Counting: Algorithms

An algorithm is a precise step by step method to solve a problem.

A pseudo-code algorithm is a way to describe an algorithm that is not in a form that can be directly executed by a computer.

Here is a simple pseudo-code algorithm.

get a blank piece of paper FOR EACH decimal number x from 0 to 255 DO start a new line write the number in decimal write the number in hexadecimal: divide x by 16 write the quotient in hex write the remainder in hex write the number in binary: convert the first hex digit to binary convert the second hex digit to binary END

Besides being fundamental to computers, another goal of learning base conversion is to learn to appreciate the concept of an algorithm and the use of programming to automate tasks.

Programs = Data structures + Algorithms

Programs = Data + Code

program numbers; const digits: array[0..15] of char = '0123456789ABCDEF'; binary: array[0..15] of string[4] = ( '0000','0001','0010','0011', '0100','0101','0110','0111', '1000','1001','1010','1011', '1100','1101','1110','1111'); var i,h0,h1: word;

begin writeln; write(' Dec Hex Bin'); for i:= 0 to 255 do begin writeln; write(' ',i:4,'d'); h1:= i div 16; h0:= i mod 16; write(' ',digits[h1],digits[h0],'h'); write(' ',binary[h1],binary[h0],'b'); end; end.

The first 17 lines of output:

Dec Hex Bin 0d 00h 00000000b 1d 01h 00000001b 2d 02h 00000010b 3d 03h 00000011b 4d 04h 00000100b 5d 05h 00000101b 6d 06h 00000110b 7d 07h 00000111b 8d 08h 00001000b 9d 09h 00001001b 10d 0Ah 00001010b 11d 0Bh 00001011b 12d 0Ch 00001100b 13d 0Dh 00001101b 14d 0Eh 00001110b 15d 0Fh 00001111b 16d 10h 00010000b

The last 8 lines of output:

248d F8h 11111000b 249d F9h 11111001b 250d FAh 11111010b 251d FBh 11111011b 252d FCh 11111100b 253d FDh 11111101b 254d FEh 11111110b 255d FFh 11111111b

For those interested, here is the entire list.

