Design of Digital Circuits Reading: Binary Numbers

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Design of Digital Circuits Reading: Binary Numbers

Required Reading for Week 1 23-24 February 2017 Spring 2017

Carnegie Mellon

Binary Numbers Design of Digital Circuits 2016 Srdjan Capkun Frank K. Gürkaynak http://www.syssec.ethz.ch/education/Digitaltechnik_16

Adapted from Digital Design and Computer Architecture, David Money Harris & Sarah L. Harris ©2007 Elsevier

2

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In This Lecture 

How to express numbers using only 1s and 0s



Using hexadecimal numbers to express binary numbers



Different systems to express negative numbers



Adding and subtracting with binary numbers

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Number Systems 1's column 10's column 100's column 1000's column

Binary Numbers 

Decimal Numbers 

537410 =

1's column 2's column 4's column 8's column

11012 =

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Number Systems 

Decimal Numbers 1's column 10's column 100's column 1000's column

537410 = 5 × 103 + 3 × 102 + 7 × 101 + 4 × 100 five thousands



three hundreds

seven tens

four ones

Binary Numbers 1's column 2's column 4's column 8's column

11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 1310 one eight

one four

no two

one one

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Powers of two 20

=

28

=

21

=

29

=

22

=

210

=

23

=

211

=

24

=

212

=

25

=

213

=

26

=

214

=

27

=

215

=

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Powers of two 20

=

1

28

=

256

21

=

2

29

=

512

22

=

4

210

=

1024

23

=

8

211

=

2048

24

=

16

212

=

4096

25

=

32

213

=

8192

26

=

64

214

=

16384

27

=

128

215

=

32768

Handy to memorize up to 215 7

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Binary to Decimal Conversion 

Convert 100112 to decimal

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Binary to Decimal Conversion 

Convert 100112 to decimal

24

× 1 + 23 × 0 + 22 × 0 + 21 × 1 + 20 × 1 =

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Binary to Decimal Conversion 

Convert 100112 to decimal

24

× 1 + 23 × 0 + 22 × 0 + 21 × 1 + 20 × 1 =

16 × 1 + 8 × 0 + 4 × 0 + 2 × 1 + 1 × 1 = 16

+ 0

+ 0

+ 2

+ 1

= 1910

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Decimal to Binary Conversion 

Convert 4710 to binary

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Decimal to Binary Conversion 

Convert 4710 to binary  Start with 26 = 64  Now 25 = 32

is 64 ≤ 47 ?

no

do nothing

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Decimal to Binary Conversion 

Convert 4710 to binary       

Start with 26 = 64 Now 25 = 32 Now 24= 16 Now 23= 8 Now 22= 4 Now 21= 2 Now 20= 1

is 64 ≤ 47 ? is 32 ≤ 47 ? is 16 ≤ 15 ? is 8 ≤ 15 ? is 4 ≤ 7 ? is 2 ≤ 3 ? is 1 ≤ 1 ?

no yes no yes yes yes yes

do nothing subtract 47 – 32 =15 do nothing subtract 15 – 8 = 7 subtract 7-4 = 3 subtract 3-2 =1 we are done

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Decimal to binary conversion 

Convert 4710 to binary       



Start with 26 = 64 Now 25 = 32 Now 24= 16 Now 23= 8 Now 22= 4 Now 21= 2 Now 20= 1

is 64 ≤ 47 ? is 32 ≤ 47 ? is 16 ≤ 15 ? is 8 ≤ 15 ? is 4 ≤ 7 ? is 2 ≤ 3 ? is 1 ≤ 1 ?

no yes no yes yes yes yes

0 1 0 1 1 1 1

do nothing subtract 47 – 32 =15 do nothing subtract 15 – 8 = 7 subtract 7-4 = 3 subtract 3-2 =1 we are done

Result is 01011112

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Binary Values and Range 

N-digit decimal number  How many values?  Range?  Example: 3-digit decimal number  



10N [0, 10N - 1]

103 = 1000 possible values Range: [0, 999]

N-bit binary number  How many values?  Range:  Example: 3-digit binary number 



2N [0, 2N - 1]

23 = 8 possible values Range: [0, 7] = [0002 to 1112] 15

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Hexadecimal (Base-16) Numbers Decimal

Hexadecimal

Binary

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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Hexadecimal (Base-16) Numbers Decimal

Hexadecimal

Binary

0

0

0000

1

1

0001

2

2

0010

3

3

0011

4

4

0100

5

5

0101

6

6

0110

7

7

0111

8

8

1000

9

9

1001

10

A

1010

11

B

1011

12

C

1100

13

D

1101

14

E

1110

15

F

1111

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Hexadecimal Numbers 

Binary numbers can be pretty long.



