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Numbering Systems

Numbering Systems. Decimal (Denary) base 10. Clumsy when dealing with computers. Other systems Binary Octal Hexadecimal Convenient when dealing with computers. Decimal. Familiar system. Can use are fingers to help. Count 0 – 9 in any column, gives 10 separate values.

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Numbering Systems

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  1. Numbering Systems • Decimal (Denary) base 10. • Clumsy when dealing with computers. • Other systems • Binary • Octal • Hexadecimal • Convenient when dealing with computers

  2. Decimal • Familiar system. • Can use are fingers to help. • Count 0 – 9 in any column, gives 10 separate values. • When a count exceeds 9 we start a new column. • Example: • 9 + 1 = 10 • 99 + 1 =100 • Mathematically each column represents a power of ten (Base 10).

  3. Decimal • 45610= 4 x102 + 5 x101 + 6 x 100 • We call: • 100 Units • 101 Tens • 102 Hundreds (x 10) • ………. And so on.

  4. Binary • Computers don’t have ten fingers. • They work in bits • A bit can only have 2 values • 0 and 1 • So when we exceed 1 we must start a new column as we did in decimal.

  5. Binary • Examples : • 12 + 12 = 102 • 02 + 12 = 12 • 101112 + 112 = 110102 • Binary works in powers of 2. (x 2) • Columns are 128 64 32 16 8 4 2 1 • 1012 = 1 x 22 + 0 x 21 + 1 x 20 • 22 = 410 • 21 = 210 • 20 = 110 • Binary number 1012 is therefore 510 in decimal.

  6. Octal • For large numbers Binary is cumbersome to humans. • Is there a compromise? • It groups 3 binary columns together. • One is octal. • Octal has 8 values 0 – 7.

  7. Octal • When we reach 7 we start a new column. • Examples : • 178 + 18 = 208 • 7778 + 18 = 10008 • Octal works in powers of 8. (x 8) • Columns are … 512 64 8 1 • 1118 = 1 x 82 + 1 x 81 + 1 x 80 • 82 = 64 • 81 = 8 • 80 = 1 • Octal number 1118 is therefore 7310 in decimal.

  8. Hexadecimal • For large numbers Binary is cumbersome to humans. • Is there a better compromise? • It groups 4 binary columns together. • Hexadecimal has 16 values 0 – F. (0 –15 in decimal) • We need to add 6 more symbols to represent the 2 column numbers 10 –15. We use A – F.

  9. Hexadecimal • When we reach F we start a new column. • Examples : • 1F16 + 116 = 2016 • FFF16 + 116 = 100016 • Hexadecimal works in powers of 16. (x 16) • Columns are … 4096 256 16 1 • 111 = 1 x 162 + 1 x 161 + 1 x 160 • 162 = 256 • 161 = 16 • 160 = 1 • Hexadecimal number 11116 is therefore 27310 in decimal.

  10. How does Hexadecimal help? • Consider the 4 bit (column) binary number • 11112 = 1510 = F16 • That is, the largest number that can be represented by 4 bits in binary can be represented by a single value F16. • All the values in between are then 0 – F. • Octal does the same for 3 bit binary.

  11. How does Hexadecimal help? • We can divide long binary numbers into groups of 4 and convert them directly in binary. • Example the 16 bit number: • 0010 1111 0001 0100 2 is 2F1416. • Computers deal in multiples of 8 (and hence 4 bits).

  12. Convertions • Binary, octal, Hex(adecimal) to decimal: • Add up powers as we did earlier. • Examples 1118 = 1 x 82 + 1 x 81 + 1 x 80 = 6410 + 810 + 110 = 73. 11116 = 1 x 162 + 1 x 161 + 1 x 160 = 25610 + 1610 + 110

  13. Convertions • Decimal to binary. • Find the largest power of 2 (column in binary number) that is less than or equal to the number we want to convert. • Mark a one in that column. • Subtract the power of 2 from the number we want to convert. • Repeat the above on the remainder. • Repeat until there is no remainder. • Fill missing columns with zeros.

  14. Convertions • Decimal to binary. • Example • Convert 67 to binary: 128 64 32 16 8 4 2 1 1 64 is largest column less than or equal to 67 leaving remainder 67 – 64 = 3 128 64 32 16 8 4 2 1 1 1 2 is largest column less than or equal to the remainder 3 . Leaving 1. 1 is largest column less than or equal to remainder 1. Leaving no remainder. 128 64 32 16 8 4 2 1 1 1 1 Fill with zeros. So 6710 is0100 00112

  15. Convertions • Decimal to hexadecimal. • Find the binary number and separate into groups of four. Express each group of 4 in hexadecimal. • Example 6710 is 0100 00112 is 4316. • Find the largest multiple of each power of 16, less than or equal than your number. • Mark the multiple in that column • Repeat for the remainders as before..

  16. Convertions • Decimal to Hexadecimal. • Example • Convert 67 to hexadecimal: 4096 256 16 1 4 4 x 16 = 64 is largest multiple less or equal to 67. Remainder 67 – 64 = 3 4096 256 16 1 4 3 3 x 1 is largest multiple less than or equal to the remainder 3. Leaving no remainder. Fill with zeros. (not necessary here) So 6710 is 4316

  17. Convertions • Hexadecimal to binary. • Take each hex symbol in turn. • Convert (or look up) to decimal. • Convert decimal number to binary. • Replace hex symbol with 4 bit binary number. • Sounds tedious, but only 6 different from decimal so becomes easy.

  18. Convertions • Hexadecimal to binary. • Example: • Convert 2F to binary. • Take the 2. • 2 is 2 in decimal. • Convert to binary or look up • Gives 0010 • Take the F • F is 15 in decimal • Convert to binary or look up • Gives 1111 • So 2F in binary is 0010 1111

  19. Convertion Table • Hexadecimal Decimal Binary. 0 0 0000 1 1 0001 2 2 0010 3 3 0011 4 4 0100 5 5 0101 6 6 0110 7 7 0111

  20. Convertion Table • Hexadecimal Decimal Binary. 8 8 1000 9 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111

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