2's column4's column8's column1's columnnnnn

(1)

Codes and number systems

Introduction to Computerp Yung-Yu Chuang

with slides by Nisan & Schocken (www.nand2tetris.org) and Harris & Harris (DDCA)

(2)

Coding

• Assume that you want to communicate with your friend with a flashlight in a night what your friend with a flashlight in a night, what will you do?

light painting?

What’s the problem?

(3)

Solution #1

• A: 1 blink B 2 bli k

• B: 2 blinks

• C: 3 blinks :

• Z: 26 blinks

Wh t’ th bl ? What’s the problem?

• How are you? = 131 blinks

(4)

Solution #2: Morse code

Hello

(5)

Lookup

• It is easy to translate into Morse code than reverse Why?

reverse. Why?

(6)

Lookup

(7)

Lookup

Useful for

checking the checking the correctness/

d d

redundency

(8)

Braille

(9)

Braille

(10)

What’s common in these codes?

• They are both binary codes.

(11)

Binary representations

• Electronic Implementation

E t t ith bi t bl l t – Easy to store with bistable elements

– Reliably transmitted on noisy and inaccurate wires

0 1 0

2.8V 3.3V

0.0V 0.5V

(12)

Number systems

(13)

Number Systems

• Decimal numbers

10's colum100's colum

1000's colu 1's column

5374₁₀ =

nmn

umn

• Binary numbers

2's column4's column

8's column 1's column

1101₂ ⁼

nn

n n

Chapter 1 <13>

(14)

Number Systems

• Decimal numbers

10's colum100's colum

1000's colu 1's column

5374₁₀ ^{= 5 ? 10}³^{+ 3 ? 10}²^{+ 7 ? 10}¹^{+ 4 ? 10}⁰

five

nmn

umn

three seven four

thousands hundreds tens ones

• Binary numbers

2's column4's column

8's column 1's column

1101₂ = 1 ? 2³ + 1 ? 2² + 0 ? 2¹ + 1 ? 2⁰= 13₁₀

one eight

nn

n

one four

no two

one one

n

Chapter 1 <14>

(15)

Binary numbers

• Digits are 1 and 0

( bi di it i ll d bit) (a binary digit is called a bit) 1 = true

0 = false

• MSB –most significant bit

• LSB –least significant bit

MSB LSB

• Bit numbering: 1 0 1 1 0 0 1 0 1 0 0 1 1 1 0 0

MSB LSB

A bit string could have different interpretations

0 15

• A bit string could have different interpretations

(16)

Powers of Two

• 2

⁰

= • 2

⁸

=

• 2

¹

=

• 2

²

=

• 2

⁹

=

• 2

¹⁰

=

• 2 =

• 2

³

=

• 2 =

• 2

¹¹

=

• 2

⁴

=

• 2

⁵

=

• 2

¹²

=

• 2

¹³

=

• 2

⁵

=

• 2

⁶

=

• 2

¹³

=

• 2

¹⁴

=

• 2

⁷

= • 2

¹⁵

=

Chapter 1 <16>

(17)

Powers of Two

• 2

⁰

= 1 • 2

⁸

= 256

• 2

¹

= 2

• 2

²

= 4

• 2

⁹

= 512

• 2

¹⁰

= 1024

• 2 = 4

• 2

³

= 8

• 2 = 1024

• 2

¹¹

= 2048

• 2

⁴

= 16

• 2

⁵

= 32

• 2

¹²

= 4096

• 2

¹³

= 8192

• 2

⁵

= 32

• 2

⁶

= 64

• 2

¹³

= 8192

• 2

¹⁴

= 16384

• 2

⁷

= 128

Hand to memori e p to 2

⁹

• 2

¹⁵

= 32768

Chapter 1 <17>

• Handy to memorize up to 2

⁹

(18)

Unsigned binary integers

• Each digit (bit) is either 1 or 0

• Each bit represents a power of 2: ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹

2⁷ 2⁶ 2⁵ 2⁴ 2³ 2² 2¹ 2⁰

Every binary number is a

f

sum of powers of 2

(19)

Translating binary to decimal

Weighted positional notation shows how to Weighted positional notation shows how to

calculate the decimal value of each binary bit:

d (D 2^{n 1}) (D 2^{n 2}) (D 2¹) (D dec = (D_n-1  2^n-1)  (D_n-2  2^n-2)  ...  (D₁  2¹)  (D₀

 2⁰)

D = binary digit

binary 00001001 = decimal 9:

(1  2³) + (1  2⁰) = 9

(20)

Translating unsigned decimal to binary

binary

• Repeatedly divide the decimal integer by 2. Each remainder is a binary digit in the translated value:

remainder is a binary digit in the translated value:

