TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New

Delhi-75Affiliated Institution of G.G.S.IP.U, Delhi

DIGITAL ELECTRONICSPaper Code :- BCA 106

Keywords:Number System, Binary, Decimal, Octal, Hexa

By :-HARI MOHAN JAIN

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

2

Number System

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

3Positional Notation•Value of number is determined by multiplying each digit by a weight and then summing.

•The weight of each digit is a POWER of the RADIX (also called BASE) and is determined by position.

22

11

00

11

22

33

210123 .

rararararara

aaaaaa

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

4Radix (Base) of a Number System

• Decimal Number System (Radix = 10)Eg:- 7392 = 7x103+3x102+9x101+2x100

• Binary Number System (Radix = 2)Eg:- 101.101 = 1x22+0x11+1x20+1x2-1+0x2-11x2-2

• Octal number system (radix = 8)

• Hexadecimal number system (radix = 16)

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

5Radix (Base) of a Number System

• When counting upwards in base-10, we increase the units digit until we get to 10 when we reset the units to zero and increase the tens digit.

• So, in base-n, we increase the units until we get to n when we reset the units to zero and increase the n-s digit.

• Consider hours-minutes-seconds as an example of a base-60 number system:– Eg. 12:58:43 + 00:03:20 = 13:02:03NB. The base of a number is often indicated by a subscript.

E.g. (123)10 indicates the base-10 number 123.

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

6Decimal Number Systems • Base 10– Ten digits, 0-9– Columns represent (from right to left) units, tens,

hundreds etc.

123

1102 + 2101 + 3100

or 1 hundred, 2 tens and 3 units

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

7Binary Number System• Base 2– Two digits, 0 & 1– Columns represent (from right to left) units, twos,

fours, eights etc.1111011

126 + 125 + 124 + 123 + 022 + 121 + 120

= 164 + 132 + 116 + 18 + 04 + 12 + 11

= 123

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

8

Decimal to Binary Conversion

123 ¸ 2 = 61 remainder 1 61 ¸ 2 = 30 remainder 1 30 ¸ 2 = 15 remainder 0 15 ¸ 2 = 7 remainder 1 7 ¸ 2 = 3 remainder 1 3 ¸ 2 = 1 remainder 1 1 ¸ 2 = 0 remainder 1

Least significant bit (LSB) (rightmost)

Most significant bit (MSB) (leftmost)

Answer : (123)10 = (1111011)2

Example – Converting (123)10 into binary

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

9

Decimal to Binary Conversion

The quotient is divided by 2 until the new quotient becomes 0

Integer Remainder 41 20 1 10 0 5 0 2 1 1 0 0 1

101001 answer

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

10Converting Decimal to Binary• To convert a fraction, keep multiplying the fractional part by 2 until it becomes

0. Collect the integer parts in forward order• Example: 162.375:• So, (162.375)10 = (10100010.011)2

162 / 2 = 81 rem 0 81 / 2 = 40 rem 1 40 / 2 = 20 rem 0 20 / 2 = 10 rem 0 10 / 2 = 5 rem 0 5 / 2 = 2 rem 1 2 / 2 = 1 rem 0 1 / 2 = 0 rem 1

0.375 x 2 = 0.7500.750 x 2 = 1.5000.500 x 2 = 1.000

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

11Binary to Decimal Conversion• Each bit represents a power of 2

• Every binary number is a sum of powers of 2

• Decimal Value = (dn-1 2n-1) + ... + (d1 21) + (d0 20)

• Binary (10011101)2 = 27 + 24 + 23 + 22 + 1 = 157

1 0 0 1 1 1 0 1

27 26 25 24 23 22 21 20

01234567

Some common powers of 2

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

12

Converting Binary to Decimal

• For example, here is 1101.01 in binary:1 1 0 1 . 0 1 Bits 23 22 21 20 2-1 2-2 Weights (in base 10)

(1 x 23) + (1 x 22) + (0 x 21) + (1 x 20) + (0 x 2-1) + (1 x 2-2) =

8 + 4 + 0 + 1 + 0 + 0.25 = 13.25

(1101.01)2 = (13.25)10

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

13Binary and Octal Conversions• Converting from octal to binary: Replace each octal digit with

its equivalent 3-bit binary sequence

= 6 7 3 . 1 2 = 110 111 011 . 001 010=

11170113110601021015001110040000

BinaryOctalBinaryOctal

8)12.673(

2)001010.110111011(

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

14Binary and Octal Conversions• Converting from binary to octal: Make groups of 3 bits,

starting from the binary point. Add 0s to the ends of the number if needed. Convert each bit group to its corresponding octal digit.

10110100.0010112 = 010 110 100 . 001 0112

= 2 6 4 . 1 38

11170113110601021015001110040000

BinaryOctalBinaryOctal

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

15Binary and Hex Conversions

• Converting from hex to binary: Replace each hex digit with itsequivalent 4-bit binary sequence

261.3516 = 2 6 1 . 3 516 =0010 0110 0001 . 0011 01012

1111F1011B01117001131110E1010A01106001021101D1001901015000111100C100080100400000

BinaryHexBinaryHexBinaryHexBinaryHex

TRINITY INSTITUTE OF PROFESSIONAL STUDIESSector – 9, Dwarka Institutional Area, New Delhi-75

16Binary and Hex Conversions

• Converting from binary to hex: Make groups of 4 bits, starting from the binary point. Add 0s to the ends of the number if needed. Convert each bit group to its corresponding hex digit

10110100.0010112 = 1011 0100 . 0010 11002

= B 4 . 2 C16

1111F1011B01117001131110E1010A01106001021101D1001901015000111100C100080100400000

BinaryHexBinaryHexBinaryHexBinaryHex