Lect 18 - Binary Addition and Subtraction

7/27/2019 Lect 18 - Binary Addition and Subtraction

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9/15/09 - L15 Decoders,Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU1

Binary additon &

subtraction


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9/15/09 - L15 Decoders,Multiplexers Copyright 2009 - Joanne DeGroat, ECE, OSU 2

Class 18Subtraction Binary Addition and Subtraction

Subtraction circuits

Incrementer, Decrementer

Material from section 4-3 and 4-4 of text


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Binary Subtraction Have previously looked at the subtraction operation. A

quick review.

Just like subtraction in any other base Minuend 10110

Subtrahand - 10010

Difference 00100

And when a borrow is needed. Note that the borrow givesus 2 in the current bit position.

.



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And a full example And more ripple -



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In General 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.

If a borrow into the msb does occur, then thesubtrahend is larger than the minuend. This was

seen back in lecture 2.



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Consider Now do the operation 46

Correct difference is -2 or -0010

Different because 2n was brought in and made

the operation M-N+2n



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Desired 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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Twos compliment 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. Have the rs complement

Have the (r-1)s complement For base 2 have 2s complement and

1s complement



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1s Complement 1s complement of N is defined as (2n -1)-N.

If n=4 have (2n -1) being 1 0000 - 1 = 1111

So for n=4 would subtract any 4-bit binarynumber from 1111.

This is just inverting each bit.

Example: 1s compliment of 1011001

is 0100110



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2s complement The 2s complement is defined as 2n-N

Can be done by subtraction of N from 2n or

adding 1 to the 1s complement of a number. For 6 = 0110

The 1s complement is 1001

The 2s complement is 1010



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Operation with 2s complement

Add 4 and -6

Will use the 2s complement of-6 or 1010

4 0100

-6 1010

1110

And taking the 2s complement of 1110 get0001 + 1 = 0010



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A 2s complement table for 4 bits

Listing the

values

represented.



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A circuit that does +/-

A general adder subtractor

OP=0 for addition/ =1 for subtraction



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Another number format

Signed magnitude

use the MSB to

indicate the sign. Theremaining bits

indicate the

magnitude.

9/15/09 - L15 Decoders,

Multiplexers

Copyright 2009 - Joanne DeGroat, ECE, OSU 14


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Overflow

When adding 2 n-bit numbers it is possilbe toget a n+1 bit result if there is a carry out.

On paper it is easy just add another bit. In 2s complement add a msb 0 for a positive

or a msb 1 for a negative.

In a computer the number of bits that can beused is fixed.

9/15/09 - L15 Decoders,

Multiplexers



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Overflow indication.

In 8-bit 2s complement notation the range

that can be represented is -127 to +127.

Then the operation to add +70 to +80 is Carries 0 1

+70 0 100 0110

+80 0 101 0000

+150 1 001 0110

Also look at the addition of -70 and -809/15/09 - L15 Decoders,

Multiplexers



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The other addition The addition of -70 and -80

Carries 1 0

-70 1 011 1010

-80 1 011 0000

-150 0 110 1010

The ruleif the carry into the msb position differs fromthe carry out from the msb position then an overflow hasoccurred.

The circuit

.

9/15/09 - L15 Decoders,

Multiplexers



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Class 18 assignment

Covered sections 4-3 through 4-4

Problems for hand in

none Problems for practice

4-3, 4, 5, 6, 7, 8,16

Reading for next class: sections 5-1, 5-2

9/15/09 - L15 Decoders,

Multiplexers


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Lect 18 - Binary Addition and Subtraction