37 = 100101 37 = 100101

(21)

Number Conversion

• Decimal to binary conversion: y

– Convert 10011₂ to decimal

• Decimal to binary conversion:

– Convert 47₁₀ to binary

Chapter 1 <21>

(22)

Number Conversion

• Decimal to binary conversion: y

– Convert 10011₂ to decimal

– 16×1 + 8×0 + 4×0 + 2×1 + 1×1 = 19₁₀

i l bi i

• Decimal to binary conversion:

– Convert 47₁₀ to binary

32 1 + 16 0 + 8 1 + 4 1 + 2 1 + 1 1 101111 – 32×1 + 16×0 + 8×1 + 4×1 + 2×1 + 1×1 = 101111₂

Chapter 1 <22>

(23)

Binary Values and Range

• N‐digit decimal number g

– How many values?

– Range?Range?

– Example: 3‐digit decimal number:

• N‐bit binary number

– How many values?

– Range:

– Example: 3‐digit binary number:

Chapter 1 <23>

(24)

Binary Values and Range

• N‐digit decimal number g

– How many values? 10^N – Range? [0, 10^N ‐ 1]

E l 3 di it d i l b

– Example: 3‐digit decimal number:

• 10³ = 1000 possible values

• Range: [0, 999]

• N‐bit binary number

How many values? 2^N – How many values? 2^N – Range: [0, 2^N ‐ 1]

– Example: 3‐digit binary number:p g y

• 2³ = 8 possible values

• Range: [0, 7] = [000₂ to 111₂]

Chapter 1 <24>

(25)

Integer storage sizes

byte

16 8

Standard sizes: ¹⁶

32 word

doubleword

64 quadword

Standard sizes:

64 quadword

Practice: What is the largest unsigned integer that may be stored in 20 bits?

(26)

Bits, Bytes, Nibbles…

• Bits 10010110 10010110

least significant

bit most

significant bit

• Bytes & Nibbles

10010110

byte

bit bit

10010110

nibble

• Bytes y CEBF9AD7

least significant

byte most

significant byte

Chapter 1 <26>

byte byte

(27)

Large Powers of Two

• 2

¹⁰

= 1 kilo

^≈

1000 (1024)

• 2

²⁰

= 1 mega

^≈

1 million (1,048,576)

• 2

³⁰

1 giga 1 billion (1 073 741 824)

• 2

³⁰

= 1 giga

^≈

1 billion (1,073,741,824)

Chapter 1 <27>

(28)

Estimating Powers of Two

• What is the value of 2

²⁴

?

• How many values can a 32‐bit variable represent?

Chapter 1 <28>

(29)

Estimating Powers of Two

• What is the value of 2

²⁴

?

‐ 2

⁴

× 2

²⁰

≈ 16 million

• How many values can a 32‐bit variable represent?

‐2

²

× 2

³⁰

≈ 4 billion

‐2 × 2 ≈ 4 billion

Chapter 1 <29>

(30)

Large measurements

• Kilobyte (KB), 2¹⁰ bytes M b (MB) 2²⁰ b

• Megabyte (MB), 2²⁰ bytes

• Gigabyte (GB), 2³⁰ bytes

• Terabyte (TB), 2⁴⁰ bytes

• Petabyte

• Exabyte Z tt b t

• Zettabyte

• Yottabyte

(31)

Hexadecimal Numbers

Hex Digit Decimal Equivalent Binary Equivalent

0 0

1 1

2 2

3 3

4 4

5 5

6 6

7 7

8 8

9 9

A 10

B 11

C 12

D 13

E 14

Chapter 1 <31>

F 15

(32)

Hexadecimal Numbers

Hex Digit Decimal Equivalent Binary Equivalent

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

A 10 1010

B 11 1011

C 12 1100

D 13 1101

E 14 1110

Chapter 1 <32>

F 15 1111

(33)

Hexadecimal Numbers

• Base 16

• Shorthand for binary

Chapter 1 <33>

(34)

Translating binary to hexadecimal Translating binary to hexadecimal

• Each hexadecimal digit corresponds to 4 binary bits.

• Example: Translate the binary integer

000101101010011110010100 to hexadecimal:

(35)

Converting hexadecimal to decimal

• Multiply each digit by its corresponding f 16

power of 16:

dec = (D₃  16³) + (D₂  16²) + (D₁  16¹) + (D₀  16⁰)

H 1234 l (1 16³) + (2 16²) + (3 16¹) + (4

• Hex 1234 equals (1  16³) + (2  16²) + (3  16¹) + (4

 16⁰), or decimal 4,660.

