How much is one divided by zero? Don’t ask…

The chairman of the press council of India, Justice Markandey Katju, wrote an article in The Hindu on September 3 on education.

Titled ‘Professor, heal thyself’, it contained this paragraph:

The level of intellect of many teachers is low, because many of them have not been appointed on merit but on extraneous considerations. To give an example, when I was a judge of Allahabad High Court I had a case relating to a service matter of a mathematics lecturer in a university in Uttar Pradesh.

Since the teacher was present in court I asked him how much one divided by zero is equal to.

He replied, “Infinity.”

I told him that his answer was incorrect, and it was evident that he was not even fit to be a teacher in an intermediate college. I wondered how had he become a university lecturer (In mathematics it is impermissible to divide by zero. Hence anything divided by zero is known as an indeterminate number, not infinity).

Not surprisingly, two wise readers of The Hindu have corrected the press council chief through letters to the editor:

In his article “Professor, teach thyself” (Sept. 3), chairman of the Press Council of India, Markandey Katju, has cited an incident that took place when he was a judge of the Allahabad High Court. He says he chided a mathematics lecturer, whose case he was hearing, and told him that he was not fit to be even a teacher because he (the lecturer) said one divided by zero was infinity.

Justice Katju claims that anything divided by zero is indeterminate. He is wrong and the lecturer was right because any non-zero number divided by zero is infinity. It is zero divided by zero that is indeterminate.

While I can understand the plight of the poor lecturer who did not have the courage to correct the judge hearing his case, I am appalled at the timidity of “some of the top senior academicians” of Jawaharlal Nehru University, to whom Justice Katju narrated the incident. I wonder why they let his fallacy pass unchallenged. Justice Katju must seek out the mathematics lecturer and apologise to him.

Kanan Vihari Jaswal, Noida

I would like to digress from the primary point made in the article — with which I completely agree — and talk about the mathematics lecturer’s answer. “Infinity” is indeed the correct answer to the question posed by Justice Katju to the lecturer. 0/0 is indeterminate because it can take multiple values depending on the limit being calculated (for example 2x/x; x->0 is 2 , 5x/x; x->0 is 5) whereas any finite number divided by 0 (eg 1/0) is an impermissible operation, which is just another way of saying that the result is infinite (an absurdly large number).

Siddharth Tiwari, Kanpur


Also read: ‘I have a poor opinion of most media people’

Editors’ Guild of India takes on Press Council chief

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28 Responses to “How much is one divided by zero? Don’t ask…”

  1. Sapna Says:

    Justice Katju is correct. 1/0 is ‘undefined’ not infinity. As the denominator tends to zero, the answer tends to infinity but that is not the same as saying 1/0 is infinity.

    An impermissible operation = infinity… Nice theory

  2. Rastrakoota Says:

    The answer being Infinite or indertimate both hold the same meaning here. The answer could incline to a positive or a negative indeterminate number and technically called infinite I guess.

    Katju is being ultra rigid on this for an answer. I pity the lecturer.

  3. Deepak Says:

    Katju thinks too much of himself. Speaks on all subjects under the earth and tries to poke his nose everywhere. He may have been a good judge, but he is a disastrous PCI chairman.

  4. the colonel Says:

    please go to:-

  5. Bk chowla Says:

    I am generally not in agreement with Justice Kataju’s on various issues

  6. Mysore Peshva Says:

    Justice Katju likes to speak outside of his areas of expertise. He also likes to generalize to a fallacy. As a result, he sometimes sounds like an imbecile. Hahaha. But i believe he is a good and noble jurist.

  7. Emptymind Says:

    Lecturer is correct. Katju is partly correct.
    The term “Infinity” is just one of many “Undefined” statement in mathematics. still there are many “Undefined” concepts like
    squ(-ve), point in the space etc. you know the “Point” do not exist in ultimate reality!

  8. Nastika Says:

    That is the difference between Law & Science – Law wants to say what is right & what is wrong. In science, sit back, enjoy and watch the fun when “limit of one by x, as x tends to zero”.


  9. Nastika Says:

    Infinity is undefined too.


  10. Emptymind Says:

    here the definition of undefined:

  11. Emptymind Says:

    How can impermissible operation like :- sqrt(-x) where x is any +ve real number equals infinity.
    please enlighten me!

