Extra Problems Worksheet (Answers at End)
1. Convert the following:
Base 2/8/16 / Decimal101111012
1110011112
3778
7568
D816
3EC16
Decimal / Binary / Octal / Hexidecimal
19910
33510
2. Do the following binary operations (unsigned numbers):
1 0 1 0 1 0 1 0
+ 0 1 1 0 - 0 1 1 0
1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1
AND 0 1 1 1 0 1 0 1 XOR 0 1 1 1 0 1 0 1
Hex value after rotating A4 left 3 bits?
Hex value after arithmetic shifting B2 right 2 bits?
3. Fill in the following ASCII text chart:
Text / Binary ASCII / Octal ASCII / Hex ASCIIM+5
130 117 122 / 58 4F 52
4. Convert the following decimal numbers to signed 8-bit numbers:
Decimal Numbers: / -68 / -91Sign/magnitude
1's complement
2's complement
5. What range of values can be represented using a:
a) 6-bit unsigned number to ______
b) 6-bit sign/magnitude number to ______
c) 6-bit 2's complement number to ______
6. Perform the indicated operations on these 2's complement signed 6-bit values. Check your answers by converting to decimal. Circle unrepresentable values.
0 0 1 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1
+0 0 1 1 0 1 +1 0 0 1 0 0 +0 1 0 1 1 1
0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 0 1
-0 0 1 0 1 1 -1 0 1 1 0 0 -0 1 0 0 1 1
7. Convert the following numbers:
Decimal Number / Floating Point Representation44.187510
-319.562510
10 / 0 10000111 11101011000000000000000
10 / 1 01111101 11000000000000000000000
8. Add floating point numbers 4.625 and 5.5625.
9. Multiply floating point numbers 32.5 and 2.25
10. Use Booth's Algorithm to multiply 12 x –12 using 5-bit signed integers.
ANSWERS
1.
101111012 / 189101110011112 / 46310
3778 / 25510
7568 / 49410
D816 / 21610
3EC16 / 100410
Decimal / Binary / Octal / Hexidecimal
19910 / 11000111 / 307 / C7
33510 / 101001111 / 517 / 14F
2. 1 0 1 0 1 0 1 0
+ 0 1 1 0 - 0 1 1 0
1 0 0 0 0 0 1 0 0
1 0 1 1 0 0 1 1 1 0 1 1 0 0 1 1
AND 0 1 1 1 0 1 0 1 XOR 0 1 1 1 0 1 0 1
0 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0
Hex value after rotating A4 left 3 bits?
A4 = 10100100
Rotate à 00100101
Convert back to hex = 25
Hex value after arithmetic shifting B2 right 2 bits?
B2 = 10110010
ASH à 11101100
Convert back to hex = EC
3.
Text / Binary ASCII / Octal ASCII / Hex ASCIIM+5 / 1001101 0101011 0110101 / 115 53 65 / 4D 2B 35
XOR / 1011000 1001111 1010010 / 130 117 122 / 58 4F 52
4.
Decimal Numbers: / -68 / -91Sign/magnitude / 11000100 / 11011011
1's complement / 10111011 / 10100100
2's complement / 10111100 / 10100101
5. Range for 6-bit unsigned: 0 to 63
Range for 6-bit sign/magnitude: -31 to 31
Range for 6-bit 2's complement: -32 to 31
6.
0 0 1 0 1 0 (+10) 1 1 0 0 1 0 (-14) 1 0 1 1 0 1 (-19)
+0 0 1 1 0 1 (+13) +1 0 0 1 0 0 (-28) +0 1 0 1 1 1 (+23)
0 1 0 1 1 1 (+23) (-42) 0 0 0 1 0 0 ( +4)
(-42) unrepresentable
0 1 1 1 1 0 (+30) 1 1 1 1 1 0 ( -2) 1 1 1 0 0 1 ( -7)
-0 0 1 0 1 1 (+11) -1 0 1 1 0 0 (-20) -0 1 0 0 1 1 (+19)
0 1 0 0 1 1 (+19) 0 1 0 0 1 0 (+18) 1 0 0 1 1 0 (-26)
7.
44.187510 / 101100.0011 ==> 0 10000100 01100001100000000000000-319.562510 / -100111111.1001 ==> 1 10000111 00111111100100000000000
49110 / 0 10000111 11101011000000000000000 ==> 1.11101011 x 28
-0.437510 / 1 01111101 11000000000000000000000 ==> -1.11 x 2-2
8. Add: 4.625 + 15.5625 = 20.1875
Convert to binary:
4.625 = 100.101
15.5625 = 1111.1001
Normalize:
1.00101 x 22
1.1111001 x 23
Align decimal points:
0.100101 x 23
1.1111001 x 23
Add:
0.1001010
1.1111001
------
10.1000011 x 23
De-normalize:
10100.0011
Convert back to decimal:
20.1875
9. Multiply 32.5 x 6.25 = 203.125
Convert to binary:
32.5 = 100000.1
6.25 = 110.01
Normalize:
1.000001 x 25
1.1001 x 22
Add exponents:
5 + 2 = 7
Multiply significands:
1.000001 x 1.1001 = 1.1001011001
Combine Results:
1.1001011001 x 27
Remove exponent:
11001011.001
Convert to decimal
203.125
10. Booth's Algorthim: 12 x -12
Multiplier = 12 (decimal) = 01100 (5-bit binary)
Multiplicand = -12 (decimal) = 10100 (5-bit binary 2’s complement)
Initial Product = Multiplier w/five leading zeros (for 5-bit operands)
00000 01100
(1a) Always use 0 as your initial previous LSB (pLSB). That means
the LSB & previous LSB above are 00, so there is no arithmetic
operation (no-op).
(1b) ASR
00000 00110 0
(2a) LSB and pLSB above are again 00 - no-op
(2b) ASR
00000 00011 0
(3a) LSB and previous LSB above are 10 - so subtract the
multiplicand from left half:
00000 - 10100 = 01100 (with phantom borrow)
Product is now: 01100 00011
(3b) ASR
00110 00001 1
(4a) LSB and previous LSB above are 11 - no-op
(4b) ASR
00011 00000 1
(5a) LSB and pLSB above are 01 - add the multiplicand to left half
00011 + 10100 = 10111
Product is now: 10111 00000
(5b) ASR
11011 10000
We have completed 5 passes, so answer is 1101110000.
1101110000 is a negative 2’s complement number, equivalent
to -144, which is the correct product of 12 x -12.