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rmsgrey
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Mathematical errors
« on: Nov 3rd, 2011, 4:24pm » |
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This is more of a matter of opinion than a solvable problem, but which of the two schoolboy errors is worse, and why?
1) 1/(a+b) = 1/a + 1/b
2) (a+b)2 = a2 + b2
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Noke Lieu
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Re: Mathematical errors
« Reply #1 on: Nov 3rd, 2011, 8:10pm » |
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wow! Great question.
For me, it's the first one.
Why? Fractions are introduced much much earlier, and whilst are more procedurally complex. This mistake often demonstrates a lack of instructional, prodecural and relational understanding.
I would further wager that the second error would likely crop up in the face of algebra more than the equivalent problem in arithmetic... hence your average schoolboy is a bit unsettled- and can well be due to a slip of the mind/pen.
I'd love to see Sir Col's take on this question, but I guess that's unlikely to happen
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Grimbal
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Re: Mathematical errors
« Reply #2 on: Nov 4th, 2011, 5:40am » |
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The first one is worse because adding something to the denominator makes it smaller, while adding something to the whole fraction makes it larger. Intuition should tell we are moving in the wrong direction. At least with positive values.
In other words:
1/(a+b) is smaller than 1/a, while 1/a + 1/b is larger.
So 1/(a+b) and 1/a + 1/b cannot be the same
I don't see a similar reasoning in the 2nd case.
Anyway, both are wrong. Period.
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SWF
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Re: Mathematical errors
« Reply #3 on: Nov 4th, 2011, 6:18am » |
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They are both the same mistake:
(a+b)n = an + bn
Maybe it is worse the further n is from 1, since the above is true for n=1.
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pex
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Re: Mathematical errors
« Reply #4 on: Nov 4th, 2011, 6:32am » |
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on Nov 4th, 2011, 6:18am, SWF wrote:They are both the same mistake:
(a+b)n = an + bn
Maybe it is worse the further n is from 1, since the above is true for n=1. |
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I think this would actually be worst for n=0. 
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rmsgrey
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Re: Mathematical errors
« Reply #5 on: Nov 4th, 2011, 8:49am » |
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on Nov 4th, 2011, 5:40am, Grimbal wrote:| Anyway, both are wrong. Period. |
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As general identities, sure, both are wrong. But what about if they're specific equations?
What can you say about a and b in each case?
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SWF
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Re: Mathematical errors
« Reply #6 on: Nov 4th, 2011, 5:19pm » |
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Yes, n=0 is not good, so n being close to 1 is not a measure. How about: 1) is a less serious error because it was a simple sign error on n. I could see somebody trying for partial credit on a test with that type of reasoning.
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pex
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Re: Mathematical errors
« Reply #7 on: Nov 5th, 2011, 4:37am » |
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on Nov 4th, 2011, 8:49am, rmsgrey wrote:But what about if they're specific equations?
What can you say about a and b in each case? |
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2) is easy: at least one of a,b must be 0.
In this respect, maybe 1) is a more serious mistake: it has no solutions with both of a,b real. In complex numbers, it requires that a3 = b3 but a =/= b. That is, one is exp(2 pi i/3) times the other.
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rmsgrey
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Re: Mathematical errors
« Reply #8 on: Nov 5th, 2011, 10:30am » |
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on Nov 4th, 2011, 5:19pm, SWF wrote:| Yes, n=0 is not good, so n being close to 1 is not a measure. How about: 1) is a less serious error because it was a simple sign error on n. I could see somebody trying for partial credit on a test with that type of reasoning. |
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How likely the reasoning is to convince anyone depends on the notation involved - confusing 1/a and a is a lot less excusable than confusing a-1 and a1
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malchar
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Re: Mathematical errors
« Reply #9 on: Nov 19th, 2011, 6:36pm » |
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At least equation 2 works just fine in binary.
(a + b)^2 = a^2 + b^2 + 2ab = a^2 + b^2 + 0ab
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rmsgrey
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Re: Mathematical errors
« Reply #10 on: Nov 20th, 2011, 6:46am » |
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on Nov 19th, 2011, 6:36pm, malchar wrote:At least equation 2 works just fine in binary.
(a + b)^2 = a^2 + b^2 + 2ab = a^2 + b^2 + 0ab |
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Working in binary:
If a=b=10
a+b = 10+10 = 100
a2 = b2 = 10+10 = 100
(a+b)2 = 100*100 = 10000
a2+b2 = 100+100 = 1000
10000>1000
If you work modulo 2, rather than base 2, then equation 2 holds.
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