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Page 1: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.1

Page 2: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.2

••••• unsigned char short int long

128 64 32 16 8 4 2 1

1 0 0 1 0 0 1 1 = +147

-128 64 32 16 8 4 2 1

1 0 0 1 0 0 1 1 = -109

9 3 = +1470x

Page 3: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.3

•––––

–

Page 4: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.4

• + - * /

•––

•

(10)

(5)

(5)

0 1 1 1

+ 0 0 1 1

1 0 1 0

(7)

(3)

(10)

1 1 1

8 4 2 1

1 0 1 0

- 0 1 0 1

0 1 0 1

0 0

8 4 2 1

11

Page 5: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.5

Page 6: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.6

•

––

•

••

•–

Page 7: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.7

•

•–

–•••

CS:APP 2.3.3

Page 8: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.8

•

––

•

010011

101100

+ 1

101101

-32 16 8 4 2 1

Page 9: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.9

101010

010101

+ 1

010110

-32 16 8 4 2 1

101001

+ 1

101010

0000

1111

+ 1

0000

1000

0111

+ 1

1000

1 2

3

4 0001 => 1111

Page 10: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.10

Page 11: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.11

12 1239

1011

487 56

12 1239

1011

487 56

00 010203

9899

04...

.

00 010203

9899

04...

.

00000001

0010

0011

1110

1111

0100...

.

00000001

0010

0011

1110

1111

0100...

.

10’s complement04 - 02 =

04 + (100 - 02) =04 + 98

2’s complement0100 - 0010 =

0100 + (10000 - 0010) =0100 + 1110

Clock Analogy4 - 2 =

4 + (12 - 2) =4 + 10

When using modulo arithmetic, subtraction can always be converted to addition.

Page 12: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.12

•–––

••

–

––

•

CS:APP 2.3.1CS:APP 2.3.2

Page 13: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.13

••

0011

+ 0010

0101

(3)

(2)

(5)

1101

+ 0010

1111

(-3)

(2)

(-1)

0011

+ 1110

0001

(3)

(-2)

(1)

1101

+ 1110

1011

(-3)

(-2)

(-5) -8+2+1

0000 0000

1110 Drop final carry-out 1100

Page 14: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.14

•

–•

1001+ 0011

1100

11(9)(3)

(12)

(-7)(3)

(-4)

If unsigned If signed

Page 15: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.15

•

•

0011- 0010

(+3)(+2)

Drop final carry-out

1111_00111101

+ 10001

Bit flip of +2

Add 1

1101- 1110

(-3)(-2)

1_11010001

+ 11111

Bit flip of -2

Add 1

Page 16: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.16

•

––

•

(12)(2)

(-4)(2)

If unsigned If signed

1100- 0010

11_1_11001101

+ 11010

~B

Add 1

A

If unsigned If signed

(10) (-6)

Page 17: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.17

•…

–

–

Page 18: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.18

Page 19: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.19

•

•

––

CS:APP 2.3.1

CS:APP 2.3.2

Page 20: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.20

00000001

0010

0011

0100

0101

01100111

1000

1111

1110

1101

1100

1011

1010

1001

0+1

+2

+3

+4

+5

+6

+7+8

+9

+10

+11

+12

+13

+14

+15

10

Plus 7

Page 21: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.21

00000001

0010

0011

0100

0101

01100111

1000

1111

1110

1101

1100

1011

1010

1001

0+1

+2

+3

+4

+5

+6

+7-8

-7

-6

-5

-4

-3

-2

-1

Page 22: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.22

•

•––

1101+ 0100

0001

11(13)(4)

(17)

(-3)(4)

(+1)

0110+ 0101

1011

10(6)(5)

(11)

(6)(5)

(-5)

Page 23: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.23

•

•––

(7)(8)

(-1)

(7)(-8)(15)

0111- 1000

0111_01110111

+ 11111 (15) (-1)

Page 24: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.24

Page 25: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.25

•char v

–• v = 1;

–• v = 0xf0;

–• v = 0;

–•

––––

? ? ? ? ? ? ? ?

? ? ? ? ? ? ? 1

1 1 1 1 ? ? ? ?

