0/0 and n/0 where n is not equal to zero

Today while arbitrarily browsing through some notes posted by some of the mathematics enthusiasts, I came across the following question:"What is 0/0?"

Well, division by zero is region of great obscurity in mathematics, especially for Mathematics students. Many physics books declare that division by zero yields infinity, while many mathematics books write that something divided by zero is undefined. Students just stow their brains with these terms and notions, without experiencing their true flavors. Well, in this note, I will try to explain the meaning of division by zero through a very simple discussion.

Well, what is division? What to you mean by dividing a number 'a' by a number 'b'? It means simply subtracting 'b' from 'a' repeatedly until you get either a 0, or a positive number which is smaller than 'b'. The number of times you perform this process is your quotient, and the resulting non-negative number is your remainder. Let us divide 15 by 5 using this definition. We see that 15-5=10, and 10-5=5, and 5-5=0,and we stop the process. Clearly, the number of subtractions performed here is 3, which is our quotient here and 0 is our remainder here. Similarly let us divide, say, 16 by 3. I am writing directly: 16-3-3-3-3-3=1, where 0<1<3. Thus here quotient is 5 and remainder is 1. These results exactly tally with the results which we can arrive at, using the well known long division process. It has to be so, as long division is essentially the repeated subtraction in disguise. This can be proved easily using the Euclid's division Lemma (well, I am discussion the same Lemma in a very casual fashion, without going into the mathematical rigor). Now let us try to divide a non zero number, say, 6, by 0. We see, 6-0=6, 6-0=6, 6-0=6..ad infinitum. Clearly, this process will never end and so we will never get a quotient, neither will we get a remainder. This is because, after every subtraction what remains is 6 itself. So, such a non-zero number divided by zero will simply give us no value, and hence is meaningless, or in mathematical terminology, "undefined". However, in case of 0/0, we see, in the first step only,0-0 gives 0. Thus we have to perform subtraction only once to get a zero as our remainder. But does that mean quotient is 1? We see that, if subtract 0 from the 0 obtained as a remainder in the very first step once again, we will again get a 0. So that would mean that the quotient is 2. Using the same logic, one can argue that the quotient is 3, or, 4, or 5,or.....

Well, that's the fundamental difference between non-zero/zero and zero/zero. While the first one gives no value, the second one has no FIXED value. That is why we call the first one UNDEFINED (i.e., meaningless), while the second one is called INDETERMINATE (i.e.,lacking a fixed value). And oh, yes, one more thing, why do physicists declare non-zero/zero as infinity? Simply because what they call 0 in the denominator is actually a "vanishingly small quantity of infinitesimal magnitude". Suppose we divide 6 by such a small quantity. Then at every step, 6 will be decreased by an infinitely small quantity, and after infinitely many such subtractions, 6 will finally reduce to 0. Hence the number of subtractions performed being infinity, non-zero divided by infinitesimal will give us an infinity. Now see how easy everything becomes after an explanation.

"Omne Ignotum Pro Magnifico!"

Word of Caution: Do not write 6/0=undefined, or 0/0=indeterminate. We use the "is equal to" symbol only when both the right and the left hand sides have real values. But "undefined" is not a real value. It is only another way of saying "I don't know what that means". So don't use "=" sign here.

Note by Kuldeep Guha Mazumder
5 years, 3 months ago

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As far as I'm able to tell, there's little distinction between "infinity", and "arbitrarily large value" in physics. I don't see any rigor applied to "infinity" in physics, nor see where it ever matters if that distinction is made.

- 5 years, 3 months ago

And on this context I allude to this nice piece of humor that I found on the internet:

An engineer, a physicist, and a mathematician are shown a pasture with a herd of sheep, and told to put them inside the smallest possible amount of fence. The engineer is first. He herds the sheep into a circle and then puts the fence around them, declaring, that a circle will use the least fence for a given area, so this is the best solution. The physicist is next. He creates a circular fence of infinite radius around the sheep, and then draws the fence tight around the herd, declaring, this will give the smallest circular fence around the herd. The mathematician is last. After giving the problem a little thought, he puts a small fence around himself and then declares,"I define myself to be on the outside."

