6.111 Lecture # 12
Binary arithmetic: most operations are familiar
Each place in a binary number has value 2
n
5 =
00000101 = 1 + 4
19 =
00010011 = 1+2+16
5 +
00000101
19
00010011
=
00011000
24 = 16+8
19 
00010011
5 =
00000101
00001110
14 = 8+4+2
What happens if we do this operation:
5
00000101
19
00010011
=
11110010
Note two things about this operation:
Addition often
requires a ‘carry’
Subtraction may
require a ‘borrow’
1. We had to invent a ‘borrow’ bit from the left
2. What is left is the two’s complement representation of 14:
14 =
00001110
14 =
11110001
+1
=
11110010
Representation of negative numbers: there are a number of ways we
might do this:
1. Use of a ‘sign bit’ (this is just like having a sign for the
number)
5 =
10000101
Note that addition and subtraction are somewhat complex
(and multiplication and division). Generally must strip the
sign bit, do the operation, then figure out the sign of the
result.
2. ‘One’s Complement’: invert each bit. We won’t have much to
say about this.
3. ‘Two’s Complement’: invert each bit and add one.
1
Two’s complement is consistent and reversible:
5 =
00000101±
5 =
11111010 +1
= 11111011±
5 =
00000100 +1
= 00000101±
Addition and Subtraction between two’s complement numbers works:
5
11111011±
+(19)
11101101±
=
11101000
(which is 24)±
00010111+1 = 00011000 = 16+8
3
4
2
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View Full Document5.5
=
00000101.1
5.0
=
00000101.0
5.0
=
11111010.1
+
1
=
11111101.0
In many cases we want to
extend a number: to employ
more ‘binary places’ to
represent a number. How do
we do this extension?
To extend a number (represent with more places) without changing
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 Fall '02
 Prof.DonTroxel
 Addition, Negative Numbers, Elementary arithmetic, MSB

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