1. Computer Architecture
Data Representation
Mark S. Staveley

2. Binary Coded Decimal Representation
Octal
Hexadecimal
3127
6271
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

3. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6271
B38
3217/2 = R 1 (Least Significant Bit)
1608/2 = R 0
804/2 = R 0
402/2 = R 0
201/2 = R 1
100/2 = R 0
50/2 = R 0
25/2 = R 1
12/2 = R 0
6/2 = R 0
3/2 = R 1
1/2 = R 1 (Most Significant Bit) =
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

4. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6271
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

5. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6271
B38
3217/8 = R 1 (Least Significant Bit)
402/8 = R 2
50/8 = R 2
6/8 = R 6 (Most Significant Bit)
= 62218
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

6. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
6271
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

7. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
6271
B38
3217/16 = R 1 (Least Significant Bit)
201/16 = R 6
2/16 = R C (1210 Most Significant Bit)
= C6116
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

8. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
C91
6271
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

9. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
C91
6271
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

10. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
6271
B38
B3816
B16 – 10112
316 – 00112
816 – 10002
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

11. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
6271
B38
Split on 3-bits (base 8)
1012 – 58
1002 – 48
1112 – 78
0002 – 08
54708
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

12. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
6271
2872
5470
B38 =
0 x X X X x x x x x x x x 211 =
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

13. Binary Coded Decimal Representation
Octal
Hexadecimal
3217
6221
C91
3281
6321
CD1
3257
6271
CB9
2872
5470
B38
Complete the following table. Each row represents a specific unsigned integer value in the different radix forms listed in the table. For example, 1210 can be written as 11002, 14­8 or C16.

14. +21 -17 +14 -32 +31 Decimal Sign & Magnitude One’s Complement
Two’s Complement
Excess 31
+21
-17
+14
-32
+31
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

15. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

16. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
+21 (convert to Sign & Magnitude)
Sign = + = 1
21 convert to 5-bit representation = 10101
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

17. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
+21 (convert to One’s Complement)
Result = same as normal because it is positive =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

18. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
+21 (convert to Two’s Complement)
Result = same as normal because it is positive =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

19. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
+21 (Convert to Excess 31)
Positive and negative representations of a number are obtained by adding a bias to the two’s complement representation, ignoring any carry out from the most significant digit.
21 in Two’s Complement =
Bias = 31 = =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

20. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

21. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-32 (convert to Sign & Magnitude)
Sign = - = 1
32 convert to 5-bit representation = Error
Why? Greatest number represented with 5 bits is 31 (11111) 32 is out of range.
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

22. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-32 (One’s Complement)
Sign = - = 1
32 convert to 5-bit representation = Error
Why? Greatest number represented with 5 bits is 31 (11111) 32 is out of range.
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

23. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-32 (convert to Two’s Complement)
Minimum 2's complement value = -2n-1
Maximum 2's complement value = 2n-1 – 1
n = 6 = Max = +31, Min = -32
-32 converted =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

24. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-32 (Convert to Excess 31)
Largest Negative Number = = - 31
Largest Positive Number = = + 32
Out of Range = N/A
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

25. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

26. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-17 (convert to Sign & Magnitude)
Sign = - = 1
17 convert to 5-bit representation = 10001
Result =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

27. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-17 (One’s Complement)
Sign = - = 1
17 convert to 5-bit representation = 10001
Complement each bit = 01110
Result =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

28. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-17 (convert to Two’s Complement)
Add One to One’s Complement
= 01111
Result =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

29. Decimal
Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21
010101
110100
-17
110001
101110
101111
001110
+14
101101
-32
N/A
100000
+31
011111
111110
-17 (Convert to Excess 31)
Positive and negative representations of a number are obtained by adding a bias to the two’s complement representation, ignoring any carry out from the most significant digit.
17 in Two’s Complement =
Bias = 31 = =
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess Each row in the table is to show a specific numerical value in the different data representations listed in the table. Note that for each numeric value listed below, it may not be possible to represent that value in all the data representations – in these cases just specify ‘N/A’.

30. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.

31. IEEE-754 Floating Point Standard
Developed in It can be supported in hardware, or a mixture of hardware and software.
There are also single extended, and double extended formats (80 bits wide, 15-bit exponent, and 64-bit fraction).
Excess-127
Hidden bit
Excess-1023
Hidden bit

32. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
a) Convert –17.5 to base 2:
b) Express the value from (a) in binary scientific notation:
c) Convert the exponent from (b) to excess 127:
d) IEEE single point precision representation:

33. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
Convert –17.5 to base 2: –

34. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
b) Express the value from (a) in binary scientific notation:
– * 24

35. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
c) Convert the exponent from (b) to excess 127:
= =

36. Fill in the following steps to find the representation for –17
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
d) IEEE single point precision representation:
Sine Bit = Negative = 1
Exponent = = =
Fraction = (leading 1 of fraction is hidden) =

37. Convert the following floating point numbers represented in IEEE single precision floating point representation to both binary and decimal representations in scientific notation, where feasible.
= 16110;
161 – 127 = 34;
– * 234

38. Convert the following floating point numbers represented in IEEE single precision floating point representation to both binary and decimal representations in scientific notation, where feasible.
= 12210;
122 – 127 = –5;
1.02 * 2–5 = * 2–5 = = * 10-2

39. Convert the following floating point numbers represented in IEEE single precision floating point representation to both binary and decimal representations in scientific notation, where feasible.
= 12910;
129 – 127 = 2;
* 22 = = 6.5 = 6.5 * 100

40. Convert the following floating point numbers represented in IEEE single precision floating point representation to both binary and decimal representations in scientific notation, where feasible.
= 22410;
224 – 127 = 97;
– * 297

41. Do the following using two’s complement arithmetic and indicate the carry (C) and overflow (V) values.
(e.g., “C = 1” if there is a carry, “C = 0” otherwise).
a) b) c)
C = C = C =
V = V = V =
d) e) f)
C = C = C =
V = V = V =

42. Do the following using two’s complement arithmetic and indicate the carry (C) and overflow (V) values.
(e.g., “C = 1” if there is a carry, “C = 0” otherwise).
a) b) c) =
C = 0 (no carry) C = C = 1
V = 1 (sum out of range) V = 0 V = 0
Note: When the CPU adds two binary integers, if their sum is out of range when interpreted in the two’s complement representation, then V is set to 1. Otherwise V is cleared to 0

43. Do the following using two’s complement arithmetic and indicate the carry (C) and overflow (V) values.
(e.g., “C = 1” if there is a carry, “C = 0” otherwise).
d) e) f)
C = 1 C = 1 C = 1
V = 0 V = 1 V = 0