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Number Systems

Number Systems. Today. Decimal Hexadecimal Binary Unsigned Binary 1’s Complement Binary 2’s Complement Binary. 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 ).

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Number Systems

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  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: 100 = 1101 = 10102 = 100103 = 1000104 = 10000105 = 100000 . . . . Example: Decimal 1024 = (1024)10 Binary: 20 = 121 = 222 = 423 = 824 = 1625 = 3226 = 6427 = 12828 = 25629 = 512210 = 1024 = 1 Kb220 = 1 Mb230 = 1 Gb240 = 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 < base --> 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 < base --> 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 Numbers1 <-- 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

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