Finding n(n-1)/2 in the Real WorldDate: 09/14/2002 at 10:33:49 From: john cook Subject: Explain a formula Dear Dr. Math, How can n(n-1)/2 be explained in common everyday language? When is it used? Thanks, John Date: 09/14/2002 at 11:40:59 From: Doctor Ian Subject: Re: Explain a formula Hi John, One place it might come from the real world is this. Suppose you know that your company is going to buy some computers, and you need to have a direct connection from each one to each other one. (You would normally use something like the Internet to hook them up, but direct connections are more secure.) How many connections will you need? If you have one computer, you need zero connections. If you add a second computer, you have to add a connection to the first one: A ..... B If you add a third computer, you have to add a connection to each of the ones you already had: A ..... B . . . . C If you make a table, Computer Additional connections -------- ---------------------- 1 0 2 1 3 2 4 3 . . . . n n-1 So if you buy N computers, the number of connections you'll need is 1 + 2 + 3 + ... + (N-1) Now, is there a way to figure this out without doing a zillion additions? In fact, there is; see the Dr. Math archives: Sum of Numbers 1-500 http://mathforum.org/library/drmath/view/57990.html But we have to make one little adjustment. If you add the first N numbers, you get N(N+1) 1 + 2 + 3 + ... + N = ------ 2 which isn't quite the same formula. But we're adding, not the first N numbers, but the first (N-1) numbers. So N becomes (N-1), and (N+1) becomes N, which gives us (N-1)((N-1)+1) 1 + 2 + 3 + ... + (N-1) = -------------- 2 (N-1)N = ------ 2 Note that this kind of 'connections' problem arises in lots of different contexts: How many direct flights can you have between various cities? How many exchange rates can you have between various currencies? How many translators do you need between various native speakers? Is this what you wanted to know? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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