2’s Complement
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2’s Complement • Another system that lets us represent negative numbers • MSB is STILL the sign bit, but there is no negative zero • Negative numbers count backwards and wrap around • Calculating 2’s complement (Pos Neg) 1. Flip the bits ( 01 and 10 ) 2.Add 1
description
2’s Complement. Another system that lets us represent negative numbers MSB is STILL the sign bit, but there is no negative zero Negative numbers count backwards and wrap around Calculating 2’s complement (Pos Neg) Flip the bits ( 0 1 and 10 ) Add 1. Example ( Pos Neg ). - PowerPoint PPT Presentation
Transcript of 2’s Complement
2’s Complement
• Another system that lets us represent negative numbers
• MSB is STILL the sign bit, but there is no negative zero
• Negative numbers count backwards and wrap around
• Calculating 2’s complement (Pos Neg)
1. Flip the bits ( 01 and 10 )
2. Add 1
Example ( Pos Neg )
110 -110
00012 11112
1. Flip bits: 1110
2. Add 1: 1110+0001 1111
Another Example ( Pos Neg )
2510 -2510
000110012 11100111 2
1. Flip bits: 11100110
2. Add 1: 11100110+00000001 11100111
Your Turn
• Assuming an 8-bit restriction, what is -2110 in 2’s complement?
1. Flip bits
2. Add 1
Answer: 111010112
Your Turn
• Assuming an 8-bit restriction, what is -3010 in 2’s complement?
1. Flip bits
2. Add 1
Answer: 111000102
Example ( Neg Pos )
-410 410
11002 01002
1. Flip bits: 0011
2. Add 1: 0011+0001 0100
Another Example ( Neg Pos )
-2910 2910
111000112 000111012
1. Flip bits: 00011100
2. Add 1: 00011100+00000001 00011101
Your Turn
• Assuming 2’s complement, what is the decimal value of 111110012?
1. Flip bits
2. Add 1
Answer: -710
Your Turn
• Assuming 2’s complement, what is the decimal value of 111010102?
1. Flip bits
2. Add 1
Answer: 22
2’s Complement ChartBinary Decimal
00000111 7
00000110 6
00000101 5
00000100 4
00000011 3
00000010 2
00000001 1
00000000 0
11111111 -1
11111110 -2
11111101 -3
11111100 -4
11111011 -5
11111010 -6
11111001 -7
11111000 -8
Binary Decimal
0111 7
0110 6
0101 5
0100 4
0011 3
0010 2
0001 1
0000 0
1111 -1
1110 -2
1101 -3
1100 -4
1011 -5
1010 -6
1001 -7
1000 -8
SHORTCUT!
1. Find the 1 on the farthest right
2. Flip all the bits to the left of the 1 (DO NOT FLIP THE 1)
Example:
4210 -4210
001010102 110101102
Awesomeness of 2’s Complement
• No more negative zero
• Lower minimum value: -(2N-1)
• So what’s the big deal?– Everything is addition– No need for special hardware to do
subtraction
2’s Complement Addition
• Just like normal positive binary addition
• You MUST restrict the number of bits
• IGNORE any overflow bits– maintain bit-size restriction
Positive Addition Example
1210 + 410 = 1610
Assuming 2’s complement
000010102 1210
+000000102 + 410
000011002 1610
Negative Addition Example
-1210 + -410 = -1610
111101002 -1210
+111111002 + -410
111100002 -1610
NOTE: We ignored the last overflow bit on the left!
Your Turn
• Show the binary addition of -14 + -3 = -17
Subtraction Example
1610 – 410 = 1610 + -410 = 1210
000100002
+111111002
000011002
NOTE: We ignored the last overflow bit on the left!
Your Turn
• Show the binary subtraction of 23 – 10 = 13
Overflow / Underflow Problem
• Addition and bit-size restriction allow for possible overflow / underflow
• Overflow – when the addition of two binary numbers yields a result that is greater than the maximum possible value
• Underflow – when the addition/subtraction of two binary numbers yields a result that is less than the minimum possible value
Overflow Example
• Assume 4-bit restriction and 2’s complement• Maximum possible value: 24-1 – 1 = 7
610 + 310 = 910
01102 610
+00112 +310
10012 -710 not good!
Underflow Example
• Assume 4-bit restriction and 2’s complement• Minimum possible value: -(24-1) = -8
-510 + -510 = -1010
10112 -510
+10112 +-510
01102 610 not good!
