Appendix A. Arithmetic Tables for a 4-Bit Machine

In the tables in Appendix A, underlining denotes signed overflow.

Table A-1.. Addition

 

 

 

 

 

 

 

 

 

 

-8

-7

-6

-5

-4

-3

-2

-1

 

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

0

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

1

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

10

 

2

2

3

4

5

6

7

8

9

A

B

C

D

E

F

10

11

 

3

3

4

5

6

7

8

9

A

B

C

D

E

F

10

11

12

 

4

4

5

6

7

8

9

A

B

C

D

E

F

10

11

12

13

 

5

5

6

7

8

9

A

B

C

D

E

F

10

11

12

13

14

 

6

6

7

8

9

A

B

C

D

E

F

10

11

12

13

14

15

 

7

7

8

9

A

B

C

D

E

F

10

11

12

13

14

15

16

-8

8

8

9

A

B

C

D

E

F

10

11

12

13

14

15

16

17

-7

9

9

A

B

C

D

E

F

10

11

12

13

14

15

16

17

18

-6

A

A

B

C

D

E

F

10

11

12

13

14

15

16

17

18

19

-5

B

B

C

D

E

F

10

11

12

13

14

15

16

17

18

19

1A

-4

C

C

D

E

F

10

11

12

13

14

15

16

17

18

19

1A

1B

-3

D

D

E

F

10

11

12

13

14

15

16

17

18

19

1A

1B

1C

-2

E

E

F

10

11

12

13

14

15

16

17

18

19

1A

1B

1C

1D

-1

F

F

10

11

12

13

14

15

16

17

18

19

1A

1B

1C

1D

1E

The table for subtraction (Table A-2) assumes that the carry bit for a - b is set as it would be for a + b?/span> + 1, so that carry is equivalent to "not borrow."

Table A-2.. Subtraction (row - column)

 

 

 

 

 

 

 

 

 

 

-8

-7

-6

-5

-4

-3

-2

-1

 

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

0

10

F

E

D

C

B

A

9

8

7

6

5

4

3

2

1

 

1

11

10

F

E

D

C

B

A

9

8

7

6

5

4

3

2

 

2

12

11

10

F

E

D

C

B

A

9

8

7

6

5

4

3

 

3

13

12

11

10

F

E

D

C

B

A

9

8

7

6

5

4

 

4

14

13

12

11

10

F

E

D

C

B

A

9

8

7

6

5

 

5

15

14

13

12

11

10

F

E

D

C

B

A

9

8

7

6

 

6

16

15

14

13

12

11

10

F

E

D

C

B

A

9

8

7

 

7

17

16

15

14

13

12

11

10

F

E

D

C

B

A

9

8

-8

8

18

17

16

15

14

13

12

11

10

F

E

D

C

B

A

9

-7

9

19

18

17

16

15

14

13

12

11

10

F

E

D

C

B

A

-6

A

1A

19

18

17

16

15

14

13

12

11

10

F

E

D

C

B

-5

B

1B

1A

19

18

17

16

15

14

13

12

11

10

F

E

D

C

-4

C

1C

1B

1A

19

18

17

16

15

14

13

12

11

10

F

E

D

-3

D

1D

1C

1B

1A

19

18

17

16

15

14

13

12

11

10

F

E

-2

E

1E

1D

1C

1B

1A

19

18

17

16

15

14

13

12

11

10

F

-1

F

1F

1E

1D

1C

1B

1A

19

18

17

16

15

14

13

12

11

10

For multiplication (Tables A-3 and A-4), overflow means that the result cannot be expressed as a 4-bit quantity. For signed multiplication (Table A-3), this is equivalent to the first five bits of the 8-bit result not being all 1's or all 0's.

Table A-3.. Signed Multiplication

 

 

 

 

 

 

 

 

 

 

-8

-7

-6

-5

-4

-3

-2

-1

 

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

1

0

1

2

3

4

5

6

7

F8

F9

FA

FB

FC

FD

FE

FF

 

2

0

2

4

6

8

A

C

E

F0

F2

F4

F6

F8

FA

FC

FE

 

3

0

3

6

9

C

F

12

15

E8

EB

EE

F1

F4

F7

FA

FD

 

4

0

4

8

C

10

14

18

1C

E0

E4

E8

EC

F0

F4

F8

FC

 

5

0

5

A

F

14

19

1E

23

D8

DD

E2

E7

EC

F1

F6

FB

 

