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Agenda

Agenda. Shortcuts converting among numbering systems Binary to Hex / Hex to Binary Binary to Octal / Octal to Binary Signed and unsigned binary numbers Addition / Subtraction of Binary Numbers Two’s Complement. Numbering System Shortcuts.

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Agenda

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  1. Agenda • Shortcuts converting among numbering systems • Binary to Hex / Hex to Binary • Binary to Octal / Octal to Binary • Signed and unsigned binary numbers • Addition / Subtraction of Binary Numbers • Two’s Complement

  2. Numbering System Shortcuts It is very simple to convert binary numbers to octal or hexadecimal numbers since 8 is 2^3, and 16 is 2^4 In other words: • 1 Octal digit = 3 binary digits • 1 Hex digit - 4 binary digits

  3. Binary to Octal Notice the Pattern: • Largest 3 digit binary is 111 • 1 octal digit will represent a 3 digit binary number • Highest Octal digit is 7 • Therefore: 1112 = 78

  4. Binary to Octal Relationship: Octal Binary 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 Does this table look familiar?

  5. Practical Example • applying octal values of rwx the chmod command (e.g., chmod 751). • chmod 777 • chmod 755 • chmod 711 • chmod 644

  6. Binary to Hexadecimal Notice the Pattern: • Largest 4 digit binary is 1111 • 1 hex digit will represent a 4 digit binary number • Highest hex digit is F • Therefore: 11112 = F16

  7. Binary to Hexadecimal Relationship: Hexadecimal Binary Hexadecimal Binary 0 0000 8 1000 1 0001 9 1001 2 0010 A 1010 3 0011 B 1011 4 0100 C 1100 5 0101 D 1101 6 0110 E 1110 7 0111 F 1111

  8. Convert Hex to Binary Steps: • Convert Hex number to groups of powers of 2. • Convert to Binary number (Remember to drop leading zeros for first set of binary numbers - i.e. first left set)

  9. Convert Hex to Binary 11F616 • 1 1 F 6 • 000(1) 000(1) (8)(4)(2)(1) 0(4)(2)0 • 1 0001 1111 0110 • 10001111101102 Drop Leading zeros 000

  10. Convert Binary to Hex Steps: • Separate into 4 bit groups starting from the right. • Calculate decimal equivalent (in placeholders in powers of 2) • Convert to Hexadecimal number

  11. Convert Binary to Hex 10001111101102 • 1 0001 1111 0110 • 0001 0001 1111 0110 • 1 1 (8)(4)(2)(1) 0(4)(2)0 • 1 1 15 6 • 11F616

  12. Converting Octal to Hexadecimal • The easiest method to convert between Octal and Hexadecimal is to convert to binary as an intermediate step • Regroup binary in groups of 4 for hexadecimal and 3 for octal

  13. Storing Numbers • Numeric information (stored as a non negative number) is often store in a computer in binary. Eg. • 1 byte (0 - 255 numbers) • 2 bytes (0 - 65535) • 4 bytes (0-4294967295)

  14. Data Formats Unsigned Binary • Data stored as a binary number, with no way to express a negative quantity

  15. Data Formats Signed Binary • Data stored as a binary number, but using a leading zero to represent a positive number, and the two’s complement of a binary number for a negative number

  16. Adding / Subtracting Binary Numbers • Addition: 0 + 0 = 0 0 + 1 = 1, 1 + 0 = 1 1 + 1+ = 10 • Subtraction: 0 - 0 = 0 1 - 1 = 0, 1 - 0 = 1 0 - 1 (Must borrow from next placeholder) Therefore 10 - 1 = 1

  17. Two’s Complement • Simple method of subtracting two binary numbers by adding. • Two Complement • Flip binary numbers (0 becomes 1, visa versa) • then add 1 • Result becomes negative • Therefore, short-hand method of representing negative integers

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