Subtracting and Borrowing in a Column
Date: 09/05/2002 at 10:23:33 From: Brad Hendrickson Subject: Subtraction borrowing. Hi, I teach high school chemistry and teach significant figures. This was the problem. 6.9 - 7.92. I worked the problem as usual, getting -1.02, reported to -1.0 with significant figures. A student of mine brought up a point that I had never thought about before, and boggled me. He borrowed as follows 6.9 - 7.92 ---- to get - 2.98 Why won't borrowing work with this number?? Thanks for your help.
Date: 09/05/2002 at 12:35:42 From: Doctor Peterson Subject: Re: Subtraction borrowing Hi, Brad. It's really not the borrowing that doesn't work, but the whole idea of subtracting in a column. Let's take a case where there's no borrowing: 6.58 - 7.43 ------ -1.15 But 6.58 - 7.43 = -(7.43 - 6.58) = -.85. So this is wrong. Why? The problem is that the minus sign in our answer really applies only to the first digit, the 1. What we've done is this: (6 + .58) - (7 + .43) = (6 - 7) + (.58 - .43) = -1 + .15 This is the correct value, -.85; but it can't be written as -1.15, as we did by writing a digit in each column, because that would mean -(1 + .15) rather than -1 + .15. So when you subtract this way (which I never do) you have to do one more step: subtract the .15 from the 1, giving .85, and apply the negative sign to the whole thing. Let's try this on your problem. Your student got "-2.98"; this is really -2 + .98, so you have to subtract .98 from 2 and get 1.02, so that the answer is -1.02, the correct answer. I should note that it is only the leftmost digit that has the negative sign on it, not the whole part of the number as you might be imagining. If we subtracted 658 - 743 ----- -115 we would finish by subtracting 15 from 100, and get -85. These ideas are closely related to the 'twos-complement' method used by computers to handle negative numbers. That method allows the machine to add or subtract 'in a column' without worrying about whether the answer will be positive or negative. If you have any further questions, feel free to write back. This was an interesting one! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 09/05/2002 at 17:21:58 From: Brad Hendrickson Subject: Thank you (Subtraction borrowing) Hi! Thanks so much for your response! That certainly clarifies the situation, and my students and I thank you tremendously. It's so neat when students bring up points you've never thought about. Thanks for the help! All the best, Brad Hendrickson
Date: 09/06/2002 at 08:31:43 From: Doctor Peterson Subject: Re: Thank you (Subtraction borrowing) Hi, Brad. One of the things I enjoy as a Math Doctor is the chance to answer these off-the-wall questions that I'd never think of on my own, the wrong ideas that clarify the right ideas. Thanks for writing. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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