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        riddles >> general problem-solving / chatting / whatever >> Mathematical errors
        
(Message started by: rmsgrey on Nov 3^rd, 2011, 4:24pm)

Title: Mathematical errors
Post by rmsgrey on Nov 3^rd, 2011, 4:24pm

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

Title: Re: Mathematical errors
Post by Noke Lieu on Nov 3^rd, 2011, 8:10pm

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

Title: Re: Mathematical errors
Post by Grimbal on Nov 4^th, 2011, 5:40am

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.

Title: Re: Mathematical errors
Post by SWF on Nov 4^th, 2011, 6:18am

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.

Title: Re: Mathematical errors
Post by pex on Nov 4^th, 2011, 6:32am

on 11/04/11 at 06:18:03, 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.

I think this would actually be worst for n=0. ;)

Title: Re: Mathematical errors
Post by rmsgrey on Nov 4^th, 2011, 8:49am

on 11/04/11 at 05:40:21, Grimbal wrote:

Anyway, both are wrong. Period.

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?

Title: Re: Mathematical errors
Post by SWF on Nov 4^th, 2011, 5:19pm

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.

Title: Re: Mathematical errors
Post by pex on Nov 5^th, 2011, 4:37am

on 11/04/11 at 08:49:29, rmsgrey wrote:

But what about if they're specific equations?

What can you say about a and b in each case?

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 [hide]a³ = b³ but a =/= b. That is, one is exp(2 pi i/3) times the other[/hide].

Title: Re: Mathematical errors
Post by rmsgrey on Nov 5^th, 2011, 10:30am

on 11/04/11 at 17:19:21, SWF wrote:

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

Title: Re: Mathematical errors
Post by malchar on Nov 19^th, 2011, 6:36pm

At least equation 2 works just fine in binary.

(a + b)^2 = a^2 + b^2 + 2ab = a^2 + b^2 + 0ab

Title: Re: Mathematical errors
Post by rmsgrey on Nov 20^th, 2011, 6:46am

on 11/19/11 at 18:36:12, malchar wrote:

At least equation 2 works just fine in binary.

(a + b)^2 = a^2 + b^2 + 2ab = a^2 + b^2 + 0ab

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.