- 5 years, 3 months ago

You are absolutely correct Sir. In Physics they make no distinction between infinity and a considerably large value. But in mathematics they do and this post is indeed a humble attempt to point out that subtle difference between the two disciplines.

- 5 years, 3 months ago

I should add that as a practical matter, math software makes a distinction between "indeterminate", "undefined", "directed infinity", and "complex infinity", and your ordinary "infinity". But you said that "many physics books declare that division by zero yields infinity", you're really referring to elementary physics texts, which I agree are misguided. Papers in theoretical physics don't make this declaration--they are usually careful to note the distinctions between different kinds of infinities and indeterminacies.

- 5 years, 3 months ago

And as for the concept of infinity, we have our subject of surreal numbers, which addresses infinity in a totally different way. And, yes, we have our complex infinity, directed infinity, etc., as you have mentioned. We have our infinite-dimensional complex spaces like the Hilbert space, we have our Kuiper's theorem on contractibility of infinite spheres. Infinity is used innumerable times in innumerable areas of Mathematics with varied notions. Infinity is indeed a gray area of mathematics. But the infinity that I have mentioned in my note is nothing of that kind. It simply means "bigger than the biggest one can imagine", the basic notion of infinity that a school-level student should have.

- 5 years, 3 months ago

I believe the main point you were making is the distinction between n/0 and 0/0, which are "undefined" and "undeterminate", which I don't think is a point of dispute. Just the same, some math software return "complex infinity" for n/0.

Meanwhile, somewhere buried in my pile of books, I have the book "Surreal Numbers" by Donald Knuth. I can only dimly remember the endless labyrinthine details about such numbers, which, as I understand, still baffles some mathematicians today.

- 5 years, 3 months ago

Yes yes..that was the central idea of my note..and yes, the study of surreal numbers is a relatively novel and labyrinthine one..I only have a basic idea about the subject..and the most catchy term that I have encountered in the subject is 'birthday of a surreal number'.

- 5 years, 3 months ago

Certainly Sir. I am only talking of the school level textbooks that confuse the students. I have devoted this note to school level students who have certain obscurities regarding the subject of division by zero, owing to such erroneous declarations in such school level textbooks.

- 5 years, 3 months ago

A little more rigor:

The division theorem says that all numbers can be represented uniquely as $n=qx+r$ for integers $q, x$ and $0\le r\le q-1$.

In the case of $0/0$, we have $n=x=0$. Then $r=0$ so we have no remainder. But what is $q$? Indeed, we can place any value of $q$ in the expression, real or complex, and get a valid equation. Since a number clearly cannot take an infinite number of values, $0/0$ is considered indeterminate.

- 5 years, 3 months ago

Daniel I am afraid you need to correct the statement of division lemma. Apropos of the equation that you have written, I would like to mention that given an integer n (the dividend) and a positive integer x (the divisor) the lemma ensures the existence of UNIQUE integers q (the quotient) and r (the remainder) where r is non-negative and less than x. But your approach is worthy of appreciation. However the uniqueness is redundant to mention as soon as we impose the restriction that r is non-negative and less than x. We can then prove the uniqueness.

- 5 years, 3 months ago

This infinity, undefined and indeterminate really confused me. Thanks for the post!

- 5 years, 3 months ago

The note is a bit mundane, I admit it..

- 5 years, 3 months ago

No, it's really interesting:)

- 5 years, 3 months ago

The "best of mine" part of the problems posted by you are good. But I found them easy and solved four of them even in my most somnolent state (well, I feel extremely sleepy now, it is 2.10 am under the clock) and will solve the rest tomorrow.

- 5 years, 3 months ago

Thanks for checking my set. It's been a long time since I edited it, anyway thank you for your comments!

- 5 years, 3 months ago

Welcome..actually today I am down with fever and so I couldn't check your other problems, but I will surely solve them by tomorrow..

- 5 years, 3 months ago

Thank you..you know I love to stimulate the minds of people with seemingly trivial but difficult-to-answer problems like this one..that is why I am here. You can check my other posts also..

- 5 years, 3 months ago