A neat trick is to use base 16



How many binary digits represent a hexadecimal digit? 4 (since 24 = 16)



Example 32 bit number: 0101 1101 0111 0001 1001 1111 1010 0110

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Hexadecimal Numbers 

Binary numbers can be pretty long.



A neat trick is to use base 16



How many binary digits represent a hexadecimal digit? 4 (since 24 = 16)



Example 32 bit number: 0101 1101 0111 0001 1001 1111 1010 0110 5 D 7 1 9 F A 6

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Hexadecimal Numbers 

Binary numbers can be pretty long.



A neat trick is to use base 16



How many binary digits represent a hexadecimal digit? 4 (since 24 = 16)



Example 32 bit number: 0101 1101 0111 0001 1001 1111 1010 0110 5 D 7 1 9 F A 6



The other way is just as simple C

E

2

8

3

5

4

B 20

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Hexadecimal Numbers 

Binary numbers can be pretty long.



A neat trick is to use base 16



How many binary digits represent a hexadecimal digit? 4 (since 24 = 16)



Example 32 bit number: 0101 1101 0111 0001 1001 1111 1010 0110 5 D 7 1 9 F A 6



The other way is just as simple C E 2 8 3 5 4 B 1100 1110 0010 1000 0011 0101 0100 1011

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Hexadecimal to Decimal Conversion 

Convert 4AF16 (or 0x4AF) to decimal

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Hexadecimal to decimal conversion 

Convert 4AF16 (or 0x4AF) to decimal

162

× 4 + 161

×A

256

× 4 + 16

× 10 + 1

1024

+ 160

+ 160 × F

+ 15

=

× 15 = = 119910

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Bits, Bytes, Nibbles… 10010110 most significant bit

least significant bit

byte

10010110 nibble

CEBF9AD7 most significant byte

least significant byte

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Powers of Two 

210 = 1 kilo



1000



220 = 1 mega



1 million (1,048,576)



230 = 1 giga



1 billion (1,073,741,824)

(1024)

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Powers of Two (SI Compatible) 

210 = 1 kibi



1000



220 = 1 mebi



1 million (1,048,576)



230 = 1 gibi



1 billion (1,073,741,824)

(1024)

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Estimating Powers of Two 

What is the value of 224?



How many values can a 32-bit variable represent?

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Estimating Powers of Two 

What is the value of 224?

24 × 220 ≈ 16 million



How many values can a 32-bit variable represent? 22 × 230 ≈ 4 billion 28

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Addition 



Decimal

Binary

11 3734 + 5168 8902

carries

11 1011 + 0011 1110

carries

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Add the Following Numbers

1001 + 0101

1011 + 0110

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Add the Following Numbers

1 1001 + 0101 1110

111 1011 + 0110 10001 OVERFLOW !

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Overflow 

Digital systems operate on a fixed number of bits



Addition overflows when the result is too big to fit in the available number of bits



See previous example of 11 + 6

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Overflow (Is It a Problem?) 

Possible faults



Security issues

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Binary Values and Range 

N-digit decimal number  How many values?  Range?  Example: 3-digit decimal number  



10N [0, 10N - 1]

103 = 1000 possible values Range: [0, 999]

N-bit binary number  How many values?  Range:  Example: 3-digit binary number 



2N [0, 2N - 1]

23 = 8 possible values Range: [0, 7] = [0002 to 1112] 34

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Signed Binary Numbers 

Sign/Magnitude Numbers



One’s Complement Numbers



Two’s Complement Numbers

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Sign/Magnitude Numbers 

1 sign bit, N-1 magnitude bits



Sign bit is the most significant (left-most) bit  Positive number: sign bit = 0  Negative number: sign bit = 1

