1. Number Systems

2. Today Decimal Hexadecimal Binary –Unsigned Binary –1’s Complement Binary –2’s Complement Binary

3. Decimal (base 10) ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ) Binary (base 2) ( 0, 1 ) Hexadecimal (base 16) ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F )

4. How do we represent numbers? Decimal: 10 0 = 1 10 1 = 10 10 2 = 100 10 3 = 1000 10 4 = 10000 10 5 = 100000.... Example: Decimal 1024 = (1024) 10 Binary: 2 0 = 1 2 1 = 2 2 2 = 4 2 3 = 8 2 4 = 16 2 5 = 32 2 6 = 64 2 7 = 128 2 8 = 256 2 9 = 512 2 10 = 1024 = 1 Kb 2 20 = 1 Mb 2 30 = 1 Gb 2 40 = 1 Tb

5. Notes: Counting Chart ( 99 ) 10 + 1 = ?? ( FF ) 16 + 1 = ?? ( 100 ) 10 ( 100 ) 16

6. Conversion Decimal (Base 10) --> Binary (Base 2) Step 1 -Divide the Number by 2 Step 2 - If the result has a remainder, --> add 1 as the current MSB Otherwise --> add 0 as the current MSB Step 3 -Finish when result Add Quotient as the final MSB Example: Convert (1000) 10 to Binary (base 2)

7. Conversion Decimal (Base 10) -> Hexadecimal (Base 16) Step 1 -Divide the Number by 16 Step 2 - Take the remainder as the current MSB Step 3 -Finish when result Add Quotient as the final MSB Example: Convert (1000) 10 to Hexadecimal (base 16)

8. Bits & Bytes (Side Note) Bit A bit is a single binary digit, a ‘1’ or a ‘0’ Byte A series of 8 bits ( 8 bits = 1 Byte ) Examples:( 1010 1010 ) 2 ( AA ) 16

9. Conversion Binary (Base 2) --> Hexadecimal (Base 16) Step 1 -Make groups of 4 bits, starting from the LSB Step 2 - Directly convert each group into Hexadecimal Example: Convert (1111101000) 2 to Hexadecimal (base 16)

10. Binary Addition Example: Add (10011011) 2 and (1110) 2

11. Signed Binary MSB is the sign bit 0 <-- Positive Numbers 1 <-- Negative Numbers

12. 2’s Complement Binary Example: Convert (-100) 10 into 2’s comp Example: Binary Addition

13. 2’s Complement Binary Why? –Simplifying the implementation of arithmetic on computer hardware. –Allows the addition of negative operands without a subtraction circuit or a circuit that detects the sign of a number. –Moreover, an addition circuit can also perform subtraction by taking the two's complement of a number