thusthe total range of word codes is from to unfortunatelythere are , words in the dictionaryso there aren' enough index numbers to go around each array element will need to hold about words ( , divided by clearlythis presents problems if we're thinking in terms of our one word-per-array element scheme maybe we could put subarray or linked list of words at each array element unfortunatelysuch an approach would seriously degrade the access speed accessing the array element would be quickbut searching through the words to find the one we wanted would be slow so our first attempt at converting words to numbers leaves something to be desired too many words have the same index (for examplewastingivetendmoantickbailsdredgeand hundreds of other words add to as cats does we conclude that this approach doesn' discriminate enoughso the resulting array has too few elements we need to spread out the range of possible indices multiplying by powers let' try different way to map words to numbers if our array was too small beforelet' make sure it' big enough what would happen if we created an array in which every wordin fact every potential wordfrom to zzzzzzzzzzwas guaranteed to occupy its own unique array elementto do thiswe need to be sure that every character in word contributes in unique way to the final number we'll begin by thinking about an analogous situation with numbers instead of words recall that in an ordinary multi-digit numbereach digit-position represents value times as big as the position to its right thus , really means * * * * orwriting the multipliers as powers of * * * * (an input routine in computer program performs similar series of multiplications and additions to convert sequence of digitsentered at the keyboardinto number stored in memory in this system we break number into its digitsmultiply them by appropriate powers of (because there are possible digits)and add the products in similar way we can decompose word into its lettersconvert the letters to their numerical equivalentsmultiply them by appropriate powers of (because there are possible charactersincluding the blank)and add the results this gives unique number for every word