the integer part

by the radix and write the remainders (R)

right to left

— Right of the separator: repeatedly multiply

the fractional part by the radix and write

the integer portion (I) of the result left to

right

Example: Decimal to binary

22.8125 = ?(2

22.8125 = 10110.1101(2

22.8125 = 10110(2.

11 R 0

5 R 1

2 R 1

1 R 0

0 R 1

22.8125 = .1101(2

1.625 I 1

1.25 I 1

0.5 I 0

1.0 I 1

0 I 0

Other examples

1234 = ?(8

1234 = 2322(8

154 R 2

19 R 2

2 R 3

0 R 2

1234 = ?(16

1234 = 4D2(16

77 R 2

4 R 13 = D

0 R 4

The remaining conversions

  To convert between binary, octal, and

hexadecimal, memorize and use the

conversion table (next slide):

— Octal is 3 bits for each digit

— Hexadecimal is 4 bits for each digit

  Use binary as the intermediate to convert

between octal and hexadecimal

0110100110.1(2 = ?(8 = ?(16

000 110 100 110.100 0001 1010 0110.1000

6 4 6 .4 1 A 6 .8

0110100110.1(2 = 646.4(8 = 1A6.8(16

Conversion table Decimal Base 10

Binary Base 2

Octal Base 8

Hexadecimal Base 16

0(10 0000(2 0(8 0(16

1(10 0001(2 1(8 1(16

2(10 0010(2 2(8 2(16

3(10 0011(2 3(8 3(16

4(10 0100(2 4(8 4(16

5(10 0101(2 5(8 5(16

6(10 0110(2 6(8 6(16

7(10 0111(2 7(8 7(16

8(10 1000(2 10(8 8(16

9(10 1001(2 11(8 9(16

10(10 1010(2 12(8 A(16

11(10 1011(2 13(8 B(16

12(10 1100(2 14(8 C(16

13(10 1101(2 15(8 D(16

14(10 1110(2 16(8 E(16

15(10 1111(2 17(8 F(16

Key points of this lecture

  Because binary numbers are the basis for

all data representation in digital systems,

it is important that you become proficient

with the binary system to understand the operation of all computer components as

well as the design of computer architectures

  The binary system is the most important positional numbering system for

computers

Key points (cont-d)

  It is, however, difficult to read long strings of

bits, and even a modestly-sized decimal

number becomes a very long binary number

— For example, 1359510 =

11010100011011(2

— For compactness and ease of reading, binary

values are usually expressed using the octal

or hexadecimal system

  To convert among the different systems, use

the conversion algorithms and, when

appropriate, the conversion table

Homework

 Read these slides again

 Read slides for the next lecture and do the self-preparation assignments

 Learn the vocabulary

 Consult, when necessary, the textbook

  Representation of data in memory

(2) - Basic computer operation

Next class

  Write your name (in English), student ID

number, and solve the following problems

(write only answers):

1. 

Test 01