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Numbering System

Numbering System. Base Conversion. Number systems. Decimal – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Binary – 0, 1 Octal – 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal system – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Why different number systems?.

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Numbering System

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  1. Numbering System Base Conversion

  2. Number systems • Decimal – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Binary – 0, 1 • Octal – 0, 1, 2, 3, 4, 5, 6, 7 • Hexadecimal system – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

  3. Why different number systems? • Binary number result in quite a long string of 0s and 1s • Easier for the computer to interpret input from the user

  4. Base Conversion • In daily life, we use decimal (base 10) number system • Computer can only read in 0 and 1 • Number system being used inside a computer is binary (base 2) • Octal (base 8) and hexadecimal (base 16) are used in programming for convenience

  5. Base Conversion • Conversion • Binary number, • Octal number, • Hexadecimal number, and • Decimal number.

  6. Base Conversion

  7. Base Conversion For example: 62 = 111110 = 76 = 3E decimal binary octal hexadecimal 1 For Decimal: 62 = 6x101 + 2x100 2 For Binary: 111110 = 1x25 + 1x24 + 1x23 + 1x22 + 1x21 + 0x20 3 For Octal: 76 = 7x81+ 6x80 4 For Hexadecimal: 3E = 3x161 + 14x160 • Since for hexadecimal system, each digit contains number from 1 to 15, thus we use A, B, C, D, E and F to represent 10, 11, 12, 13, 14 and 15.

  8. Binary and decimal system • Binary to decimal • X . 27 + X . 26+ X . 25+ X . 24 + X . 23+ X . 22+ X . 21 + X . 20 • Decimal to binary • Keep dividing the number by two and keep track of the remainders. • Arrange the remainders (0 or 1) from the least significant (right) to most significant (left) digits

  9. Octal and Hexadecimal system • Binary to Octal (8 = 23) • Every 3 binary digit equivalent to one octal digit • Binary to Hexadecimal (16 = 24) • Every 4 binary digit equivalent to one hexadecimal digit • Octal to binary • Every one octal digit equivalent to 3 binary digit • Hexadecimal to binary • Every one hexadecimal digit equivalent to 4 binary digits

  10. Base Conversion • How to convert the decimal number to other number system • e.g. convert 1810 in binary form 2 |18 ----0 2 |09 ----1 2 |04 ----0 2 |02 ----0 1 • 1810 = 100102

  11. Base Conversion • e.g. convert 1810 in octal form • Since for octal form, one digit is equal to 3 digits in binary number, we can change binary number to octal number easily. e.g. 10010 = 010010 2 2 Thus, 100102 = 228

  12. Base Conversion • e.g. convert 1810 in hexadecimal form • Similarly, for hexadecimal form, one digit is equal to 4 digits in binary number. e.g. 10010 = 00010010 1 2 Thus, 100102 = 1216

  13. Numbering System Addition & Subtraction

  14. Decimal Addition • What is going on? • 1 1 1 (carry) • 3 7 5 8 • + 4 6 5 7 • 14 11 15 • 10 10 10 (subtract the base) • 8 4 1 5 111 3758 + 4657 8415

  15. Binary Addition Rules. • 0 + 0 = 0 • 0 + 1 = 1 • 1 + 0 = 1 • 1 + 1 = 2 = 102 = 0 with 1 to carry • 1 + 1 + 1 = 3 = 112 = 1 with 1 to carry

  16. Binary Addition • Verification • 5510 • + 2810 • 8310 • 64 32 16 8 4 2 1 • 1 0 1 0 0 1 1 • = 64 + 16 + 2 +1 • = 8310 1 1 1 1 1 1 0 1 1 1 + 0 1 1 1 0 0 2 3 2 2 - 2 2 2 2 1 0 1 0 0 1 1

  17. Binary Addition ex Verification 1 0 0 1 1 1 + 0 1 0 1 1 0 + ___ ___________ 128 64 32 16 8 4 2 1 = =

  18. Octal Addition 1 1 6 4 3 78 + 2 5 1 08 9 9 - 8 8 (subtract Base (8)) 1 1 1 4 78

  19. Octal Addition ex 3 5 3 68 + 2 4 5 78 - (subtract Base (8))

  20. Hexadecimal Addition 1 1 7 C 3 916 + 3 7 F 216 20 18 11 - 16 16 (subtract Base (16)) B 4 2 B16

  21. Hexadecimal Addition 8 A D 416 + 5 D 616 - (subtract Base (16)) 16

  22. Decimal Subtraction 7 13 10 8 4 1 15 - 4 6 5 7 3 7 5 8 • How it was done? • ( add the base 10 when borrowing) • 1010 • 73010 • 8 41 5131015 • - 4 6 5 7 • 3 7 5 8

  23. Binary Subtraction • Verification • 8310 • - 2810 • 5510 • 64 32 16 8 4 2 1 • 1 1 0 1 1 1 • = 32 + 16 + + 4 + 2 +1 • = 5510 1 2 1 02 0 2 2 1 0 1 0 0 1 1 - 0 1 1 1 0 0 1 1 0 1 1 1

  24. Binary Subtraction ex Verification 1 0 0 1 1 1 - 0 1 0 1 1 0 - ___ ___________ 128 64 32 16 8 4 2 1 = =

  25. Octal Subtraction 8 0 0 8 1 11 4 78 89 - 6 4 3 78 2 5 1 08

  26. Octal Subtraction ex 3 5 3 68 - 2 4 5 78

  27. Hexadecimal Subtraction B 16 7 C 3 916 19 - 3 7 F 216 4 4 4 716

  28. Hexadecimal Subtraction 8 A D 416 - 5 D 616 16

  29. Let’s do some exercises! Octal, Hexadecimal, Binary Addition & Subtraction

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