• Hex 3BA4 equals (3Hex 3BA4 equals (3  16 ) + (11 16 ) + (10  16 )  16³) + (11 * 16²) + (10  16¹) + (4  16⁰), or decimal 15,268.

(36)

Hexadecimal to Binary Conversion

• Hexadecimal to binary conversion: y

– Convert 4AF₁₆ (also written 0x4AF) to binary

• Hexadecimal to decimal conversion:

– Convert 0x4AF to decimal

Chapter 1 <36>

(37)

Hexadecimal to Binary Conversion

• Hexadecimal to binary conversion: y

– Convert 4AF₁₆ (also written 0x4AF) to binary – 0100 1010 1111₂

• Hexadecimal to decimal conversion: Hexadecimal to decimal conversion:

– Convert 4AF₁₆ to decimal

– 16²×4 + 16¹×10 + 16⁰×15 = 1199₁₀

Chapter 1 <37>

(38)

Powers of 16

Used when calculating hexadecimal values up to 8 digits long:

(39)

Converting decimal to hexadecimal

decimal 422 = 1A6 hexadecimal

(40)

Addition

carries

• Decimal

11

3734 5168 +

carries 11

5168 +

8902

11 carries

• Binary

1011 0011 +

1110

Chapter 1 <40>

(41)

Binary Addition Examples

• Add the following 1001

1001 0101 +

g 4-bit binary

numbers numbers

• Add the following 1011 1011 0110 +

• Add the following 4-bit binary

b

numbers

Chapter 1 <41>

(42)

Binary Addition Examples

1001

• Add the following 1

1001 0101 +

g 4-bit binary

numbers numbers 1110

111 1011

• Add the following 1011 0110 +

10001

• Add the following 4-bit binary

b 10001

numbers

Overflow!

Chapter 1 <42>

(43)

Overflow

• Digital systems operate on a fixed number of g y p bits

• Overflow: when result is too big to fit in the

• Overflow: when result is too big to fit in the available number of bits

• See previous example of 11 + 6

Chapter 1 <43>

(44)

Hexadecimal addition

Divide the sum of two digits by the number base (16) Th ti t b th l d (16). The quotient becomes the carry value, and the remainder is the sum digit.

36 28 28 6A

1 1

36 28 28 6A

42 45 58 4B

78 6D 80 B5

Important skill: Programmers frequently add and subtract the addresses of variables and instructions

addresses of variables and instructions.

(45)

Signed Binary Numbers

• Sign/Magnitude Numbers g g

• Two’s Complement Numbers

Chapter 1 <45>

(46)

Signed integers

The highest bit indicates the sign. 1 = negative, 0 i i

0 = positive

sign bit sign bit

1 1 1 1 0 1 1 0

Negative

0 0 0 0 1 0 1 0 Positive

If the highest digit of a hexadecmal integer is > 7, the value is negative Examples: 8A C5 A2 9D

negative. Examples: 8A, C5, A2, 9D

(47)

Sign/Magnitude Numbers

• 1 sign bit, N-1 magnitude bits

• Sign bit is the most significant (left-most) bit

– Positive number: sign bit = 0os t ve u be : s g b t 0 A:



a_N_1,a_N_2,a a a2, ,1 0



– Negative number: sign bit = 1

 

1

1 2 2 1 0

2

( 1) ⁿ 2

N N

n

a i

A ^ ai

 





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

0 i

+6 = - 6 =

R f N bit i / it d b

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

Chapter 1 <47>

(48)

Sign/Magnitude Numbers

• 1 sign bit, N-1 magnitude bits

• Sign bit is the most significant (left-most) bit

– Positive number: sign bit = 0os t ve u be : s g b t 0 A:



a_N_1,a_N_2,a a a2, ,1 0



– Negative number: sign bit = 1

 

1

1 2 2 1 0

2

( 1) ⁿ 2

N N

n

a i

A ^ ai

 





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

0 i

+6 = 0110 - 6 = 1110

R f N bit i / it d b

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

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

Chapter 1 <48>

(49)

Sign/Magnitude Numbers

• Problems:

– Addition doesn’t work, for example -6 + 6:

1110

1110 + 0110

10100 (wrong!)

