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Understanding Number Systems and Conversions: A Comprehensive Overview

Explore decimal, binary, hexadecimal, and octal systems along with conversion methods. From place value to electronic translators, grasp the foundation of numeric representation.

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Understanding Number Systems and Conversions: A Comprehensive Overview

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  1. Chapter 1 Number System RGGP, Narwana

  2. Counting in Decimal and Binary Place Value Binary to Decimal Conversion Decimal to Binary Conversion Electronic Translators Hexadecimal Numbers Octal Numbers CHAPTER 1 PREVIEW

  3. COUNTING IN DECIMAL AND BINARY • Number System - Code using symbols that refer to a number of items. • Decimal Number System - Uses ten symbols (base 10 system) • Binary System - Uses two symbols (base 2 system)

  4. Introduction to Numbering Systems • We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: • Binary  Base 2 • Octal  Base 8 • Hexadecimal  Base 16

  5. PLACE VALUE • Numeric value of symbols in different positions. • Example - Place value in binary system: 2s Place Value 8s 4s 1s Binary Yes Yes No No Number 1 1 0 0 RESULT: Binary 1100 = decimal 8 + 4 + 0 + 0 = decimal 12

  6. BINARY TO DECIMAL CONVERSION Convert Binary Number 110011 to a Decimal Number: Binary 1 1 0 0 1 1 32 + 16 + 0 + 0 + 2 + 1 = 51 Decimal

  7. TEST Convert the following binary numbers into decimal numbers: 9 Binary 1001 = 15 Binary 1111 = 2 Binary 0010 =

  8. DECIMAL TO BINARY CONVERSION Divide by 2 Process Decimal # 13 ÷ 2 = 6 remainder 1 6 ÷ 2 = 3 remainder 0 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1 1 1 0 1

  9. TEST Convert the following decimal numbers into binary: 1011 Decimal 11 = 0100 Decimal 4 = 10001 Decimal 17 =

  10. HEXADECIMAL NUMBER SYSTEM Uses 16 symbols -Base 16 System 0-9, A, B, C, D, E, F Binary 0001 1001 1010 1111 10000 Hexadecimal 1 9 A F 10 Decimal 1 9 10 15 16

  11. HEXADECIMAL AND BINARY CONVERSIONS • Hexadecimal to Binary Conversion HexadecimalC 3 Binary1100 0011 • Binary to Hexadecimal Conversion Binary 1110 1010 Hexadecimal E A

  12. DECIMAL TO HEXADECIMAL CONVERSION Divide by16Process Decimal # 47÷16 = 2remainder15 2÷16 =0 remainder 2 2 F

  13. HEXADECIMAL TO DECIMAL CONVERSION Convert hexadecimal number 2DB to a decimal number 256s 16s 1s Place Value 2 D B Hexadecimal (256 x 2) (16 x 13) (1 x 11) 512 + 208 + 11 =731 Decimal

  14. TEST Convert Hexadecimal numberA6to Binary A6 = 1010 0110 (Binary) Convert Hexadecimal number16to Decimal 16 = 22 (Decimal) Convert Decimal 63to Hexadecimal 63 = 3F (Hexadecimal)

  15. OCTAL NUMBERS Uses 8 symbols -Base 8 System 0, 1, 2, 3, 4, 5, 6, 7 Decimal 1 6 7 8 9 Binary 001 110 111 001 000 001 001 Octal 1 6 7 10 11

  16. Binary-Coded Decimal (BCD) • Four bits per digit • Note: the following bit patterns are not used: • 1010 1011 1100 1101 1110 1111

  17. Example • 709310 = ? (in BCD) • 7 0 9 3 • 0111 0000 1001 0011

  18. ASCII Features • 7-bit code • 8th bit is unused (or used for a parity bit) • 27 = 128 codes • Two general types of codes: • 95 are “Graphic” codes (displayable on a console) • 33 are “Control” codes (control features of the console or communications channel)

  19. ASCII Chart

  20. Addition (decimal)

  21. Addition (binary)

  22. Addition (binary)

  23. Addition (binary)

  24. Multiplication (decimal)

  25. Multiplication (binary)

  26. Multiplication (binary) • It’s interesting to note that binary multiplication is a sequence of shifts and adds of the first term (depending on the bits in the second term. • 110100 is missing here because the corresponding bit in the second terms is 0.

  27. Methods for representing signed ints. • signed magnitude • 1’s complement (diminished radix complement) • 2’s complement (radix complement)

  28. Signed magnitude • Ex. 4-bit signed magnitude • 1 bit for sign • 3 bits for magnitude

  29. 1’s Complement • 1’s complement of N is defined as (2n -1)-N. • If n=4 have (2n -1) being 1 0000 - 1 = 1111 • So for n=4 would subtract any 4-bit binary number from 1111. • This is just inverting each bit. • Example: 1’s compliment of 1011001 • is 0100110

  30. 2’s complement • The 2’s complement is defined as 2n-N • Can be done by subtraction of N from 2n or adding 1 to the 1’s complement of a number. • For 6 = 0110 • The 1’s complement is 1001 • The 2’s complement is 1010

  31. Operation with 2’s complement • Add 4 and -6 • Will use the 2’s complement of -6 or 1010 • 4 0100 • -6 1010 • 1110 • And taking the 2’s complement of 1110 get 0001 + 1 = 0010

  32. ELECTRONIC TRANSLATORS Devices that convert from decimal to binary numbers and from binary to decimal numbers. Encoders - translates from decimal to binary Decoders - translates from binary to decimal

  33. 3 ELECTRONIC ENCODER - DECIMAL TO BINARY Binary output Decimal input 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 Decimal to Binary Encoder 7 5 0 • Encoders are available in IC form. • This encoder translates from decimal input to binary (BCD) output.

  34. ELECTRONIC DECODING: BINARY TO DECIMAL Binary input Decimal output 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 Binary-to- 7-Segment Decoder/ Driver • Electronic decoders are available in IC form. • This decoder translates from binary to decimal. • Decimals are shown on an 7-segment LED display. • This decoder also drives the 7-segment display.

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