6

0

6

C

12

18

1E

24

2A

D0

D6

DC

E2

E8

EE

F4

FA

 

7

0

7

E

15

1C

23

2A

31

C8

CF

D6

DD

E4

EB

F2

F9

-8

8

0

F8

F0

E8

E0

D8

D0

C8

40

38

30

28

20

18

10

8

-7

9

0

F9

F2

EB

E4

DD

D6

CF

38

31

2A

23

1C

15

E

7

-6

A

0

FA

F4

EE

E8

E2

DC

D6

30

2A

24

1E

18

12

C

6

-5

B

0

FB

F6

F1

EC

E7

E2

DD

28

23

1E

19

14

F

A

5

-4

C

0

FC

F8

F4

F0

EC

E8

E4

20

1C

18

14

10

C

8

4

-3

D

0

FD

FA

F7

F4

F1

EE

EB

18

15

12

F

C

9

6

3

-2

E

0

FE

FC

FA

F8

F6

F4

F2

10

E

C

A

8

6

4

2

-1

F

0

FF

FE

FD

FC

FB

FA

F9

8

7

6

5

4

3

2

1

 

Table A-4.. Unsigned Multiplication

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

2

0

2

4

6

8

A

C

E

10

12

14

16

18

1A

1C

1E

3

0

3

6

9

C

F

12

15

18

1B

1E

21

24

27

2A

2D

4

0

4

8

C

10

14

18

1C

20

24

28

2C

30

34

38

3C

5

0

5

A

F

14

19

1E

23

28

2D

32

37

3C

41

46

4B

6

0

6

C

12

18

1E

24

2A

30

36

3C

42

48

4E

54

5A

7

0

7

E

15

1C

23

2A

31

38

3F

46

4D

54

5B

62

69

8

0

8

10

18

20

28

30

38

40

48

50

58

60

68

70

78

9

0

9

12

1B

24

2D

36

3F

48

51

5A

63

6C

75

7E

87

A

0

A

14

1E

28

32

3C

46

50

5A

64

6E

78

82

8C

96

B

0

B

16

21

2C

37

42

4D

58

63

6E

79

84

8F

9A

A5

C

0

C

18

24

30

3C

48

54

60

6C

78

84

90

9C

A8

B4

D

0

D

1A

27

34

41

4E

5B

68

75

82

8F

9C

A9

B6

C3

E

0

E

1C

2A

38

46

54

62

70

7E

8C

9A

A8

B6

C4

D2

F

0

F

1E

2D

3C

4B

5A

69

78

87

96

A5

B4

C3

D2

E1

Tables A-5 and A-6 are for conventional truncating division. Table A-5 shows a result of 8 with overflow for the case of the maximum negative number divided by -1, but on most machines the result in this case is undefined, or the operation is suppressed.

Table A-5.. Signed Short Division (row ?column)

 

 

 

 

 

 

 

 

 

 

-8

-7

-6

-5

-4

-3

-2

-1

 

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

0

-

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

1

-

1

0

0

0

0

0

0

0

0

0

0

0

0

0

F

 

2

-

2

1

0

0

0

0

0

0

0

0

0

0

0

F

E

 

3

-

3

1

1

0

0

0

0

0

0

0

0

0

F

F

D

 

4

-

4

2

1

1

0

0

0

0

0

0

0

F

F

E

C

 

5

-

5

2

1

1

1

0

0

0

0

0

F

F

F

E

B

 

6

-

6

3

2

1

1

1

0

0

0

F

F

F

E

D

A

 

7

-

7

3

2

1

1

1

1

0

F

F

F

F

E

D

9

-8

8

-

8

C

E

E

F

F

F

1

1

1

1

2

2

4

8

-7

9

-

9

D

E

F

F

F

F

0

1

1

1

1

2

3

7

-6

A

-

A

D

E

F

F

F

0

0

0

1

1

1

2

3

6

-5

B

-

B

E

F

F

F

0

0

0

0

0

1

1

1

2

5

-4

C

-

C

E

F

F

0

0

0

0

0

0

0

1

1

2

4

-3

D

-

D

F

F

0

0

0

0

0

0

0

0

0

1

1

3

-2

E

-

E

F

0

0

0

0

0

0

0

0

0

0

0

1

2

-1

F

-

F

0

0

0

0

0

0

0

0

0

0

0

0

0

1

 

