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Number Systems

Rayat Shikshan Sanstha’s S.M. Joshi College Hadapsar-028 Department of Electronics Science. Number Systems. Presented by- Dr. Bhalerao S.P. Number Systems. Numbers. Each number system is associated with a base or radix The decimal number system is said to be of base or radix 10

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Number Systems

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  1. RayatShikshanSanstha’s S.M. Joshi College Hadapsar-028 Department of Electronics Science Number Systems Presented by- Dr. Bhalerao S.P

  2. Number Systems

  3. Numbers • Each number system is associated with a base or radix • The decimal number system is said to be of base or radix 10 • A number in base r contains r digits 0,1,2,...,r-1 • Decimal (Base 10): 0,1,2,3,4,5,6,7,8,9 • Numbers are usually expressed in positional notation • MSD: most significant digit • LSD: least significant digit

  4. Numbers • The value of the number is given in the polynomial form

  5. Numbers In addition to decimal, three other number systems are also important: Binary, Octal, and Hexadecimal

  6. Unsigned Binary Numbers • The binary number system: Base-2 • Two digits: 0 and 1 • The digits in a binary number are called bits • MSB: most significant bit • LSB: least significant bit

  7. Unsigned Binary Numbers

  8. Unsigned Binary Numbers • For a computer with the word size of 32-bit • - 4 data-bit unit – nibble (half byte) • - 8 data-bit unit - byte • - 16 data-bit unit – two bytes (half-word) • - 32 data-bit unit – word (four bytes) • - 64 data-bit unit – double-word

  9. Converting Binary to Decimal • For example, here is 1101.01 in binary: (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

  10. 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 Converting Decimal to Binary • To convert a decimal integer into binary, keep dividing by 2 until thequotient is 0. Collect the remainders in reverse order • To convert a fraction, keep multiplying the fractional part by 2 until itbecomes 0. Collect the integer parts in forward order • Example: 162.375: • So, (162.375)10 = (10100010.011)2

  11. Why does this work? • This works for converting from decimal to any base • Why? Think about converting 162.375 from decimal todecimal • Each division strips off the rightmost digit (theremainder). The quotient represents the remaining digits in the number • Similarly, to convert fractions, each multiplication strips off the leftmost digit (the integer part). The fraction represents the remaining digits 162 / 10 = 16 rem 2 16 / 10 = 1 rem 6 1 / 10 = 0 rem 1 0.375 x 10 = 3.750 0.750 x 10 = 7.500 0.500 x 10 = 5.000

  12. Octal and Hexadecimal Numbers • The octal number system: Base-8 • Eight digits: 0,1,2,3,4,5,6,7 • The hexadecimal number system: Base-16 • Sixteen digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F • For our purposes, base-8 and base-16 are most useful as a “shorthand”notation for binary numbers

  13. Numbers with Different Bases You can convert between base-10 base-8 and base-16 using techniques like the ones we just showed for converting between decimal and binary

  14. Octal Binary Octal Binary 0 000 4 100 1 001 5 101 2 010 6 110 3 011 7 111 Binary and Octal Conversions • Converting from octal to binary: Replace each octal digit with its equivalent 3-bit binary sequence • 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. = 673 . 12 = 110111011 . 001010 = 10110100.0010112 = 010110100. 0010112 = 264 . 138

  15. Hex Binary Hex Binary Hex Binary Hex Binary 0 0000 4 0100 8 1000 C 1100 1 0001 5 0101 9 1001 D 1101 2 0010 6 0110 A 1010 E 1110 3 0011 7 0111 B 1011 F 1111 Binary and Hex Conversions • Converting from hex to binary: Replace each hexdigit with its equivalent 4-bit binary sequence • Converting from binary to hex:Make groups of 4 bits, starting fromthe binary point. Add 0s to the ends of the number if needed. Convert each bit group to its corresponding hex digit 261.3516 = 261 . 3516 = 001001100001 . 001101012 10110100.0010112 = 10110100 . 001011002 = B4 . 2C16

  16. Unsigned Binary Coded Decimal (BCD)

  17. Number Systems Summary • Computers are binary devices • We’re forced to think in terms of base 2. • We learned how to convert numbers between binary, decimal, octaland hexadecimal • We’ve already seen some of the recurring themes of architecture: • We use 0 and 1 as abstractions for analog voltages. • We showed how to represent numbers using just these two signals.

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