For example, remember the famous old story about the reward asked of a sultan by the inventor of the game of chess? He asked for a grain of rice to be placed on the upper left hand square of a chessboard, for two grains to be placed on the next square over, four grains on the third square, and so on -- doubling the quantity of rice grains with each square, for a total of 64 squares. So how much rice did he ask for? The question calls for an appropriate input. You are starting with 20 = 1 grain of rice, then asking the sum of 20 + 21 + 22 + 23 grains, plus . . . on up to 263 grains on the final square. So you want the sum of the terms of the geometric progression 2n, where n ranges from 0 to 63. Mathematica makes this extremely simple to program as a question, and in a fraction of a second you have your answer: it tells you that the inventor asked for a total of 18,446,744,073,709,551,615 grains of rice.
Now you need to apply some practical wisdom to the result you have been given. First, name that number: it is eighteen quintillion, four hundred forty-six quadrillion, seven hundred forty-four trillion, seventy-three billion, seven hundred nine million, five hundred fifty-one thousand and six hundred and fifteen grains of rice. Next, take a rough guess as to how many grains of rice are in a sack weighing one hundred pounds -- if one gram of long-grain rice has about 50 grains in it, then one pound (equals 453.6 grams) has 22,680 grains, and so a 100-pound sack would have 2,268,000 grains of rice in it. Now divide the eighteen quintillion number by 2,268,000 (again, Mathematica makes it simple, and gives you the answer in a split second): you get 8,133,485,041,318.14 hundred-pound sacks of rice -- that is more than eight trillion such sacks, weighing a total of more than eight hundred trillion pounds. Obviously that quantity of rice was not to be had in even the richest of kingdoms, and so the inventor's request would have bankrupted the sultan.
In the TED talk given below, Stephen Wolfram explains his lifelong love of making information computable, and the way in which he has brought his dream to fruition through the amazing Website called Wolfram Alpha, which builds on all the programming wizardry he has developed and incorporated over the years into Mathematica. For example, the chess-rice problem can be computed at the Wolfram Alpha site by copying this simple formula into its box, and hitting "Enter":
sum (2^n), 0 <=n <64
--- where the notation "2^n" stands for what we write in mathematics as 2n, the word "sum" tells the engine to add up all the individual terms of the series in the parentheses, and the expression "0 <=n <64 " instructs the engine to compute the sum letting the variable n range through all the values from zero up to (but not including) 64.
As you watch what Wolfram Alpha can do, I guarantee you will be astonished, and will not want to delay to see what questions of yours it can answer, and what relationships among data it can display:
Here is a link to more information about Stephen Wolfram; I have given the links to his software and to the Wolfram Alpha site above. You may watch his talk in its high-resolution version at this link, and you may download his talk in that and other formats from this page. Enjoy!