Table A-6.. Unsigned Short Division (row ?column)

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

-

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

-

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

-

2

1

0

0

0

0

0

0

0

0

0

0

0

0

0

3

-

3

1

1

0

0

0

0

0

0

0

0

0

0

0

0

4

-

4

2

1

1

0

0

0

0

0

0

0

0

0

0

0

5

-

5

2

1

1

1

0

0

0

0

0

0

0

0

0

0

6

-

6

3

2

1

1

1

0

0

0

0

0

0

0

0

0

7

-

7

3

2

1

1

1

1

0

0

0

0

0

0

0

0

8

-

8

4

2

2

1

1

1

1

0

0

0

0

0

0

0

9

-

9

4

3

2

1

1

1

1

1

0

0

0

0

0

0

A

-

A

5

3

2

2

1

1

1

1

1

0

0

0

0

0

B

-

B

5

3

2

2

1

1

1

1

1

1

0

0

0

0

C

-

C

6

4

3

2

2

1

1

1

1

1

1

0

0

0

D

-

D

6

4

3

2

2

1

1

1

1

1

1

1

0

0

E

-

E

7

4

3

2

2

2

1

1

1

1

1

1

1

0

F

-

F

7

5

3

3

2

2

1

1

1

1

1

1

1

1

Tables A-7 and A-8 give the remainder associated with conventional truncating division. Table A-7 shows a result of 0 for the case of the maximum negative number divided by -1, but on most machines the result for this case is undefined, or the operation is suppressed.

Table A-7.. Remainder for Signed Short Division (row ?column)

 

 

 

 

 

 

 

 

 

 

-8

-7

-6

-5

-4

-3

-2

-1

 

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

 

0

-

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 

1

-

0

1

1

1

1

1

1

1

1

1

1

1

1

1

0

 

2

-

0

0

2

2

2

2

2

2

2

2

2

2

2

0

0

 

3

-

0

1

0

3

3

3

3

3

3

3

3

3

0

1

0

 

4

-

0

0

1

0

4

4

4

4

4

4

4

0

1

0

0

 

5

-

0

1

2

1

0

5

5

5

5

5

0

1

2

1

0

 

6

-

0

0

0

2

1

0

6

6

6

0

1

2

0

0

0

 

7

-

0

1

1

3

2

1

0

7

0

1

2

3

1

1

0

-8

8

-

0

0

E

0

D

E

F

0

F

E

D

0

E

0

0

-7

9

-

0

F

F

D

E

F

0

9

0

F

E

D

F

F

0

-6

A

-

0

0

0

E

F

0

A

A

A

0

F

E

0

0

0

-5

B

-

0

F

E

F

0

B

B

B

B

B

0

F

E

F

0

-4

C

-

0

0

F

0

C

C

C

C

C

C

C

0

F

0

0

-3

D

-

0

F

0

D

D

D

D

D

D

D

D

D

0

F

0

-2

E

-

0

0

E

E

E

E

E

E

E

E

E

E

E

0

0

-1

F

-

0

F

F

F

F

F

F

F

F

F

F

F

F

F

0

 

Table A-8.. Remainder for Unsigned Short Division (row ?column)

 

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

0

-

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

-

0

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

-

0

0

2

2

2

2

2

2

2

2

2

2

2

2

2

3

-

0

1

0

3

3

3

3

3

3

3

3

3

3

3

3

4

-

0

0

1

0

4

4

4

4

4

4

4

4

4

4

4

5

-

0

1

2

1

0

5

5

5

5

5

5

5

5

5

5

6

-

0

0

0

2

1

0

6

6

6

6

6

6

6

6

6

7

-

0

1

1

3

2

1

0

7

7

7

7

7

7

7

7

8

-

0

0

2

0

3

2

1

0

8

8

8

8

8

8

8

9

-

0

1

0

1

4

3

2

1

0

9

9

9

9

9

9

A

-

0

0

1

2

0

4

3

2

1

0

A

A

A

A

A

B

-

0

1

2

3

1

5

4

3

2

1

0

B

B

B

B

C

-

0

0

0

0

2

0

5

4

3

2

1

0

C

C

C

D

-

0

1

1

1

3

1

6

5

4

3

2

1

0

D

D

E

-

0

0

2

2

4

2

0

6

5

4

3

2

1

0

E

F

-

0

1

0

3

0

3

1

7

6

5

4

3

2

1

0