Chapter : Data Representation In Computer Memory, PPT, Semester, Engineering - Electronics and Communication Engineering (ECE) PDF Download

DATA REPRESENTATION
IN COMPUTER MEMORY

SUMMARY: This topic introduces the numbering systems: decimal, binary, octal and hexadecimal. The topic covers the conversion between numbering systems, binary arithmetic, one's complement, two's complement, signed number and coding system. This topic also covers the digital logic components.
CLO 2:apply appropriate method to solve arithmetic problem in numbering system (C3).

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2.1 Understand data representation on CPU.
2.1.1 Define decimal, binary, octal and hexadecimal number.
2.1.2 Perform arithmetic operation (addition and subtraction) in different number bases.
2.1.3 Convert decimal, binary, octal and hexadecimal numbers to different bases and vice-versa

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INTRODUCTION

The binary system and decimal system is most important in digital system.
Decimal - Universally used to represent quantities outside a digital system.
Its means, there will be situations decimal values must be converted to binary values before entered to digital system.
Example : Calculator / Computer

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DECIMAL NUMBERING SYSTEM

Decimal system is composed of 10 numerals or symbols.
These 10 sysmbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Using these symbols as digits of a number, it can express any quantity.

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Base number = 10
Basic number = 0,1,2,3,4,5,6,7,8,9

23410
Basic number
Base number

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Positional Values (weights)

2746.210 is from calculation below:
2746.2 = (2x103) + (7x102) + (4x101) + (6x100) + (2x10-1) = 2000 + 700 + 40 +6 +0.2 = 2746.2

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BINARY NUMBERING SYSTEM

Define Binary numbers
Binary numbers representing number in which only digits 0 or 1.
ADDITION BINARY NUMBERS
Basic binary addition rule :
0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10
1 + 1 + 1 = 11
Example : 101 + 101 = 1010
1011 + 1011 = ?

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Exercise

Ex 1:
110112 + 100012 = 1011002
Ex 2:
101112 + 1112 = ________

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Subtraction

Four conditions in binary subtraction
0 - 0 = 0
0 - 1 = 1 borrow 1
1 - 0 = 1
1 - 1 = 0
10 - 1 = 1
If a 10 being borrow a 1, what‟s left with that 10 is a 1

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Ex 1:
10012 – 102 = 1112

Ex 2:
1010112 – 11112 =__________

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Conversions of Binary Numbers

Binary to Decimal conversions

Example : 1 1 0 1 12
 24 + 23 + 22 + 21 + 20 = 16 + 8 + 2 + 1
= 2710

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Decimal to Binary conversions

Convert 2510 to binary number

Exercise: Convert 3010 to binary number

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OCTAL NUMBERING SYSTEM

The octal number system has a base of eight, meaning that it has eight possible digits: 0,1,2,3,4,5,6 and 7.
The digit positions in an octal number have weights as follows :

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Octal number - (Subtraction - Pengurangan)

Ex:

524₈ – 167₈ = 335₈
167₈ – 24₈ = _________

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Octal –to-decimal conversion

Convert 3728 to decimal number
3728 = 3 x (82) + 7 x (81) + 2 x (80)
= (3 x 64) + (7x 8) + (2 x 1)
= 25010

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Decimal-to-Octal Conversion

Decimal integer can be converted to octal by using the same repeated-division method with a division factor of 8.

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Octal –to- Binary conversion

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Binary to Octal conversion

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HEXADECIMAL NUMBERING SYSTEM

The hexadecimal number system uses base 16.
It has 16 possible digit symbols.
It uses the digits 0 through 9 plus the letters A, B, C, D, E and F as the 16 digit symbols.

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Hexadecimal number - Addition (Penambahan)

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Hexadecimal number - Subtraction (Pengurangan)

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Hexadecimal-To-Decimal Conversion

A hexadecimal number can be converted to its decimal equivalent by using the fact that each hex digit position has a weight that is a power of 16.

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Decimal-To-Hexadecimal Conversion

Decimal to hex conversion can be done using repeated division by 16.

Ex: Convert 2010 to hex

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Hexadecimal-to-Binary Conversion

Like the octal number system, the hexadecimal number system is used primarily as a “shorthand” method for representing binary numbers.
It is a relatively simple matter to convert a hex number to binary .
Each hex digit is converted to its four-bit binary equivalent.

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Binary-to-Hexadecimal Conversion

The binary number is grouped into groups of four bits, and each group is converted to its equivalent hex digit.
Zero are added, as needed to complete a four-bit group.

Ex:
101₂ = 0101
= 5₁₆

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Summary

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2.1.4 Describe the coding system

a.Sign and magnitude b. 1‟s Complement and 2‟s Complement c. Binary Coded Decimal (BCD system) d. ASCII and EBCDIC

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Describe the coding system

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One’s Complements and Two’s Complements

One‟s Complements
One‟s complements is used in binary number.
The one‟s complement of a binary number is obtained by changing each 0 to 1 and 1 to a 0.
Only change negative number
In other words, change each bit in the number to its complement.

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Exp:

10011001 – original binary number
01100110 – complement each bit to form 1‟s complement
Thus, we say that the 1‟s complement of 10011001 is 01100110.