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: 111 2 = 7 8

4. Binary to Octal Relationship: OctalBinary 0000 1001 2010 3011 4100 5101 6110 7111 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: 1111 2 = F 16

7. Binary to Hexadecimal Relationship: HexadecimalBinary 0000081000 1000191001 20010A1010 30011B1011 40100C1100 50101D1101 60110E1110 70111F1111

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 11F6 16 = 1 1 F 6 =000(1) 000(1) (8)(4)(2)(1) 0(4)(2)0 =1 0001 1111 0110 =1000111110110 2 000 Drop Leading zeros

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 1000111110110 2 =1 0001 1111 0110 =0001 0001 1111 0110 =1 1 (8)(4)(2)(1) 0(4)(2)0 =1 1 15 6 =11F6 16

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