DIGITAL ELECTRONICS- Number System
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The document discusses number systems, focusing on positional notation and various bases including decimal, binary, octal, and hexadecimal. It explains how to convert between these systems and details specific methods for decimal to binary and binary to decimal conversions, as well as conversions between binary and octal or hexadecimal. Overall, it provides foundational knowledge on how numbers operate within different bases.
DIGITAL ELECTRONICS- Number System
- 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 . rararararara aaaaaa
- 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.
- 6. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 9, Dwarka Institutional Area, New Delhi-75 6 Decimal 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
- 7. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 9, Dwarka Institutional Area, New Delhi-75 7 Binary 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
- 8. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 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
- 9. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 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
- 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
- 11. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 9, Dwarka Institutional Area, New Delhi-75 11 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
- 12. TRINITY INSTITUTE OF PROFESSIONAL STUDIES 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
- 15. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 9, Dwarka Institutional Area, New Delhi-75 15 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
- 16. TRINITY INSTITUTE OF PROFESSIONAL STUDIES Sector – 9, Dwarka Institutional Area, New Delhi-75 16 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