Number Systems

Number Systems Table for Use in Simple Conversions 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 Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F Conversions between Octal, Binary & Hexadecimal Table Lookup 7658 == 111 110 101 == 0001 1111 0101 == 1F516 111 110 101 0001 1111 0101 groups of three groups of four Conversion of Base 10 to Any Other Base Successive Division of the Base 10 Number by the Base Number of the Target Base Collecting the Remainders in Reverse Order to Form the Target Base Number, e.g., 76810 = _____8 8|768 8|96 8|12 14 76810 = 0 0 14008 Restatement: Conversion of Decimal Number to Any Base n, i.e., Successive Divisions of the Decimal Number by n, preserving the remainders 6510 = X5 5 | 65 5 |13 2 0 3 6510 = 2305 Conversion of Any Base to Base 10 Polynomial Expansion of the Number, i.e., Multiply the Coefficient by the Base Raised to the Power of the Exponent, e.g., 14008 = _______10 14008 = 1*83 + 4*82 + 0*81 + 0*80 = 83 + 4*82 = 82 * (8 + 4) = 64 * 12 = 76810 3210 Indexes Base == 8 Conversion of Any Base n Number to a Decimal (Base 10) Number Polynomial Expansion 2305 = 2 3 0 5 = 2*52 + 3*51 +0*50 = 50 + 15 + 0 = 6510 210 Coefficient * BaseIndex + Coefficient * BaseIndex + Coefficient * BaseIndex + … Addition Base n (1) dump the bucket when it has n stones in it; (2) add one stone to the bucket on the left 1 1 n-1 n-1 Subtraction “Take Away” When bucket is empty for Base n (1) remove one stone from the bucket on the left (2) place n stones in the bucket that was empty 1 1 n Primitive Symbols 1, 2, 3, … , A, B, C, … , etc. Composite Symbols 143, AC9, 1011, 75, etc. n