ECE101: Digital System Design

Unit I: Lecture 2
Number Systems

Dr.R.EZHILARASIE
Assistant Professor
School of Computing
SASTRA Deemed to be University

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 Objectives

• To comprehend the different number System and its weighting structure

– Binary, Decimal, Octal and Hexadecimal

• To convert the given number from one number system to other number system
Eg. Binary → Decimal , Decimal → Binary , Binary→ Hexadecimal, Decimal → Octal etc.,

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 Objectives

• To comprehend the different number System and its weighting structure

– Binary, Decimal, Octal and Hexadecimal

• To convert the given number from one number system to other number system
Eg. Binary → Decimal , Decimal → Binary , Binary→ Hexadecimal, Decimal → Octal etc.,

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 Outcomes

• List different Number systems

• Explain the weight structure of each number system

• Perform number conversion

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 Number System

• Used to represent the information

• Commonly used Number System

– Decimal, Binary, Octal and Hexadecimal

• Any number system there is an ordered set of symbols : Digits

• Collection of these digits make a number

• Contains two part: Integer & Fractional and separated by a radix point

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 Decimal Number System
• Decimal number system contain ten digits: 0 through 9

• Ten distinct digits: base of 10

• Value of each digit in a number can be determined using

– Position of the digit in the number

– Base of the number system

• Positional Weights in Decimal Number System:

. …

• E.g. 34.23

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 Binary Number System
• Uses two digits 0 and 1 : Base two system

• 0 and 1:bits

• Positional weights in Binary system:

…
. …

• E.g. 1101.011

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 Octal Number System
• Uses eight digits 0 through 7 : Base eight system

• Used extensively in early microcomputers

• Sets of 3 bit binary numbers can be represented by a single octal number

• Positional weights in Octal system:

…
. …

• E.g. (73.25)8 = 7*8^1+3*8^0 (.) + 2*8^-1+5*8^-2 = 7*8+3*1+2*0.125+5*0.015625

= 56+3+0.25+0.078

= 59.3210
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 Hexadecimal Number System
• Uses 16 digits 0 through 9 plus A,B,C,D,E,F : Base sixteen system

• Sets of 4 bit binary numbers can be represented by single hex number

• Used extensively in memory organization

• Positional weights in hexadecimal system:

…
. …

• E.g. 2A.31 = 2∙161+10∙160+3∙16-1+1∙16-2 = 2*16+10*1+3*0.063+1*0.004

= 32+10+0.189+0.004= 42.19310

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 Number Conversion
• Human uses decimal number system but computer uses binary number system: Decimal to Binary

• Binary to Decimal :when computer displays the result

• Large quantity of binary bits: inconvenient

– Binary → Octal or Hexadecimal

• Number base conversions

– Decimal → Binary / Octal / Hexadecimal

– Binary →Decimal / Octal / Hexadecimal

– Octal → Decimal / Binary / Hexadecimal

– Hexadecimal → Decimal / Binary /Octal

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 Decimal to Binary / Octal / Hexadecimal
Steps:
1. Divide the integer part of decimal number by desired base , store the quotient and remainder
2. Consider quotient as a new decimal number and repeat step 1 until quotient becomes 0
3. List the remainders in the reverse order

E.g. 34

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 Decimal to Binary / Octal / Hexadecimal

Decimal to Binary Decimal to Octal Decimal to Hexadecimal

(34)10 = (100010)2 (34)10 = (42)8 (28)10 = (?)16

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 Decimal fractions to Binary / Octal/ Hexadecimal
Steps:
1. Multiply the fractional part of decimal number by desired base
2. Record the integer part of product as carry and fractional part as new fractional part
3. Repeat steps 1 and 2 until fractional part of product becomes zero or desired digits are obtained.
4. Read Carries downwards to get desired base

E.g. 0.65

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 Decimal fractions to Binary/ Octal / Hecxadecimal

Decimal fractions to Binary Decimal fractions to Octal Decimal fractions to

Hexadecimal
(0.23)10 = (?)2 (0.23)10 = (?)8 (0.23)10 = (?)16

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 Binary to Decimal: Octal to Decimal: Hexadecimal to Decimal
Step:
• Sum of all digits multiplied by their weights (in a power of desired base) gives the
decimal equivalent

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 Binary to Octal
Octal Digit 0 1 2 3 4 5 6 7
Binary 000 001 010 011 100 101 110 111

Steps
• Make group of 3-bits starting from LSB for integer and MSB for fractional part, by adding 0s at the
end if required
• Convert each 3-bit group to the equivalent octal digit
• E.g. (11001.101011)2

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 Binary to Hexadecimal
Dec 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hex 0 1 2 3 4 5 6 7 8 9 A B C D E F
Bin 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

Steps
• Make group of 4-bits starting from LSB for integer part and MSB for fractional part, by adding 0s at
the end if required
• Convert each 4-bit group to the equivalent hexadecimal digit
• E.g.(11001.101011)2

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 Octal to Binary
Octal Digit 0 1 2 3 4 5 6 7
Binary 000 001 010 011 100 101 110 111

Steps:
• Write equivalent 3-bit binary number for each octal digit
• Remove any leading or trailing zeros
• E.g. 246.71

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 Octal to Hexadecimal
Steps:
• Convert Octal → Binary
• Convert Binary → Hexadecimal equivalent
• E.g. 246.71

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 Hexadecimal to Binary
Steps:
• Write equivalent 4-bit binary number for each hexadecimal digit
• Remove any leading or trailing zeros
• Eg: 24A

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 Hexadecimal to Octal
Steps:
• Convert Hexadecimal → Binary
• Convert Binary → Octal equivalent
• Eg: 24A

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 Summary

• List the different number system

• Weighting Structure of each number system

– Position
– Base of the System

• Number Conversions

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Thank You

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