A : aN 1 , aN 2 ,

a2 , a1 , a0 

n 2

A  ( 1)an 1  ai 2i i 0



Example, 4-bit sign/mag representations of ± 6: +6 = -6=



Range of an N-bit sign/magnitude number: 36

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Sign/Magnitude Numbers 

1 sign bit, N-1 magnitude bits



Sign bit is the most significant (left-most) bit  Positive number: sign bit = 0  Negative number: sign bit = 1

A : aN 1 , aN 2 ,

a2 , a1 , a0 

n 2

A  ( 1)an 1  ai 2i i 0



Example, 4-bit sign/mag representations of ± 6: +6 = 0110 - 6 = 1110



Range of an N-bit sign/magnitude number: [-(2N-1-1), 2N-1-1] 37

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Problems of Sign/Magnitude Numbers 

Addition doesn’t work, for example -6 + 6: 1110 + 0110 10100 wrong!



Two representations of 0 (± 0): 1000 0000



Introduces complexity in the processor design (Was still used by some early IBM computers) 38

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One’s Complement 

A negative number is formed by reversing the bits of the positive number (MSB still indicates the sign of the integer): One’s Complement

27

26

25

24

23

22

21

20

0

0

0

0

0

0

0

0

=

+0

0

0

0

0

0

0

0

0

1

=

1

1

0

0

0

0

0

0

1

0

=

2

2





















0

1

1

1

1

1

1

1

=

127

127

1

0

0

0

0

0

0

0

=

-127

128

1

0

0

0

0

0

0

1

=

-126

129





















1

1

1

1

1

1

0

1

=

-2

253

1

1

1

1

1

1

1

0

=

-1

254

1

1

1

1

1

1

1

1

=

-0

255

Unsigned

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One’s Complement 

A negative number is formed by reversing the bits of the positive number (MSB still indicates the sign of the integer): One’s Complement

27

26

25

24

23

22

21

20

0

0

0

0

0

0

0

0

=

+0

0

0

0

0

0

0

0

0

1

=

1

1

0

0

0

0

0

0

1

0

=

2

2





















0

1

1

1

1

1

1

1

=

127

127

1

0

0

0

0

0

0

0

=

-127

128

1

0

0

0

0

0

0

1

=

-126

129





















1

1

1

1

1

1

0

1

=

-2

253

1

1

1

1

1

1

1

0

=

-1

254

1

1

1

1

1

1

1

1

=

-0

255

Unsigned

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One’s Complement 

The range of n-bit one’s complement numbers is: [-2n-1-1, 2n-1-1] 8 bits: [-127,127]



Addition: Addition of signed numbers in one's complement is performed using binary addition with end-around carry. If there is a carry out of the most significant bit of the sum, this bit must be added to the least significant bit of the sum:



Example: 17 + (-8) in 8-bit one’s complement +

0001 0001

(17)

1111 0111

(-8)

1 0000 1000 +

1 0000 1001 =

(9) 41

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Two’s Complement Numbers 

Don’t have same problems as sign/magnitude numbers:  Addition works  Single representation for 0



Has advantages over one’s complement:  Has a single zero representation  Eliminates the end-around carry operation required in one's complement addition

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Two’s Complement Numbers 

A negative number is formed by reversing the bits of the positive number (MSB still indicates the sign of the integer) and adding 1: Two’s Complement

27

26

25

24

23

22

21

20

0

0

0

0

0

0

0

0

=

0

0

0

0

0

0

0

0

0

1

=

1

1

0

0

0

0

0

0

1

0

=

2

2





















0

1

1

1

1

1

1

1

=

127

127

1

0

0

0

0

0

0

0

=

-255

128

1

0

0

0

0

0

0

1

=

-254

129





















1

1

1

1

1

1

0

1

=

-3

253

1

1

1

1

1

1

1

0

=

-2

254

1

1

1

1

1

1

1

1

=

-1

255

Unsigned

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Two’s Complement Numbers 

A negative number is formed by reversing the bits of the positive number (MSB still indicates the sign of the integer) and adding 1: Two’s Complement

27

26

25

24

23

22

21

20

0

0

0

0

0

0

0

0

=

0

0

0

0

0

0

0

0

0

1

=

1

1

0

0

0

0

0

0

1

0

=

2

2





















0

1

1

1

1

1

1

1

=

127

127

1

0

0

0

0

0

0

0

=

-128

128

1

0

0

0

0

0

0

1

=

-127

129





















1

1

1

1

1

1

0

1

=

-3

253

1

1

1

1

1

1

1

0

=

-2

254

1

1

1

1

1

1

1

1

=

-1

255

Unsigned

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Two’s Complement Numbers 

Same as unsigned binary, but the most significant bit (msb) has value of -2N-1 i=n-2

