Why Does Long Division Work?
Date: 12/22/2006 at 07:03:29 From: Peter Subject: long division re-invention Hi. I've been looking everywhere and can't find a real explanation of the "long division" algorithm. I can of course use this algorithm (I learned it looooong ago in elementary school) but what I'm asking you is to "re-invent" this algorithm. I find it very confusing to know that it works but not know why.
Date: 12/22/2006 at 10:37:57 From: Doctor Peterson Subject: Re: long division re-invention Hi, Peter. Here is an attempt (I'm not sure how successful) to explain why long division works to a kid: Long Division http://mathforum.org/library/drmath/view/58499.html I'll try to repeat the same example using algebraic notation rather than manipulatives (bundles of sticks). Here's our problem: __18_ 4 ) 73 4 - 33 32 -- 1 We're dividing 73, which is 7*10 + 3, by 4. We first divide 7 by 4, giving 73 = 7*10 + 3 = (4*1 + 3)*10 + 3 Here I've used the basic idea of division, which is dividend = divisor * quotient + remainder 7 = 4 * 1 + 3 Now we rearrange this so that we have the first digit of our quotient: = 4*(1*10) + (3*10 + 3) Note that the remainder of that first division, the 3, is actually a number of tens, so I've combined it with the 3 in the dividend that we haven't touched yet. That whole bundle still has to be divided by 4; so do that, finding that 33 = 4*8 + 1. (The 8 is the quotient, and the 1 is the remainder.) Put that into what we've already done: = 4*(1*10) + 4*8 + 1 We're almost done. We just have to combine our two quotients: = 4*(1*10 + 8) + 1 There we have it: the quotient is 1*10 + 8 = 18, and the remainder is 1. If you have any further questions, feel free to write back. I'm sure I can fill in details or extend this to more complicated problems if necessary. Incidentally, if you've taken algebra you've probably seen long division of polynomials, which uses the same general algorithm, but without the need for "borrowing" between digits, so it's more straightforward. Some of the details are easier to understand in that context. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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