Dec Hex Bin 0d 00h 00000000b 1d 01h 00000001b 2d 02h 00000010b 3d 03h 00000011b 4d 04h 00000100b 5d 05h 00000101b 6d 06h 00000110b 7d 07h 00000111b 8d 08h 00001000b 9d 09h 00001001b 10d 0Ah 00001010b 11d 0Bh 00001011b 12d 0Ch 00001100b 13d 0Dh 00001101b 14d 0Eh 00001110b 15d 0Fh 00001111b 16d 10h 00010000b 17d 11h 00010001b 18d 12h 00010010b 19d 13h 00010011b 20d 14h 00010100b 21d 15h 00010101b 22d 16h 00010110b 23d 17h 00010111b 24d 18h 00011000b 25d 19h 00011001b 26d 1Ah 00011010b 27d 1Bh 00011011b 28d 1Ch 00011100b 29d 1Dh 00011101b 30d 1Eh 00011110b 31d 1Fh 00011111b 32d 20h 00100000b 33d 21h 00100001b 34d 22h 00100010b 35d 23h 00100011b 36d 24h 00100100b 37d 25h 00100101b 38d 26h 00100110b 39d 27h 00100111b 40d 28h 00101000b 41d 29h 00101001b 42d 2Ah 00101010b 43d 2Bh 00101011b 44d 2Ch 00101100b 45d 2Dh 00101101b 46d 2Eh 00101110b 47d 2Fh 00101111b 48d 30h 00110000b 49d 31h 00110001b 50d 32h 00110010b 51d 33h 00110011b 52d 34h 00110100b 53d 35h 00110101b 54d 36h 00110110b 55d 37h 00110111b 56d 38h 00111000b 57d 39h 00111001b 58d 3Ah 00111010b 59d 3Bh 00111011b 60d 3Ch 00111100b 61d 3Dh 00111101b 62d 3Eh 00111110b 63d 3Fh 00111111b 64d 40h 01000000b 65d 41h 01000001b 66d 42h 01000010b 67d 43h 01000011b 68d 44h 01000100b 69d 45h 01000101b 70d 46h 01000110b 71d 47h 01000111b 72d 48h 01001000b 73d 49h 01001001b 74d 4Ah 01001010b 75d 4Bh 01001011b 76d 4Ch 01001100b 77d 4Dh 01001101b 78d 4Eh 01001110b 79d 4Fh 01001111b 80d 50h 01010000b 81d 51h 01010001b 82d 52h 01010010b 83d 53h 01010011b 84d 54h 01010100b 85d 55h 01010101b 86d 56h 01010110b 87d 57h 01010111b 88d 58h 01011000b 89d 59h 01011001b 90d 5Ah 01011010b 91d 5Bh 01011011b 92d 5Ch 01011100b 93d 5Dh 01011101b 94d 5Eh 01011110b 95d 5Fh 01011111b 96d 60h 01100000b 97d 61h 01100001b 98d 62h 01100010b 99d 63h 01100011b 100d 64h 01100100b 101d 65h 01100101b 102d 66h 01100110b 103d 67h 01100111b 104d 68h 01101000b 105d 69h 01101001b 106d 6Ah 01101010b 107d 6Bh 01101011b 108d 6Ch 01101100b 109d 6Dh 01101101b 110d 6Eh 01101110b 111d 6Fh 01101111b 112d 70h 01110000b 113d 71h 01110001b 114d 72h 01110010b 115d 73h 01110011b 116d 74h 01110100b 117d 75h 01110101b 118d 76h 01110110b 119d 77h 01110111b 120d 78h 01111000b 121d 79h 01111001b 122d 7Ah 01111010b 123d 7Bh 01111011b 124d 7Ch 01111100b 125d 7Dh 01111101b 126d 7Eh 01111110b 127d 7Fh 01111111b 128d 80h 10000000b 129d 81h 10000001b 130d 82h 10000010b 131d 83h 10000011b 132d 84h 10000100b 133d 85h 10000101b 134d 86h 10000110b 135d 87h 10000111b 136d 88h 10001000b 137d 89h 10001001b 138d 8Ah 10001010b 139d 8Bh 10001011b 140d 8Ch 10001100b 141d 8Dh 10001101b 142d 8Eh 10001110b 143d 8Fh 10001111b 144d 90h 10010000b 145d 91h 10010001b 146d 92h 10010010b 147d 93h 10010011b 148d 94h 10010100b 149d 95h 10010101b 150d 96h 10010110b 151d 97h 10010111b 152d 98h 10011000b 153d 99h 10011001b 154d 9Ah 10011010b 155d 9Bh 10011011b 156d 9Ch 10011100b 157d 9Dh 10011101b 158d 9Eh 10011110b 159d 9Fh 10011111b 160d A0h 10100000b 161d A1h 10100001b 162d A2h 10100010b 163d A3h 10100011b 164d A4h 10100100b 165d A5h 10100101b 166d A6h 10100110b 167d A7h 10100111b 168d A8h 10101000b 169d A9h 10101001b 170d AAh 10101010b 171d ABh 10101011b 172d ACh 10101100b 173d ADh 10101101b 174d AEh 10101110b 175d AFh 10101111b 176d B0h 10110000b 177d B1h 10110001b 178d B2h 10110010b 179d B3h 10110011b 180d B4h 10110100b 181d B5h 10110101b 182d B6h 10110110b 183d B7h 10110111b 184d B8h 10111000b 185d B9h 10111001b 186d BAh 10111010b 187d BBh 10111011b 188d BCh 10111100b 189d BDh 10111101b 190d BEh 10111110b 191d BFh 10111111b 192d C0h 11000000b 193d C1h 11000001b 194d C2h 11000010b 195d C3h 11000011b 196d C4h 11000100b 197d C5h 11000101b 198d C6h 11000110b 199d C7h 11000111b 200d C8h 11001000b 201d C9h 11001001b 202d CAh 11001010b 203d CBh 11001011b 204d CCh 11001100b 205d CDh 11001101b 206d CEh 11001110b 207d CFh 11001111b 208d D0h 11010000b 209d D1h 11010001b 210d D2h 11010010b 211d D3h 11010011b 212d D4h 11010100b 213d D5h 11010101b 214d D6h 11010110b 215d D7h 11010111b 216d D8h 11011000b 217d D9h 11011001b 218d DAh 11011010b 219d DBh 11011011b 220d DCh 11011100b 221d DDh 11011101b 222d DEh 11011110b 223d DFh 11011111b 224d E0h 11100000b 225d E1h 11100001b 226d E2h 11100010b 227d E3h 11100011b 228d E4h 11100100b 229d E5h 11100101b 230d E6h 11100110b 231d E7h 11100111b 232d E8h 11101000b 233d E9h 11101001b 234d EAh 11101010b 235d EBh 11101011b 236d ECh 11101100b 237d EDh 11101101b 238d EEh 11101110b 239d EFh 11101111b 240d F0h 11110000b 241d F1h 11110001b 242d F2h 11110010b 243d F3h 11110011b 244d F4h 11110100b 245d F5h 11110101b 246d F6h 11110110b 247d F7h 11110111b 248d F8h 11111000b 249d F9h 11111001b 250d FAh 11111010b 251d FBh 11111011b 252d FCh 11111100b 253d FDh 11111101b 254d FEh 11111110b 255d FFh 11111111b