computer architecture data representation mark s. staveley [email protected]
TRANSCRIPT
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3127
110011010001
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,
14 8 or C16.
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217
110011010001
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,
14 8 or C16.
3217/2 = 1608 R 1 (Least Significant Bit)1608/2 = 804 R 0 804/2 = 402 R 0 402/2 = 201 R 0 201/2 = 100 R 1 100/2 = 50 R 0 50/2 = 25 R 0 25/2 = 12 R 1 12/2 = 6 R 0 6/2 = 3 R 0 3/2 = 1 R 1 1/2 = 0 R 1 (Most Significant Bit)
321710 = 1100100100012
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001
110011010001
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,
14 8 or C16.
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001
110011010001
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,
14 8 or C16.
3217/8 = 402 R 1 (Least Significant Bit)
402/8 = 50 R 2
50/8 = 6 R 2
6/8 = 0 R 6 (Most Significant Bit)
321710 = 62218
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221
110011010001
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,
14 8 or C16.
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221
110011010001
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,
14 8 or C16.
3217/16 = 201 R 1 (Least Significant Bit)
201/16 = 12 R 6
2/16 = 0 R C (1210 Most Significant Bit)
321710 = C6116
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221 C91
110011010001
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,
14 8 or C16.
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221 C91
110011010001
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,
14 8 or C16.
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221
110011010001
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,
14 8 or C16.
B3816
B16 – 10112
316 – 00112
816 – 10002
1011001110002
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221
110011010001
6271
101100111000 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,
14 8 or C16.
1011001110002
Split on 3-bits (base 8)
1012 – 58
1002 – 48
1112 – 78
0002 – 08
54708
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221
110011010001
6271
2872 101100111000 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,
14 8 or C16.
1011001110002 =
0 x 20 + 0 X 21 + 0 X 22 + 1 X 23 + 1 x 24 + 1 x 25 + 0 x 26 + 0 x 27 + 1 x 28 + 1 x 29 + 0 x 210 + 1 x 211
= 287210
Binary Coded Decimal Representation
Decimal Binary Octal Hexadecimal
3217 110010010001 6221 C91
3281 110011010001 6321 CD1
3257 110010111001 6271 CB9
2872 101100111000 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,
14 8 or C16.
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
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 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110
+21 (convert to One’s Complement)
Result = same as normal because it is positive = 010101
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110
+21 (convert to Two’s Complement)
Result = same as normal because it is positive = 010101
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 = 010101
Bias = 31 = 011111
010101 + 011111 = 110100
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 = 100000
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110 -32 (Convert to Excess 31)
Largest Negative Number = 000000 = - 31
Largest Positive Number = 111111 = + 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 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110 -17 (convert to Sign & Magnitude)
Sign = - = 1
17 convert to 5-bit representation = 10001
Result = 110001
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110 -17 (One’s Complement)
Sign = - = 1
17 convert to 5-bit representation = 10001
Complement each bit = 01110
Result = 101110
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 011111 111110 -17 (convert to Two’s Complement)
Add One to One’s Complement
101110 + 1 = 01111
Result = 101111
Complete the table below using 6-bit representation for sign and magnitude, ones’ complement, two’s complement, and excess 31. 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’.
Decimal Sign & Magnitude
One’s Complement
Two’s Complement
Excess 31
+21 010101 010101 010101 110100
-17 110001 101110 101111 001110
+14 001110 001110 001110 101101
-32 N/A N/A 100000 N/A
+31 011111 011111 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 = 101111
Bias = 31 = 011111
101111 + 011111 = 001110
Fill in the following steps to find the representation for –17.5 in the IEEE single-precision floating-point standard.
IEEE-754 Floating Point Standard° Developed in 1985. 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
Excess-1023 Hidden bit
Hidden bit
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:
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: –10001.12
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:
–1.000112 * 24
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:
12710 + 410 = 11111112 + 1002 = 100000112
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 = 12710+410 = 13110 = 100000112
Fraction = 1.000112 (leading 1 of fraction is hidden)
= 00011000000000000000000
1 10000011 00011000000000000000000
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.
1 1 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
101000012 = 16110;
161 – 127 = 34;
–1.110012 * 234
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.
0 0 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
011110102 = 12210;
122 – 127 = –5;
1.02 * 2–5 = 1.010 * 2
–5 = 0.03125 = 3.125 * 10-2
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.
0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
100000012 = 12910;
129 – 127 = 2;
1.10102 * 22 = 110.102 = 6.5 = 6.5 * 100
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.
1 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
111000002 = 22410;
224 – 127 = 97;
–1.011001112 * 297
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) 0 1 0 1 1 1 b) 0 1 1 0 0 0 c) 1 0 1 1 0 1
+ 0 1 0 0 1 1 - 0 1 1 1 0 0 - 1 0 1 0 0 1
C = C = C =
V = V = V =
d) 1 1 0 0 1 1 e) 1 0 1 1 0 0 f) 0 1 1 1 1 1
- 0 0 0 0 0 1 + 1 0 0 1 1 0 + 1 0 1 0 1 0
C = C = C =
V = V = V =
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) 0 1 0 1 1 1 b) 0 1 1 0 0 0 c) 1 0 1 1 0 1
+ 0 1 0 0 1 1 - 0 1 1 1 0 0 - 1 0 1 0 0 1
1 0 1 0 1 0 = 0 1 1 0 0 0 0 0 0 1 0 0
+ 1 0 0 1 0 0
1 1 1 1 0 0
C = 0 (no carry) C = 0 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
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) 1 1 0 0 1 1 e) 1 0 1 1 0 0 f) 0 1 1 1 1 1
- 0 0 0 0 0 1 + 1 0 0 1 1 0 + 1 0 1 0 1 0
1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1
C = 1 C = 1 C = 1
V = 0 V = 1 V = 0