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Dale & Lewis Chapter 2 Binary Numbers and Number Systems

Dale & Lewis Chapter 2 Binary Numbers and Number Systems. Number categories. Definition of Number Unit of a mathematical system subject to arithmetic Natural numbers 0, 1, 2, 3, 4… Negative numbers Number less then 0, i.e. -1, -2, -3, -4… Integers …-4, -3, -2, -1, 0, 1, 2, 3, 4…

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Dale & Lewis Chapter 2 Binary Numbers and Number Systems

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  1. Dale & Lewis Chapter 2Binary Numbers and Number Systems

  2. Number categories • Definition of • Number • Unit of a mathematical system subject to arithmetic • Natural numbers • 0, 1, 2, 3, 4… • Negative numbers • Number less then 0, i.e. -1, -2, -3, -4… • Integers • …-4, -3, -2, -1, 0, 1, 2, 3, 4… • Rational numbers • Fraction of integers, except division by 0, i.e. ¼, ½, -7/13…

  3. The idea of a positional number system 4357 four thousand, three hundred and fifty seven four units of a thousand (4x1000) 4000 three units of a hundred (3x100) 300 five units of ten (5x10) 50 seven units of one (7x1) 7 4 x103 + 3 x 102 + 5 x 101 + 7 x 100 5743 same digits, different positions, different number The position of each digit determines that digit’s contribution to the number.

  4. The idea of a positional number system base: b any integer > 1 digits: 0, 1, ..., b−1 number: dn-1dn-2…d2d1d0 its definition dn-1 x bn-1 + dn-2 x bn-2 + … + d2 x b2+ d1 x b1 + d0 x b0 Examples: Base Digits 2 0, 1 5 0, 1, 2, 3, 4 8 0, 1, 2, 3, 4, 5, 6, 7 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

  5. Binary, Octal and HexadecimalComparison of first natural numbers in four different bases etc...

  6. …So what base was used in the data in the Matrix movies?

  7. Arithmetic in other basesAll the familiar rules of pencil-and-paper decimal arithmetic carry over to any other base Decimal Grid Addition 46 101110 + 27 + 11011 Subtraction 5037 - 95 57 111001 - 6 - 110 Binary Grid

  8. Octal Grid Hexadecimal Grid

  9. Examples of arithmetic In Octal: 3754 76312 +6317 + 4634 In Hexadecimal: 3B6F 89FCD +5743 + 9FD In Decimal: 123 + 45 +3682 + 12 + 654

  10. Conversion between Decimal and Binary • Binary to Decimal • Use the definition of a number in a positional number system with base 2 • Evaluate the definition formula using decimal arithmetic • Example 101011 = 1 x 25 + 0 x 24+ 1 x 23+ 0x 22+ 1 x 21+ 1 x 20 = 43 (decimal)

  11. Conversion between Decimal and Binary • Decimal to Binary • Repeatedly divide by 2 • Quotient Carries • Remainder is the next digit • Binary number is developed right to left • Example (Quotient) (Remainder) (Binary) 173 ÷ 2 86 1 1 86 ÷ 2 430 01 43 ÷ 2 211 101 21 ÷ 2 101 1101 10 ÷ 2 5 0 01101 5 ÷ 2 21 101101 2 ÷ 2 1 0 0101101 1 ÷ 2 01 10101101

  12. Generalization: Conversion between Decimal and base b • Base b to Decimal • Use the definition of a number in a positional number system with base b • Evaluate the definition formula using decimal arithmetic • Decimal to base b • Repeatedly divide by b • Quotient carries • Remainder is the next digit • Base b number is developed right to left

  13. Conversion between Binary and Octal/Hexadecimal • Binary to Octal • Group bits into threes, right to left • Convert each group into an octal digit • Example 1011010111 = 001 011 010 111 = 1327 (octal) • Binary to Hexadecimal • Group bits into fours, right to left • Convert each group into a hexadecimal digit • Example 1011001011 = 0010 1100 1011 = 2CB (hexadecimal)

  14. Conversion between Binary and Octal/Hexadecimal • Octal to Binary • Convert each digit to a three-bit binary representation • Example 705 = 111 000 101 = 111000101 (binary) • Hexadecimal to Binary • Convert each digit to a four-bit binary representation • Example 10AF = 0001 0000 1010 1111 = 0001000010101111 (binary) Conversion tables can be reconstructed in the margins of a test paper when needed

  15. What about converting between Octal and Hexadecimal? • Hexadecimal is not base 8, so grouping won’t work as in binary  octal or binary  hexadecimal • Octal to Hexadecimal • Convert octal to binary and then binary to hexadecimal • Hexadecimal to Octal • Convert hexadecimal to binary and then to octal

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