  12. M Says:

    I’m not a mathematician but still remember being taught that any number divided by zero is infinity. That aside, the issue is here is not what happens when something is divided by zero but when someone happens to be a personification of zero. This Katju may think he is Zorro, but is merely a zero.

    This whole zero question is another example of judicial over-reach. Did he have the authority or the expertise to question the professional ability of a litigant; particularly when that profession happened to be other than law.

    This ‘I know everything’ attitude of judges has to be stopped.


  13. Rahul Says:

    Was this mathematical discussion germane to the case at the court or was His Honour just showing off and hectoring the lecturer.

  14. Armchair Guy Says:

    As is clear from the above discussion (and the Wiki article), the answer depends on the exact interpretation of the question: is this an algebraic operation we are talking about, or the limit of a fraction as the denominator approaches zero?

    While Katju is right on the facts, it wasn’t clear the lecturer was wrong, and certainly it wasn’t necessary to humiliate the lecturer.

    But the two commentators in the Hindu, as well as the author of this Churumuri post, display behaviour that can perhaps be described as “the blind leading the blind”. They do not really understand the math themselves, but are quick to lead criticism of Katju.

  15. Srikanta Says:

    It is infinity; becoz you dont know the largest value that needs to be multiplied to zero to make it one. zero/zero is indeterminate, one does not know what infinitseminal value that needs to be multiplied to zero to make it infinitesiminal. Any large value that cannot be comphrended is represented as infinity.

    Here is some stuff

    Purnamadah purnamidam ‘That (Almighty) is complete, this (Aatman) is complete’.
    purnat purna mudachyate – ‘From that completeness comes this completeness’
    purnasya purnamadaya – ‘If we take away this completeness from that completeness’
    purnameva vashishyate: ‘Only completeness remains.’

    Aliter on Katju.

    KAtju is infinitely stupid; you add Infinite intelligeence to Katju’s IQ; == Katjus still remains infinitely stupid.

  16. Vishal Says:

    1/0 = Infinity not indeterminate at least in the mathematical sense. However, 0/0 is indeterminate. For 0/0, we need to apply limits which then tends to 1.

  17. harkol Says:

    Well, it is our own 11th century Mathamatecian, who first propounded the theory that anything divided by zero will result in Infinity.

    So, the answer of infinity isn’t a wrong one. But, for science is not static and new interpretations and definitions keep happening.

    So, assuming Infinity is the wrong answer – what was Katju doing in a court taking a maths test of a qualified mathematician? How would Katju feel if he is tested on fundamentals of law, when he goes to get a driving license?

    How can a judge dispense justice by asking a complex question to which apparently there can be many answers?

    Katju is a pompus fool. He needs to be ignored. Instead he is given a lot of publicity.

  18. harkol Says:

    Sapna: Various mathematicians look at this differently. Bhaskara II, in his work Lilavati, expounded the various operations on zero, and says if any value is divided by zero the resulting quantity will be so large that it can only be called infinity (Anant). That is an accepted formulations for many mathematicians.

    But, there are also folks who argue otherwise, because

    6/2 = 3. If you take 2 to RHS then the equation is 6= 3 x 2. So, this works.

    However, if 1/0 = ∞ then 1 = ∞ x 0. But, that’s not true. We all know anything multiplied by zero is ZERO.

    So, some mathematicians consider the divide by zero operation logically not feasible.

    Logic being foundation of maths – You can divide something to any number of pieces. But, How to you divide something to nothing?

    So, answer isn’t indeterminate as Katju says. It is a non-logical operation, that can’t happen. But, if you want to theoretically put a value, then what Bhaskara2 said is as good a definition as anyone elses!!

    IAC, It is far too complex a question without universal agreed answer. It is unfortunate a judge sought to belittle a maths professor, who would have gone through huge grinding, on such a question.

  19. bellary brother Says:

    Katju, in his relatively new avatar as PCI chief, has been commenting on all an sundry. This discloses his stupidity, especially when he talks about areas where he has no expertise.
    To those who are unaware, he was not even a very good judge. He needlessly yelled at lawyers, some of whom were older than Katju. He also did not think much of women, but I think Indira Jaising has forced him to change this opinion.

  20. Nastika Says:

    What is infinity? It is some large number. Is it defined? No. Hence its indeterminate.

    While both the judge & the lecturer are correct (like an elephant can be like rope or a pillar, depending on which part you touch), the high-handed chiding of the lecturer wasn’t required.


