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NPP Number System, Boolean Algebra and Logic Circuits 283

3.34 Signed Integer Arithmetic 3.34 gmBÝS> nyUmªH$ A§H$J{UV
(2's Complement addition and subtrac- (2's H$m°påßb_|Q> Omo‹S> d KQ>md)
tion)
Positive Integers have unique represen- YZmË_H$ nyUmªH$ H$mo EH$ hr VarHo$ go Xem©`m Om
tation; signed-Magnitude, but negative gH$Vm h¡ ; gmBÝS> _op½ZQ²>`yS> à{V{Z{YËd Ÿ& na§Vw F$UmË_H$
numbers can be represented in three ways;
signed-magnitude, signed- 1's complement and nyUmªH$ H$mo VrZ àH$ma go Xem©`m Om gH$Vm h¡ ; gmBÝS>
signed- 2's complement representation. First _op½ZQ²>`yS>, gmBÝS> 1's H$m°påßb_|Q> VWm gmBÝS> 2's
two are rarely used because there are different H$m°påßb_|Q> & àW_ XmoZm| H$m Cn`moJ AË`§V H$_ hmoVm h¡
representations of +0 and –0. But in 2's Š`m|{H$ BZ_| + 0 VWm -0 Ho$ Xmo AbJ-AbJ ê$n hmoVo
complement representation both + 0 and – 0 h¢ & bo{H$Z 2's H$m°påßb_|Q> _| + 0 VWm - 0 Ho$ {bE
have only one representation. Therefore 2's
complement method is used extensively. Here EH$ hr à{V{Z{YËd hmoVm h¡Ÿ& Bg{bE BgH$m AË`{YH$
Binary addition and subtraction as applicable Cn`moJ {H$`m OmVm h¡& h_ ~mBZar _| Omo‹S> VWm KQ>md H$s
to signed integers will be discussed: MMm© H$a|JoŸ&
2's Complement Addition (Signed) gmBÝS> 2's H$m°påßb_|Q> Omo‹S> (gmBªS>)

This method is straight forward; represent `h ~hþV hr gab {d{Y h¡& Bg_| YZmË_H$ nyUmªH$m|
positive operands using signed magnitude H$mo gmBÝS> _op½ZQ²>`yS> go VWm F$UmË_H$ nyUmªH$m| H$mo gmBÝS>
representation and negative operands using 2's
complement representation. Add both the 2's H$m°påßb_|Q> go Xem©Vo h¢Ÿ& XmoZm| H$mo Omo‹S> XoVo h¢ VWm
binary numbers and neglect any end carry. A§{V_ hm{gb H$mo N>mo‹S> XoVo h¢& A~ EH$ CXmhaU boH$a
Consider the case in which both the numbers g_PVo h¢ {Og_| XmoZm| g§»`mE± YZmË_H$ h¢…
are positive :
(+13) + (+12)
Take sufficient number of bits to avoid the g_w{MV g§»`m _| {~Q>| boVo h¢ {Oggo Amodaâbmo H$s
problem of overflow. Starting by taking 8-bits g_ñ`m go ~Mm Om gHo$&$`hm± h_ 8-{~Q>m| _| (+13) d
to represent +13 and +12.
(+ 12) H$mo Xem©H$a Omo‹S>Vo h¢ …
+ ( 13+ ) → 0000 1 1 0 1
( + + ) 12 →+ 0000 1 1 00
000 1 1 00 1

Since MSB is 0 the result is positive integer MSB Ho$ ñWmZ na 0 h¡ Omo Xem©Vm h¡ {H$ n[aUm_
and the remaining bits show the magnitude. YZmË_H$ h¡ Ÿ& ~mH$s H$s {~Q>| CgH$m n[a_mU ~VmVr h¢, Omo
The magnitude 11001 can be evaluated to {H$ 11001 h¡ & BgH$m Xe_bd _mZ 25 AmVm h¡ Ÿ&
decimal 25. Therefore
AV… h_ {bI gH$Vo h¢ {H$
(+13) + (+12) = +25
Now consider another problem in which A~ EH$ Xygar g_ñ`m bmo {Og_| EH$ nyUmªH$ YZmË_H$
one integer is positive and another is negative. VWm Xygam F$UmË_H$ h¡ Ÿ&

(+17) + (–26)