? ? ? ? ? ? 0 0

Page 26: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.26

•••

X Y AND0 0 00 1 01 0 01 1 1 P

ass

Forc

e '0

'

X Y XOR0 0 00 1 11 0 11 1 0

Pas

sIn

vert

X Y OR0 0 00 1 11 0 11 1 1

Pas

sFo

rce

'1'

Page 27: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.27

•

0xa5AND 0xf0

0xa0

1010 0101& 1111 0000

1010 0000

0xa5OR 0xf0

0xf5

1010 0101| 1111 0000

1111 0101

0xa5XOR 0xf0

0x55

1010 0101^ 1111 0000

0101 0101

#include <stdio.h> // C-Library // for printf()

int main(){ char a = 0xa5; char b = 0xf0;

printf("a & b = %x\n", a & b); printf("a | b = %x\n", a | b); printf("a ^ b = %x\n", a ^ b); printf("~a = %x\n", ~a); return 0;}

NOT 0xa50x5a

~ 1010 01010101 1010

C bitwise operators:& = AND| = OR^ = XOR~ = NOT

Page 28: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.28

•–

•–

•–

0000 0001=T&& 0000 0010=T

0000 0001=T

0000 0001& 0000 0010

F = 0000 0000

! 0000 0001=T0000 0000=F

~ 0000 0001T = 1111 1110

#include <stdio.h> int main() { int x = 1, y = 2; int z1 = x && y; // 1 int z2 = x & y; // 0 printf("z1=%d, z2=%d\n",z1,z2); char z = 1; if(!z) { printf("L1\n"); } // F if(~z) { printf("L2\n"); } // T return 0;}

…

!! 0101 0111=T0000 0001=T

Page 29: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.29

•

•

0101 1001=x#include <stdio.h> int main() { int x = 0x59; int y = 0xd3; x = x ^ y; // x = bitwise diff y = x ^ y; // flip y if different x = x ^ y; // flip y if flipped}

1101 0011=y

0101 1001=x^ 1101 0011=y

1000 1010=x 1000 1010=x^ 1101 0011=y

0101 1001=y1000 1010=x^ 0101 1001=y

1101 0011=x

#include <stdio.h>

int main() { int x = 0x59; int y = 0xd3; int temp = x; // save x x = y; // overwrite x y = temp; // overwrite y}

Page 30: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.30

•

•

int isOdd(int x) { /* Isolate the lowest bit */ return x & 1;}

int isMultOf4(int x) { /* Check if 2 LSBs are both 0 */ return !(x & 3);}

Page 31: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.31

Page 32: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.32

•––

••

––

0 0 0 0 0 0 1 1

0 0 0 0 1 1 0 0

0 0 0 0 1 0 1 0 0 0

0 0 0 0 1 0 1 0

0 0… …

Page 33: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.33

•–

–

•–––

• …

• …

Right shift

Left shift

Right shift

Left shift

0

0

0

Copies of MSB are shifted in

Page 34: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.34

••

––

0 0 ... 0 0 1 1 ... 0 1 1 0 0 0 0 0… …

0 ... 0 1 1 0 0 = +12

= +3 = +96

0x 0000 000C

0x00000003 0x00000060

Page 35: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.35

••

–•

–

1 1 1 ... 1 1 1 1 ..1 1 0 0 0 0

… …

1 1 ... 1 1 0 0 = -4

= -1 = -16

…

0xFFFFFFFC

0xFFFFFFFF 0xFFFFFFF0

Page 36: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.36

•

•

–

•

––

int mul17(int x) { return 17*x; }

17= 1 0 0 0 12

124816

sall $4, %edx addl %edx, %eax

int mul17(int x) { int x16 = x << 4; return x16 + x; }

CS:APP 2.3.6

Page 37: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.37

▪▪

▪▪

5 = 0 1 0 1

124-8

5>>1 = 0 0 1 0

124-8

1

0.5

-5>>1 = 1 1 0 1

124-8

1

0.5

-5 = 1 0 1 1

124-8

▪▪

▪▪

Page 38: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.38

05 >> 10 0 1 0

124-8

1

0.5

0-5 >> 1

-4-8 -3 -2.5

2.52

+5>>1

124-8

1

0.5

-5>>1

1 1 0 1

0 2.52-2-2.5

Page 39: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.39

• x >> k– ≥

–

•–

–

-4>>1 = 1 1 1 0

-4 = 1 1 0 0

-5>>1 = 1 1 0 1

-5 = 1 0 1 1

-5 1 0 1 1 +1 + 1

-4 1 1 0 0-4>>1 = 1 1 1 0

Page 40: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.40

•––

•––

•––

-8 = 1 0 0 0-8>>2 = 1 1 1 0

-7 = 1 0 0 1-7>>2 = 1 1 1 0

-7 1 0 0 1 +3 + 1 1

-4 1 1 0 0-4>>2 = 1 1 1 1

-8 1 0 0 0 +3 + 1 1

-5 1 0 1 1-5>>2 = 1 1 1 0

Page 41: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.41

#define M /* mystery number 1 */#define N /* mystery number 2 */

int arith(int x, int y) { int result = x*M + y/N; return result;}

/* Translation of assembled code for a given value of M and N */

int optarith(int x, int y) { int t = x; x <<= 5; x -= t; if (y < 0) y += 3; y >>= 2; return x + y; }

