DIGITAL ELECTRONICS- Number System Positional Notation Radix (Base) of a Number System Radix (Base) of a Number System Decimal Number Systems Binary Number System Decimal to Binary Conversion Converting Decimal to Binary

1. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
Affiliated Institution of G.G.S.IP.U, Delhi
DIGITAL ELECTRONICS
Paper Code :- BCA 106
Keywords:
Number System, Binary, Decimal, Octal, Hexa
By :-HARI MOHAN JAIN

2. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
2
Number System

3. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
3
Positional 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.
2
2
1
1
0
0
1
1
2
2
3
3
210123 .
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4. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
4
Radix (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)

5. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
5
Radix (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 = 13:02:03
NB. The base of a number is often indicated by a subscript.
E.g. (123)10 indicates the base-10 number 123.

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Sector – 9, Dwarka Institutional Area, New Delhi-75
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Decimal Number Systems
• Base 10
– Ten digits, 0-9
– Columns represent (from right to left) units, tens,
hundreds etc.
123
1102 + 2101 + 3100
or
1 hundred, 2 tens and 3 units

7. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
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Binary Number System
• Base 2
– Two digits, 0 & 1
– Columns represent (from right to left) units, twos,
fours, eights etc.
1111011
126 + 125 + 124 + 123 + 022 + 121 + 120
= 164 + 132 + 116 + 18 + 04 + 12 + 11
= 123

8. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
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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

9. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
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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

10. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
10
Converting 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 = 0.750
0.750 x 2 = 1.500
0.500 x 2 = 1.000

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Sector – 9, Dwarka Institutional Area, New Delhi-75
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Binary 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

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Sector – 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

13. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
13
Binary 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
=
11170113
11060102
10150011
10040000
BinaryOctalBinaryOctal
8)12.673(
2)001010.110111011(

14. TRINITY INSTITUTE OF PROFESSIONAL STUDIES
Sector – 9, Dwarka Institutional Area, New Delhi-75
14
Binary 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
11170113
11060102
10150011
10040000
BinaryOctalBinaryOctal

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Sector – 9, Dwarka Institutional Area, New Delhi-75
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Binary and Hex Conversions
• Converting from hex to binary: Replace each hex digit with its
equivalent 4-bit binary sequence
261.3516 = 2 6 1 . 3 516
=0010 0110 0001 . 0011 01012
1111F1011B0111700113
1110E1010A0110600102
1101D100190101500011
1100C100080100400000
BinaryHexBinaryHexBinaryHexBinaryHex

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Sector – 9, Dwarka Institutional Area, New Delhi-75
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Binary 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
1111F1011B0111700113
1110E1010A0110600102
1101D100190101500011
1100C100080100400000
BinaryHexBinaryHexBinaryHexBinaryHex