ECE2030 Introduction to Computer Engineering

Lecture 2: Number System

Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean LeeSchool of Electrical and Computer EngineeringSchool of Electrical and Computer EngineeringGeorgia TechGeorgia Tech

22

Decimal Number Representation• Example: 90134 (base-10, used by Homo

Sapien)= 90000 + 0 + 100 + 30 + 4= 9*104 + 0*103 + 1*102 + 3*101 + 4*100

• How did we get it?

9013490134109013901310 44 90190110 33 909010 11 99 00

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Generic Number Representation• 90134

= 9*104 + 0*103 + 1*102 + 3*101 + 4*100

• A4 A3 A2 A1 A0 for base-10 (or radix-10)= A4*104 + A3*103 + A2*102 + A1*101 + A0*100

(A is coefficient; b is base)• Generalize for a given number N N w/

base-bbNN = An-1 An-2 … A1 A0

NN = An-1*bn-1 + An-2*bn-2 + … + A2*b2 + A0*b0

**Note that A < b**Note that A < b

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Counting numbers with base-bb

00112233445566778899

1010111112121313141415151616171718181919

Base-10

9090919192929393949495959696979798989999

…..

100100101101102102103103104104105105106106107107108108109109

How about Base-8

0011223344556677

10101111121213131414151516161717

20202121222223232424252526262727

70707171727273737474757576767777

…..

100100101101102102103103104104105105106106107107

2020212122222323242425252626272728282929

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How about base-22

0011

10101111

100100101101110110111111

1000100010011001101010101011101111001100110111011110111011111111

66

How about base-22

0 0 1 1

10101111

100100101101110110111111

1000100010011001101010101011101111001100110111011110111011111111

77

How about base-22

0 0 = 0= 01 1 = 1= 1

10 10 = 2= 211 11 = 3= 3

100 100 = 4= 4101 101 = 5= 5110 110 = 6= 6111 111 = 7= 7

1000 1000 = 8= 81001 1001 = 9= 91010 1010 = 10= 101011 1011 = 11= 111100 1100 = 12= 121101 1101 = 13= 131110 1110 = 14= 141111 1111 = 15= 15

Binary Binary == DecimalDecimal

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Derive Numbers in Base-2 • Decimal (base-10)

– (25)10

• Binary (base-2)– (11001)2

• Exercise

252521212 2 11 66 2 00 33 2 00 11 11

99

Base-2 • Decimal (base-10)

– (982)10

• Binary (base-2)– (1111010110)2

• Exercise

1010

Base 8• Decimal (base-10)

– (982)10

• Octal (base-8)– (1726)8

• Exercise

1111

Base 16• Decimal (base-10)

– (982)10• Hexadecimal (base-16)• Hey, what do we do when

we count to 10??

• 0

• 1• 2• 3• 4• 5• 6• 7• 8• 9• 10• 11• 12• 13• 14• 15

00112233445566778899aabbccddeeff

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Base 16

• (982)10 = (3d6)16

• (3d6)16 can be written as (0011 1101 0110)2

• We use Base-16 (or Hex) a lot in computer world– Ex: A 32-bit address can be written as 0xfe8a7d20 0xfe8a7d20 ((0x 0x is an abbreviation of Hex))

– Or in binary form Or in binary form 1111_1110_1000_1010_0111_1101_0010_0000 1111_1110_1000_1010_0111_1101_0010_0000

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Number Examples with Different Bases• Decimal (base-10)

– (982)10

• Binary (base-2)– (01111010110)2

• Octal (base-8)– (1726)8

• Hexadecimal (base-16)– (3d6)16

• Others examples: – base-9 = (1321)9 – base-11 = (813)11

– base-17 = (36d)17

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Convert between different bases• Convert a number base-x to base-y, e.g. (0100111)2 to

(?)6– First, convert from base-x to base-10 if x 10– Then convert from base-10 to base-y

0100111 = 026 + 125 + 024 + 023 + 122 + 121 + 120 = 39

3939666 6 33 11 00

(0100111)2 = (103)6

Base-b Addition

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Negative Number Representation• Options

– Sign-magnitude– One’s Complement– Two’s Complement (we use this in this

course)

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Sign-magnitude• Use the most significant bit

(MSB) to indicate the sign– 00: positive, 11: negative

• Problem– Representing zeros?– Do not work in computation

• We will NOT use it in this course !

+0 000+1 001+2 010+3 011-3 111-2 110-1 1010 100

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One’s Complement• Complement (flip) each bit

in a binary number • Problem

– Representing zeros?– Do not always work in

computation• Ex: 111 + 001 = 000

Incorrect !• We will NOT use it in this

course !

+0 000+1 001+2 010+3 011-3 100-2 101-1 1100 111

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Two’s Complement• ComplementComplement (flip) each bit in a

binary number and adding 1adding 1, with overflow ignored

• Work in computation perfectly• We will use it in this course !

011

100One’s complement

3

101Add 1

-3

010One’s complement

101-3

011Add 1

3

2020

Two’s Complement• ComplementComplement (flip) each bit in a

binary number and adding 1adding 1, with overflow ignored

• Work in computation perfectly• We will use it in this course !

0 000+1 001-1 111+2 010-2 110+3 011-3 101?? 100

100

011One’s complement

100Add 1

The same 100 representsboth 4 and -4 which is no good

2121

Two’s Complement• ComplementComplement (flip) each bit in a

binary number and adding 1adding 1, with overflow ignored

• Work in computation perfectly• We will use it in this course !

0 000+1 001-1 111

1+2 010-2 111

0+3 011-3 110

1--4 110

0

100

011One’s complement

100Add 1

MSB = 1 for negative Number, thus 100 represents -4

2222

Range of Numbers • An N-bit number

– Unsigned: 0 .. (2N -1)– Signed: -2N-1.. (2N-1 -1)

• Example: 4-bit

1110 (-8) 0111 (7)Signed numbers

0000 (0) 1111 (15)Unsigned numbers

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Binary Computation

010001 (17=16+1)001011 (11=8+2+1)---------------011100 (28=16+8+4)

Unsigned arithmetic 010001 (17=16+1)101011 (43=32+8+2+1)---------------111100 (60=32+16+8+4)

Signed arithmetic (w/ 2’s complement) 010001 (17=16+1)101011 (-21: 2’s complement=010101=21)---------------111100 (2’s complement=000100=4, i.e. -4)

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Binary ComputationUnsigned arithmetic 101111 (47)011111 (31)---------------001110 (78?? Due to overflow, note that 62 cannot be represented by a 6-bit unsigned number)

The carry isdiscarded

Signed arithmetic (w/ 2’s complement) 101111 (-17 since 2’s complement=010001)011111 (31)---------------001110 (14)

The carry isdiscarded

BACKUP

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Application of Two’s Complement• The first Pocket CalculatorPocket Calculator

“Curta” used Two’s complement method for subtractionsubtraction

• First complement the subtrahend – Fill the left digits to be the same

length of the minuend– Complemented number = (9 – digit)

• 4’s complement = 5• 7’s complement = 2• 0’s complement = 9

• Add 1 to the complemented number

• Perform an addition with the minuend

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Examples• 13 – 7

– Two’s complement of 07 = 92 + 1 = 93– 13 + 93 = 06 (ignore the leftmost carry digit)

• 817 – 123– Two’s complement of 123 = 876 + 1 = 877– 817 + 877 = 694 (ignore the leftmost carry

digit)• 78291 – 4982

– Two’s complement of 04982 = 95017 + 1 = 95018

– 78291 + 95018 = 73309 (ignore the leftmost carry digit)