Page 42: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

P1.42

Page 43: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.43

Page 44: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.44

• + - * /

•–––

•

Page 45: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.45

•••

0

+ 0

00

no need to carry

sum bit

0

+ 1

01

no need to carry

sum bit

1

+ 0

01

no need to carry

sum bit

1

+ 1

10

carry 1 into next column of bits

sum bit

1

Page 46: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.46

(10)

(5)

(5)

0 1 1 1

+ 0 0 1 1

1 0 1 0

(7)

(3)

(10)

1 1 1

8 4 2 1

1 0 1 0

- 0 1 0 1

0 1 0 1

0 0

8 4 2 1

11

Page 47: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.47

0110

+ 0111

(6)

(7)

( )

8 4 2 1

Page 48: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.48

0110

+ 0111

1101

(6)

(7)

(13)

110

8 4 2 1

Page 49: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.49

0110

+ 0111

1101

(6)

(7)

(13)

0

+ 1

01

0

carry bit sum bit

0110

+ 0111

1101

(6)

(7)

(13)

1

+ 1

10

10 0

carry bit sum bit

0110

+ 0111

1101

(6)

(7)

(13)

1

+ 1

11

110 1

carry bit sum bit

0110

+ 0111

1101

(6)

(7)

(13)

0

+ 0

01

110 1

carry bit sum bit

1 2

43

Page 50: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.50

•–

4 D16+ B 5161 0 216

1 1

256 16 1

Page 51: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.51

Page 52: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.52

•

–•

0 1 1 0

* 0 1 0 1

0 1 1 0

(6)

(5)

Sum of the partial products

0 0 0 0 0 1 1 0

+ 0 0 0 0

0 0 1 1 1 1 0

Partial Products

Page 53: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.53

••••

1010* 1011

10101010_

0000__+ 1010___

01101110

M (Multiplicand)Q (Multiplier)

PP(Partial Products)

P (Product)

Page 54: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.54

•

•

1001* 0110

00001001_

1001__+ 0000___00110110

= -7= +6

= +54

…

1001* 0110

000000001111001_111001__

+ 00000___11010110

= -7= +6

= -42

…

Page 55: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.55

•––

1100* 1010

000000001111100_000000__

+ 11100___11011000

= -4= -6

= -40

… …

1100* 1010

000000001111100_000000__

+ 00100___00011000

= -4= -6

= +24

Place Value: -8Multiply -4 by -1

Main Point: Signed and Unsigned Multiplication require different techniques… Thus different assembly instructions.

Page 56: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.56

•

•

10 1 0 1 1

0 1 0 1 r.1

-1 0

0 1-0 0

1 1-1 0

0 1

(2)10 (11)10

(5 r.1)10

Page 57: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

2.57

••

10 1 0 1 1

0 1 0 1 r.1

-1 0

0 1-0 0

1 1-1 0

0 1

(2)10 (11)10

(5 r.1)10

Page 58: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

3.58

10 1 0 1 1

0

10 (2) goes into 1, 0 times. Since it doesn’t, bring in the next bit.

Page 59: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

3.59

10 (2) goes into 10, 1 time. Multiply, subtract, and bring down the next bit.

10 1 0 1 1

0 1

-1 0

0 1

Page 60: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

3.60

10 (2) goes into 01, 0 times. Multiply, subtract, and bring down the next bit.

10 1 0 1 1

0 1 0

-1 0

0 1-0 0

1 1

Page 61: Unit 2 - GitHub Pages · Unit 2 Integer Operations. Summary of Unit 1 2.2 • n bits, 2n choices: [0, 2n-1] unsigned and [-2n-1, 2n-1-1] signed range ... 1 0 0 1 0 0 1 1 = -109 Unsigned

3.61

10 (2) goes into 11, 1 time. Multiply and subtract. The remainder is 1.

10 1 0 1 1

0 1 0 1 r.1

-1 0

0 1-0 0

1 1-1 0

0 1


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