Be careful in solving K-12 math problem: divison by zero
When I passed “SD percobaan” this morning, I couldn’t resist the foods sold at the front gates (it was almost 12, and I was extremely hungry). So I went there and bought 2 lehers (lekers? forgot the name) which were put in a paper container.
The paper container was made from a used math book page, or a copy of it. I won’t talk about its hygiene but about a problem in it:
Simplify: a3 x a2 : a4
It was a “problems” page so there were no answers. However I could guess that the answer deemed “correct” by the book is a3+2-4 = a1 = a. That is also the answer that most, if not all teachers would give. However that’s wrong!
First, looking at other problems on the page, they use negatives (-13, -8, etc.) and fractionals (3/2, 4/9, etc.). So it is safe to say that the variable a is a rational number (as opposed to, i.e., a natural number). Now, remember that the set of rational numbers contains 0, and this is where we need to be careful.
If a is nonzero, the expression can be simplified to a3+2-4 = a. However if a is zero, division by a4 = 0 cannot be done so the result is undefined. Therefore the correct answer is:
a, if a is nonzero
undefined, if a is zero
Saying that the answer is unconditionally a is the same as saying that 03 x 02 : 04 = 0 which is of course wrong.
As a tangential side note, I once encountered a multiple-choice national (UAN) math test question about integrals. Because the problem question maker overlooked division by zero, none of the answers were correct. A math question then becomes a moral question: Is it a sin knowing that an answer is wrong, but answer it anyway because the question maker would probably regard it as correct? Indeed, ignorance is bliss.
December 26th, 2006 at 12:14 pm
a*b=c
c/b=a
Example=
3*2=6
6/2=3
OR
3*0=0
0/0=3
December 26th, 2006 at 7:18 pm
And the necessary steps to avoid getting 0/0 = 3 is….
a * b = c
(a * b) / b = c / b, with b not 0 (we must put this restriction, because we are doing division with b)
a * (b / b) = c / b, with b not 0 (yup the restriction will stay until the end of the world)
a = c / b, with b not 0
c / b = a, with b not 0
Now you cannot plug b = 0 to the last equation, because there’s this limitation “with b not 0″
January 6th, 2007 at 9:41 pm
Perhaps a better title would have been: Be careful in posing k-12 math problems…
It is true that teachers and problem writers sometimes forget to properly restrict division by 0. More often than they should. But students should learn to distinguish between an inadvertant omission and a genuine problem, and react accordingly.
I actually consider it a measure of how well I’ve taught that students will quickly remind me if I’ve absent-mindedly failed to introduce necessary restrictions.
Jonathan