– Two representations of 0 (± 0):

1000

1000 0000

Chapter 1 <49>

(50)

Two’s Complement Numbers

• Don’t have same problems as sign/magnitude p g g numbers:

– Addition works

– Single representation for 0

Chapter 1 <50>

(51)

Two's complement notation

Steps:

Complement (reverse) each bit – Complement (reverse) each bit – Add 1

Note that 00000001 + 11111111 = 00000000

(52)

“Taking the Two’s Complement”

• Flip the sign of a two’s complement number p g p

• Method:

1 Invert the bits 1. Invert the bits 2. Add 1

l li h i f

• Example: Flip the sign of 3

₁₀

= 0011

₂

Chapter 1 <52>

(53)

“Taking the Two’s Complement”

• Flip the sign of a two’s complement number p g p

• Method:

1 Invert the bits 1. Invert the bits 2. Add 1

l li h i f

• Example: Flip the sign of 3

₁₀

= 0011

₂

1. 1100 2. + 1

1101 = -3₁₀

Chapter 1 <53>

(54)

Two’s Complement Examples

• Take the two’s complement of 6 p

₁₀₁₀

= 0110

₂₂

• What is the decimal value of 1001

₂

?

Chapter 1 <54>

(55)

Two’s Complement Examples

• Take the two’s complement of 6 p

₁₀₁₀

= 0110

₂₂

1. 1001 2. + 1

1010₂ = -6₁₀

• What is the decimal value of the two’s complement number 1001

₂

?

1. 0110 2. + 1

0111₂ = 7₁₀, so 1001₂ = -7₁₀

Chapter 1 <55>

2 10, ₂ ₁₀

(56)

Binary subtraction

• When subtracting A – B, convert B to its two's complement

complement

• Add A to (–B)

0 1 0 1 0 0 1 0 1 0 – 0 1 0 1 1 1 0 1 0 0 1 1 1 1 1 Advantages for 2’s complement:

Advantages for 2’s complement:

• No two 0’s

• Sign bit

• Remove the need for separate circuits for add and sub

(57)

Two’s Complement Addition

• Add 6 + (-6) using two’s complement

0110

( ) g p

numbers

+

0110 1010

• Add -2 + 3 using two’s complement numbers

1110

g p

+

1110 0011

Chapter 1 <57>

(58)

Two’s Complement Addition

• Add 6 + (-6) using two’s complement

0110 111

( ) g p

numbers

+

0110 1010 10000 10000

• Add -2 + 3 using two’s complement numbers

1110 111

g p

+

1110 0011 10001

Chapter 1 <58>

10001

(59)

Increasing Bit Width

• Extend number from N to M bits (M > N) :

– Sign-extension – Zero-extensionZero extension

Chapter 1 <59>

(60)

Sign‐Extension

• Sign bit copied to msb’s g

• Number value is same

• Example 1:

4 bit representation of 3 = 0011 – 4-bit representation of 3 = 0011

– 8-bit sign-extended value: 00000011

E l 2

• Example 2:

– 4-bit representation of -5 = 1011

– 8-bit sign-extended value: 11111011

Chapter 1 <60>

(61)

Zero‐Extension

• Zeros copied to msb’s

• Value changes for negative numbers

• Example 1:

4 bit value = 0011 = 3

– 4-bit value = 0011₂ = 3₁₀ – 8-bit zero-extended value: 00000011 = 3₁₀

E l 2

• Example 2:

– 4-bit value = 1011 = -5₁₀ – 8-bit zero-extended value: 00001011 = 11₁₀

Chapter 1 <61>

(62)

Number System Comparison

Number System Range

Unsigned [0, 2^N-1]

Sign/Magnitude [-(2^N-1-1), 2^N-1-1]

1 1

Two’s Complement [-2^N-1, 2^N-1-1]

For example 4-bit representation:

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

For example, 4-bit representation:

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

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

Unsigned

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

1001 1000 1010 1011 1100 1101 1110

1111 0000

0001 0010 0011 0100 0101 0110 0111 Sign/Magnitude

Chapter 1 <62>

(63)

Ranges of signed integers

The highest bit is reserved for the sign. This limits the range:

the range:

(64)

Character

• Character sets

St d d ASCII(0 127) – Standard ASCII(0 – 127) – Extended ASCII (0 – 255)

ANSI (0 255) – ANSI (0 – 255)

– Unicode (0 – 65,535)

• Null-terminated String

– Array of characters followed by a null byte

• Using the ASCII table

– back inside cover of book

(65)

(66)

Representing Instructions

int sum(int x, int y)

{ ^{Alpha sum} ^{Sun sum} ^{PC sum}

{

return x+y;

}

55 89 00

00 p

81 C3

– For this example, Alpha &

Sun use two 4-byte

E5 8B 45 30

42 01

E0 08 90

instructions

• Use differing numbers of instructions in other cases

0C 03 45 80

FA 6B

02 00 instructions in other cases 09

– PC uses 7 instructions with lengths 1, 2, and 3

08 89 EC

Diff t hi t t ll diff t

g , ,

bytes

• Same for NT and for Linux

EC 5D C3 Different machines use totally different instructions and encodings

• NT / Linux not fully binary compatible