Thursday, March 05, 2009

Zero over Zero

730am this morning.

Sean: If a fraction equals 1, that means the numerator is the same as the denominator right?

Me: Right.

Sean: So what if the fraction is zero over zero? Does that equal 1?

In the past, Sean has asked why can't anything be divided by zero, and I've explained that if you divide 1 by 1/2, you get 2, and the smaller the denominator, the bigger the answer, eg divide 1 by 1/1000, you get 1000, hence, as the denominator approaches zero, the answer would approach infinity.

But his question today is slightly different. He notices that when the numerator=denominator, the quotient is he's wondering if 0/0=1. Fair enough.

When I googled "zero over zero", there is already quite a bit of discussion about this question.

A professor even created a new number (he calls it "nullity") to deal with dividing by zero but he got panned by many.

Anyway, someone asked this:

"Hi, I was just wondering - if you have 0/0 (zero divided by zero), which law takes precedence - a) zero divided by any number is zero, or b) any number divided by zero is undefined, or c) any number divided by itself is one? Thanks."

Sean seems to think (c) take precedence from the way he asked his question.  

This entire section is devoted to questions on dividing by zero, and includes dividing zero by zero.

What I'm getting from the site is:

(Any number other than zero)/zero = Undefined
Zero/zero = Indeterminate

There, apparently, is a difference between Undefined and Indeterminate. How the site explains it is...

Undefined: (About 1/0) "What happens if you add apples to oranges? It just doesn't make sense, so the easiest thing is just to say that it doesn't make sense, or, as a mathematician would say, "it is undefined."

Indeterminate: (About 0/0) "There's a special word for stuff like this, where you could conceivably give it any number of values. That word is "indeterminate." It's not the same as undefined. It essentially means that if it pops up somewhere, you don't know what its value will be in your case. For instance, if you have the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1 in the first case and 7 in the second case. Indeterminate."

Unfortunately, I'm all alone now, the boys have gone to school, and I only understand bits of the explanations. My understanding is definitely not robust enough for me to confidently explain the difference to a 6-year old though. It seems 0/0 can be 1 and it can be any other number.



monlim said...

Sean, Auntie Monica's head is swimming. When you come back to SG, can you please give Andre a math lesson??

Lilian said...

Ha...Sean only knows Math trivia, real useful Singapore Math, must still depend on Auntie Monica aka Model Mum.

Mateusz said...

Try using l'Hopital's rule in the limit as x approaches 0 for the problem ;)

Lilian said...

L'Hopital's rule? haha...I threw out any knowledge I have of it after my first year Statistic' exams!