  21. Sapna Says:

    @Emptymind: ‘An impermissible operation = infinity’ was the readers theory not mine. ‘Nice theory’ in my comment was sarcastic!

  22. vaidya Says:

    What next? Pass judgement on the mobile OS wars? One thing to have an idea, another to blame the whole education system based on an answer that can be argued forever without being of any meaning. What use does he derive from whether 1/0 is infinity or undefined? I guess the lecturer was a tad overawed by the situation and didn’t want to argue with Katju.

    I do believe that 1/0 is infinity and 0/0 is undefined. The graphs plotting this do point that way.

  23. Vishal Says:


    1/0 = infinity. But infinity * 0 is indeterminate since it would be in 0/0 form. Your statement that anything multiplied by 0 is zero does not apply for infinity. Limit of Nr that tends to zero / Dr that tends to zero is 1.

    Anyway, I hope that the courts dont apply their mathematical skills on their own when such a need arises and instead leaves maths to mathematicians.

  24. harkol Says:

    Vishal: What you say is exactly my point. It is too deep a problem in Maths for most to know. To use that to harangue a Maths professor in a full court is downright mean. Katju is a mean spirited person, who thinks nothing of saying 90% Indians are fools.

  25. harkol Says:

    >Katjus still remains infinitely stupid

    Brilliant! ;-)

    Katju = ∞Wisdom x ∞insensitivity x ∞Mouth

    Mysore Peshva:

    Not sure he was a good jurist. I am very sure he is very intelligent.

    But, having intelligence alone doesn’t make one a good judge. You need to balance between idealism, reason, compassion, sensitivity & common-sense to judge another person.

    A man who says India will need to live with corruption for 20 years, doesn’t have idealism.
    Man who says 90% Indians are fools doesn’t have sensitivity.
    Man who berates a poor math professor in a court room doesn’t have compassion.
    Man who advocates Laxman rekha for judiciary – even when executive exceeds its Laxman Rekha and indulges in runaway corruption – has no common sense. Common sense would’ve told when one pillar collapses, another pillar takes additional load/burden.

    Wonder how many of his judgement were colored by his biases and prejudices.

  26. Gagan Says:

    My take on this issue is:
    The answer for Katju’s question leads to an answer which is still disputable. Finally its open to interpretation, just like law!

    What is more important about the episode is that Katju used this one single question to measure the competence of the Lecturer to hold the post. How can one call a person not qualified / meritorious just because he answered one question wrong? This episode happened when Katju was the Judge at of a High Court. This episode is nothing but a case of “Judicial Hyperactivism”.

  27. Reetesh Mukul Says:

    Katju is right. (If) 1, 0 are elements of a field, in which division by zero is impermissible. One can call this impressibility as ‘undefined’, that should basically mean that there is no inverse of ‘0’.

  28. New User (@saral_sahaj) Says:

    I am afraid, that on a strictly technical level (though I don’t abide much by it, and believe that the lecturer should be given the benefit of doubt), Katju is more correct than lecturer. I agree with the comment by Armchair Guy.

    The replies by the “wise readers”of Hindu are more incorrect than the lecturer, since they actually give out fallacious reasoning; for the lecturer, we at least don’t know his point of view, and can give him the benefit of doubt, depending on how he interpreted the question.

    The first reply just reiterates what the lecturer said(that anything divided by zero is infinity), which can be argued against. Any non-zero divided by zero cannot be called infinity, because the limit is different depending on whether you approach 0 (in the denominator) from the positive or -ve side, so the limit TENDS to +ve infinity from 1 direction, and -ve infinity from the other, and so technically, the limit does not exist. Also, I see that people are talking about numbers being EQUAL to infinity; technically, a quantity always TENDS to inifinity, and never equals it. This takes care of the second reply in Hindu;

    for the first, it is important to note that Katju clearly says that division by zero is impermissible (which is just another way of saying ‘undefined’), and goes on to say that it is an indeterminate NUMBER. The first reply in Hindu is talking about an indeterminate FORM in calculus(which I agree, 1/0 is certainly not, and hence the first reply is partially correct); and to my mind, katju was talking about something else, not the indeterminate form in calculus, and, taken in conjunction with his previous statement, I give him the benefit of doubt.

    I am sorry to be splitting hairs at such a minor matter, but the tone of this article compelled me to do so. The author takes the 2 replies by so-called “wise readers” to Hindu as sufficient proof that Katju is wrong, without trying to verify anything himself. A case of blind leading the blind, indeed.

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