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Base ‘b’ number • In general a number system can have any base b • the digit used are 0, 1, … , b-1 • The weight of ith place is bi • The conversion formula from base b into decimal number is i n 1 b x i i 0 i for i = 0 to n – 1 for an n digit quantity • Commonly used base are 2, 3, 8, 10, 16, ... 1 Bases 2, 8, and 16 are related Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Octal 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F 2 Conversion • From binary to octal – make groups of 3 bits from right to left 01 110 1102 1668 • From octal to binary – make each digit as 3 bits sequence 2768 010 111 1102 • From binary to hexadecimal – make groups of 4 bits from right to left 0111 01102 7616 • From hexadecimal to binary – make each digit as 4 bits sequence 3716 0011 01112 3 More on Conversion • Convert from base b1 to decimal • Convert from decimal to base b2 • Direct conversion from base b1 and base b2 – we will not pursue this anymore • Decimal to Hexadecimal – Divide by 16 recursively and collect digit from right to left from the remainders. – Example 347110 = D8F16 • 3471 divided by 16 gives 216, remains 15 (F) • 216 divided by 16 gives 13, remains 8 (8) • 13 divided by 16 gives 0, remain 13 (D) 4