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NPP Number System, Boolean Algebra and Logic Circuits 153

3.6 Conversion from any base to 3.6 {H$gr ^r AmYma go Xe_bd _| n[adV©Z
Decimal
There is a generalised way to convert any EH$ hr ì`mnH$ {d{Y Ho$ Ûmam {H$gr ^r AmYma dmbr
number into decimal. Consider the following g§»`m H$mo Xe_bd _| ~Xbm Om gH$Vm h¡. O¡gm {H$ ZrMo
number whose base is r :
{bIo Z§~a H$mo Ü`mZ go XoImo, {OgH$m ~og r h¡ …
( xyz. pqr) r
The integer part of the number is xyz and BgH$m nyUmªH$ ^mJ xyz h¡ VWm Am§{eH$ ^mJ pqr
fractional part is pqr. The decimal equivalent h¡Ÿ& BgH$m Xe_bd Vwë` Bg àH$ma go àmá {H$`m Om
can be obtained as below :
gH$Vm h¡…
x y z p q r
× × × . × × ×
r 2 r 1 r 0 r − 1 r − 2 r − 3
The decimal number will be: AV… Xe_bd g§»`m hmoJr

x × r + 2 y × r + 1 z × r + 0 p × r + − 1 q r× − 2 + r r× − 3
The procedure described can be applied to CnamoŠV ì`mnH$ {d{Y H$mo h_ {ZåZ _| go {H$gr ^r
any of the following conversions: EH$ H$ÝdO©Z Ho$ {bE Cn`moJ _| bm gH$Vo h¢:

Binary to Decimal (r = 2), Octal to Decimal ~mBZar go Xe_bd (r = 2), Am°ŠQ>b go Xe_bd
(r = 8), Hexadecimal to Decimal (r = 16).
(r = 8), hoŠgmS>o{g_b go Xe_bd (r = 16),
Or for any arbitrary value of r. `m r H$m H$moB© ^r ñd¡pÀN>H$ _mZ hmo Ÿ&
Now Consider one by one. A~ BÝh| h_ EH$-EH$ H$aHo$ CXmhaUm| H$s ghm`Vm

go g_PVo h¢ Ÿ&
Binary to Decimal Conversion ~mBZar go Xe_bd _| H$ÝdO©Z
The base of Binary number system is 2. My±{H$ ~mBZar g§»`m nÕ{V H$m AmYma 2 h¡, AV…
Therefore all the calculations are done with the gmar JUZm r = 2 boH$a H$s OmVr h¡ Ÿ&
base 2.
Problem 3.12 àíZ 3.12
Convert following binary numbers into {ZåZ{b{IV ~mBZar g§»`mAm| H$mo CZHo$ Xe_bd
their decimal equivalents. Vwë` _| ~X{bE …

(a) 1101 (b) 101.11 (c) 101011
Solution : hc :

(a) The given Binary Number is 1101. Write (a) Xr JB© ~mBZar g§»`m 1101 h¡ Ÿ& BgH$s Mmam|
the bits giving some space and their positional {~Q>m| H$mo Wmo‹S>r Xya-Xya {bI bmo Ÿ& CZHo$ R>rH$ ZrMo CZH$m
weights below. Multiply the bits with positional ^ma {bI Xmo Ÿ& A~ ^ma H$m JwUm g§~§{YV {~Q> go H$amo VWm
weights and add.
Eogo gmao JwUZ\$bm| H$mo Omo‹S>mo …