I=

∑ bi2i – bn-12n-1

i=0

 Most positive 4-bit number:  Most negative 4-bit number: 

The most significant bit still indicates the sign (1 = negative, 0 = positive)



Range of an N-bit two’s comp number:

45

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Two’s Complement Numbers 

Same as unsigned binary, but the most significant bit (msb) has value of -2N-1 i=n-2

I=

∑ bi2i – bn-12n-1

i=0

 Most positive 4-bit number:  Most negative 4-bit number:

0111 1000



The most significant bit still indicates the sign (1 = negative, 0 = positive)



Range of an N-bit two’s comp number: [-2N-1, 2N-1-1] 8 bits: [-128,127] 46

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“Taking the Two’s Complement” 

How to flip the sign of a two’s complement number:  Invert the bits  Add one



Example: Flip the sign of 310

=

00112

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“Taking the Two’s Complement” 

How to flip the sign of a two’s complement number:  Invert the bits  Add one



Example: Flip the sign of 310  Invert the bits

=

00112 11002

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“Taking the Two’s Complement” 

How to flip the sign of a two’s complement number:  Invert the bits  Add one



Example: Flip the sign of 310  Invert the bits  Add one

=

00112 11002 11012

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“Taking the Two’s Complement” 

How to flip the sign of a two’s complement number:  Invert the bits  Add one



Example: Flip the sign of 310

=

00112 11002 11012

=

110002

 Invert the bits  Add one 

Example: Flip the sign of -810

50

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“Taking the Two’s Complement” 

How to flip the sign of a two’s complement number:  Invert the bits  Add one



Example: Flip the sign of 310

=

00112 11002 11012

=

110002 001112 010002

 Invert the bits  Add one 

Example: Flip the sign of -810  Invert the bits  Add one

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Two’s Complement Addition 

Add 6 + (-6) using two’s complement numbers 0110 + 1010



Add -2 + 3 using two’s complement numbers 1110 + 0011

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Two’s Complement Addition 

Add 6 + (-6) using two’s complement numbers 111 0110 + 1010 10000



Add -2 + 3 using two’s complement numbers



111 1110 + 0011 10001 Correct results if overflow bit is ignored 53

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Increasing Bit Width 

A value can be extended from N bits to M bits (where M > N) by using:  Sign-extension  Zero-extension

54

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Sign-Extension 

Sign bit is copied into most significant bits



Number value remains the same



Give correct result for two’s complement numbers



Example 1:  4-bit representation of 3 =  8-bit sign-extended value:



0011 00000011

Example 2:  4-bit representation of -5 =  8-bit sign-extended value:

1011 11111011 55

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Zero-Extension 

Zeros are copied into most significant bits



Value will change for negative numbers



Example 1:  4-bit value = 00112 = 310  8-bit zero-extended value: 000000112 = 310



Example 2:  4-bit value = 10112 = -510  8-bit zero-extended value: 000010112 = 1110

56

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Number System Comparison Number System

Range

Unsigned

[0, 2N-1]

Sign/Magnitude

[-(2N-1-1), 2N-1-1]

Two’s Complement

[-2N-1, 2N-1-1]

For example, 4-bit representation: -8

-7

-6

-5

-4

-3

-2

-1

Unsigned

0

1

2

3

4

5

6

7

9

10

11

12

13

14

15

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111

1000 1001 1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111

1111 1110 1101 1100 1011 1010 1001

8

0000 1000

0001 0010 0011 0100 0101 0110 0111

Two's Complement Sign/Magnitude

57

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Lessons Learned 

How to express decimal numbers using only 1s and 0s



How to simplify writing binary numbers in hexadecimal



Adding binary numbers



Methods to express negative numbers  Sign Magnitude  One’s complement  Two’s complement (the one commonly used)

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Design of Digital Circuits Reading: Binary Numbers

Design of Digital Circuits Reading: Binary Numbers Required Reading for Week 1 23-24 February 2017 Spring 2017 Carnegie Mellon Binary Numbers Desi...

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