Transcript Lectures/Lect 18 - Binary Addition and Subtraction

Binary additon &
subtraction
9/15/09 - L15 Decoders,
Multiplexers
Copyright 2009 - Joanne DeGroat, ECE, OSU
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Class 18 – Subtraction
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Binary Addition and Subtraction
Subtraction circuits
Incrementer, Decrementer
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Material from section 4-3 and 4-4 of text
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Copyright 2009 - Joanne DeGroat, ECE, OSU
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Binary Subtraction
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Have previously looked at the subtraction operation. A
quick review.
Just like subtraction in any other base
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Minuend
Subtrahand
Difference
10110
- 10010
00100
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And when a borrow is needed. Note that the borrow gives
us 2 in the current bit position.
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.
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And a full example
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And more ripple -
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In General
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When there is no borrow into the msb position,
then the subtrahend in not larger than the
minuend and the result is positive and correct.
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If a borrow into the msb does occur, then the
subtrahend is larger than the minuend. This was
seen back in lecture 2.
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Consider
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Now do the operation 4 – 6
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Correct difference is -2 or -0010
Different because 2n was brought in and made
the operation M-N+2n
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Desired
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Actual desired magnitude is N-M
To get this need to do 2n – (M-N+2)= N-M
Doing the subtraction from 2n gives the
correct result.
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Two’s compliment
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But how do you represent a minus sign
electronically in a computer?
How can you represent it such that arithmetic
operations are manageable?
There are two types of compliments for each
number base system.
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Have the r’s complement
Have the (r-1)’s complement
For base 2
have 2’s complement
1’s complement
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and
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1’s Complement
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1’s complement of N is defined as (2n -1)-N.
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If n=4 have (2n -1) being 1 0000 - 1 = 1111
So for n=4 would subtract any 4-bit binary
number from 1111.
This is just inverting each bit.
Example: 1’s compliment of 1011001
is 0100110
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2’s complement
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The 2’s complement is defined as 2n-N
Can be done by subtraction of N from 2n or
adding 1 to the 1’s complement of a number.
For 6 = 0110
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The 1’s complement is 1001
The 2’s complement is 1010
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Operation with 2’s complement
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Add 4 and -6
Will use the 2’s complement of -6 or 1010
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4
-6
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0100
1010
1110
And taking the 2’s complement of 1110 get
0001 + 1 = 0010
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A 2’s complement table for 4 bits
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Listing the
values
represented.
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A circuit that does +/
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A general adder subtractor
OP=0 for addition/ =1 for subtraction
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Another number format
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Signed magnitude –
use the MSB to
indicate the sign. The
remaining bits
indicate the
magnitude.
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Overflow
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When adding 2 n-bit numbers it is possilbe to
get a n+1 bit result if there is a carry out.
On paper it is easy just add another bit.
In 2’s complement add a msb 0 for a positive
or a msb 1 for a negative.
In a computer the number of bits that can be
used is fixed.
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Overflow indication.
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In 8-bit 2’s complement notation the range
that can be represented is -127 to +127.
Then the operation to add +70 to +80 is
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Carries
+70
+80
+150
0 1
0 100 0110
0 101 0000
1 001 0110
Also look at the addition of -70 and -80
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The other addition
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The addition of -70 and -80
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Carries
-70
-80
-150
1 0
1 011 1010
1 011 0000
0 110 1010
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The rule – if the carry into the msb position differs from
the carry out from the msb position then an overflow has
occurred.
The circuit
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.
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Class 18 assignment
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Covered sections 4-3 through 4-4
Problems for hand in
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Problems for practice
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none
4-3, 4, 5, 6, 7, 8,16
Reading for next class: sections 5-1, 5-2
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