Transcript negative binary numbers

Negative Binary
Numbers
&
Binary Subtraction
2’s Complement
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2’s Complement is a way to represent negative numbers in
binary.
Recall the Odometer analogy: On a car odometer, all the
digits eventually reset to 0s. The same applies to a “binary
odometer”.
What would a binary odometer look like as it approaches
and passes the “reset” point?
(Reset)
1101
Represents -3
1110
Represents -2
1111
Represents -1
0000
0001
Represents 1
0010
Represents 2
0011
Represents 3
2’s Complement Cont’d
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In Computers, the memory is divided up into bytes – or groups
of 8 bits.
1111 1101
In byte format:
1111 1110
1111 1111
(Reset)
0000 0000
0000 0001
0000 0010
0000 0011
In all negative binary numbers, the left-most bit is always 1.
This bit is called the “sign bit”. That is, if the sign bit (left bit in
a group of 8) is 1, the number is negative.
Fortunately, there is an easy way of finding the negatives of
binary numbers – it is called the 2’s Complement Method.
Finding the 2’s Complement
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There are 2 steps:
1) Flip all the bits (all 1’s become 0, & 0’s become 1)
2) Add 1 to the new number.
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Example:
Find the negative (aka 2’s Complement) of 0010 1101
(decimal number 45)
Step 1: Flip the bits.
Step 2: Add 1.
Sign Bit
00101101
11010010
+
1
11010011
(Answer:
Proof: 128-64-0-16-0-0-2-1 = 45
-45)
Binary Subtraction
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Use the following steps:
1) Find the 2’s Complement of the number being subtracted.
2) Add the new number to the number it is being subtracted
from.
3) Discard the overflow bit.
Example: Find 01110111 – 00010111
(119 – 23)
1) 2’s Complement of 00010111: 11101000
+
1
11101001
(Flip the Bits)
(Add 1)
Binary Subtraction Cont’d
2) Add the Original Number and the 2’s Complement Number:
Overflow Bit
01110111
+11101001
1 01100000
Original Number
2’s Complement Number
3) Discard the Overflow Bit:
01100000
(Decimal 96)
(119 – 23 = 96)
(Proof: 0 + 64 + 32 + 0 + 0 + 0 + 0 + 0 = 96)