| Math Mojo - Making Math Meaningful |
Division by zero
My elementary school child is confused about the answer to these three problems. I don't really know how to answer him. He really enjoys math but is having a hard time with this.
#1) 0/0 =
#2) 0/1 =
#3) 1/0 =His teacher has answered
# 1) 0/0 = 1,
# 2) 0/1 = 0,
# 3) 0/0 = 0
His calculator says
# 1) 0/0 = not a number,
# 2) 0/1 = 0,
# 3) 1/0 = infinity
What are the real answers?
Thanks from a confused Mom.
Professor Homunculus' answer:
His teacher needs to learn some math. I think it is a shame when a $10 piece of plastic and wires has more sense than someone our public schools have hired to inspire young minds.
But I digress.
# 2 is the easiest to explain. You can generally look at most division problems as "how many groups of x are in y?"
Like for example, how many six-packs of soda are in a case of twenty-four?" Fair enough?
So, by that logic, "How many groups of one bottle of soda do you have if there are zero (none) bottles of soda?"
or:
"How many bunches of bananas to you have, if you have none?" Well, none. And none is represented by zero.
#s 1 and 3 are a little harder to explain. Because the "How many groups of ..." question is nonsensical.
"How many groups of zero are in zero?" Hmmm.
So we have to go to the blackboard.
We know (or should know) that math is just a language which humans have created to describe size, quantity, and order.
It is just a language, much like any other. Why do we call a shoe a "shoe"? Why can't we call it a "Dxinw?" Well, we can.
We won't go to jail if we do, and the shoe will still be what it is.
The problem is, that if everyone called it something different, we wouldn't get much done.
That is why people have agreed that languages must have certain rules.
So in math, we have the rule that to divide a number by another number, means to find another number such that the first number times the second number equals the third number.
That sounds difficult, but what it boils down to, is that for a division like 8/2 = 4 to be true, then 2 x 4 = 8 must also be true.
And it is.
Let's try that with 0/1 = 0. Does 1 x 0 = 0? Yes, it does. So # 2 is proven true. Wow, a teacher and a calculator got one right!
Now let's try it with 1/0 = 0. Does 0 x 0 = 1? Not on my planet! There is no number which would fit both situations.
So there is no answer. It is a nonsensical operation, therefore humans created a law to say that it is more-or-less "not allowed" to divide by 0. We call the answer "undefined" because we cannot define an answer that would work.
Strictly speaking, it is allowed, but we just don't get a useful answer when we do.This could be a great opportunity for your child (and maybe you, too!) to discover that math is not just a "bunch of formulae," but a wonderful adventure into how things work and how humans describe reality, and that there is not always "just one best answer." Math continues to evolve, just as the human mind does.
But there I go digressing again!
Now, as far as 0/0, we have a real hard nut to crack. If you answer "0", the answer WOULD fit the expression 0 x 0 = 0.
So it would seem that you CAN get a number if you divide by zero after all. But we have just determined that you CAN'T.
This gives us quite a problem.
In this case, we have to look a little deeper. Yes, 0 x 0 DOES equal zero, but then again, ANY number times 0 equals zero.
So the answer to 0/0 could be ANY number.
In the case of 1/0, there is no number which fits the reverse. But in this case (0/0) any number can fit the reverse.
Neither of these is satisfactory, but they are also not exactly the same, so we can't call them both "undefined," because, actually you can define the answer to 0/0, but the answer could be "any number, " and that is not acceptable.
Also, there is a mathematical rule, which says, "Any number divided by itself equals 1." That would make 0/0 = 1,
which is clearly impossible. There is nothing to begin with, so you can't end up with something.
Yes, the reverse would be true (1 x 0 = 0), but ANYTHING x 0 = 0, so again, this is not an acceptable answer.
Because mathematicians (who look for answers, but do not accept everything on dogma) are not like math teachers (who often try to bully you into believing that there is one right answer to everything, and that answer is THEIRS
or the (ugh!) "standardized test's"), mathematicians have come up with a word which is not quite "undefined" for 0/0.
They call it "indeterminate."
In a nutshell, the answers to your questions are:
#1) 0/0 = indeterminate
#2) 0/1 = 0
#3) 1/0 = undefined
I hope this helps.
I am sorry I disparaged your math teacher. S/he might be a fine person, and even a good teacher.
But it is important that s/he understand that asking children a question (the answer to which you are not sure of) and then expecting them to "get it right," is like asking them for a definition for a word, and then telling them the wrong definition as the answer. You have to take this stuff more seriously, because the child's future attitude towards math (and learning in general) are going to be dependent on how you guide and inspire them.
I hope you will show the teacher this (you can delete my rantings) and I hope it inspires him/her to become an inquiring teacher, so s/he can inspire inquiring students, like your child seems to be.
Best of luck to you,
Professor Homunculus.
(For a more advanced, but absolutely brilliant discussion on division by zero, zero in general, and good thinking, check out The Zero Saga, by Professor Hossein Arsham. If you are a university math instructor or professor, this link is is for you!)
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