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Number Systems

Number Systems. Ron Christensen CIS 121. Positional Notation. “Positional Notation” Value of a digit depends on the position of the digit Positional Notation permits unique representation of Integers. Positional Notation. Decimal numbers are “Base 10 positional notation”

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Number Systems

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  1. Number Systems Ron Christensen CIS 121

  2. Positional Notation • “Positional Notation” • Value of a digit depends on the position of the digit • Positional Notation permits unique representation of Integers

  3. Positional Notation • Decimal numbers are “Base 10 positional notation” • Base 10 means that the value of any position is a multiple of a power of 10 • Binary numbers are “Base 2 Positional notation” • Octal numbers are “Base 8 Positional notation” • Hexadecimal numbers are “Base 16”

  4. Positional Notation • The value of digit n is baseN • N0 = 1 no matter what N is • In decimal notation the value of digit N is 10N • “19” = 1x101 + 9x100

  5. Base 10 • Base 10 uses ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. • This is called a decimal number system and is the one with which people are most familiar.

  6. Why Binary? • Build Transistor

  7. Base 2 • Base 2 has exactly two symbols: 0 and 1. • This is called a binary system and all numbers must be formed using these two symbols.

  8. Base 8 • Base 8 uses eight symbols: 0, 1, 2, 3, 4, 5, 6, and 7. • This is called an octal number system and is a convenient shorthand for base 2 numbers. • One octal digit is the equivalent of three binary digits.

  9. Base 16 • Base 16 uses sixteen symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. • This system is called hexadecimal and can also serve as a shorthand for the binary number system.

  10. Converting Number Systems • It is sometimes necessary or more convenient to convert numbers in one system to those in another.

  11. Converting Base 2(Binary) to Base 10 (Decimal) • In binary notation the value of digit N is 2N • “1011” = 1x24 + 0x23 + 1x21 + 1x20 = 8+0+2+1 = 11 • What is the decimal value of 110110001? • 1x28 + 1x27 + 1x25 + 1x24 + 1x20 • = 256 + 128 + 32 + 16 + 1 = 433

  12. Converting Base 2to Base 8 • To convert base 2 to base 8, combine the binary digits from the right in groups of three and convert. 111101001011 111 101 001 011 7 5 1 3

  13. Converting Base 2to Base 16 • To convert base 2 to base 16, combine the binary digits from the right in groups of four and translate. 111101001011 1111 0100 1011 F 4 B

  14. Converting Base 8to Base 2 • To convert base 8 to base 2, convert each octal digit into 3 binary digits. 7513 111 101 001 011 111101001011

  15. Converting Base 8to Base 16 • To convert base 8 to base 16, first translate into base 2 then into base 16. 7513 111 101 001 011 1111 0100 1011 F 4 B

  16. Converting Base 10to Base 2 • To convert base 10 to base 2, divide the number repeatedly by 2; use remainders as answer. On your own, try: 2363.

  17. Practice • What is the binary value of 120? • First division is 120/2. Q=60, R=0 • Second division is 60/2. Q=30, R=0 • Third division is 30/2. Q=15, R=0, • Fourth division is 15/2. Q=7, R=1 • Fifth division is 7/ 2. Q=3, R=1 • Sixth division is 3/2. Q=1, R=1 • Seventh division = 1 / 2. Q=0, R=1 • Quotient is zero, so we stop. • 12010 = 11110002

  18. Converting Base 10to Base 8 • To convert base 10 to base 8, divide the number repeatedly by 8; use remainders as answer. On your own, try: 2363.

  19. Converting Base 10 to Base 16 • To convert base 10 to base 16, divide the number repeatedly by 16; use remainders as answer. On your own, try: 2363.

  20. Converting Base 16to Base 2 • To convert base 16 to base 2, translate each hexadecimal digit to 4 binary digits. On your own, try: 4BC.

  21. Converting Base 16to Base 8 • To convert base 16 to base 8, translate to base 2 then to base 8. On your own, try: 4BC.

  22. Converting Base 16to Base 10 • To convert base 16 to base 10, expand the number and translate base 16 digits to base 10. On your own, try: 4BC.

  23. Practice • The number “123DF” in decimal is • 1x164 + 2x163 + 3x162 + 13x161 + 15x160 • 65536 + 8192 + 768 + 208 + 15 = 74719

  24. Number Systems • Binary is used by hardware • Hex numbers are used in many places • HTML color codes • Decimal numbers appear in user interfaces Conversions are done with calculators

  25. Binary Encoding – Data • Binary numbers work well for positive integers • Not all data is positive integer: • Negative integers • Real numbers, e.g. decimal points • Characters

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