In response to my article published in The Hindu some persons have said that I was not correct when I wrote that anything divided by zero is an indeterminate number, not infinity.

So let me prove what I said.

Suppose 1 divided by zero = x. Then x multiplied by 0 should be 1. But we know that anything multiplied by 0 becomes 0. It follows that it is impermissible to divide by 0.

Infinity is not a number at all. Infinity can be expressed thus :

Read the article (Click Here)

So let me prove what I said.

Suppose 1 divided by zero = x. Then x multiplied by 0 should be 1. But we know that anything multiplied by 0 becomes 0. It follows that it is impermissible to divide by 0.

Infinity is not a number at all. Infinity can be expressed thus :

Limit of 1 divided by x, x tending to 0, is infinity. This merely means that if we keep making x smaller and smaller the number keeps becoming bigger and bigger without any limit. That is all that infinity means. It is fallacious to regard it as a number.

Read the article (Click Here)

Sorry sir but you are obfuscating from your responsibility - to correct yourself and issue an apology. The issue has never been about the properties of infinity.

ReplyDeleteLet us consider what happened:

1) You gave a judgement based on your understanding that "1 divided by 0 is indeterminate". Please mark that word carefully - indeterminate. You were quite sure about it. This is from your own op-ed. The poor lecturer answered the question as infinity, which was not what you were expecting. You evidently delivered your judgement based on this (wrong) understanding.

2) People obviously, the 90% of the indians or in your words - fools, identified your mistake. The very same people have written to you, to newspapers, about the mistake you made which ultimately affected an individual.

3) Now you reduce the issue to a question wide open for interpretation - "is infinity a number?". Whether or not infinity is a number doesnt matter sir. Your answer and understanding are wrong. Any number, other than zero, when divided by 0 is not indeterminate. In some branches of mathematics it is infinity, in some it is undefined. In no branch, it is indeterminate.

Limit of a number is defined based on some of the fundamental principles of mathematics. If you now tell us that use of "infinity" as number is "fallacious", then what you are actually asking for is a complete change in Theory of Calculus. Just imagine, what you are talking about.

1 divided by 0 sir, in formal mathematics is infinity. You may not accept this owing to your "great knowledge in mathematics", which is of course without any formal training.

But ponder for a moment, from your own op-ed, we know that your judgement has affected an individual's life, probably in a negative way. Is this your notion of justice?

0 multiplied by any number is not 0. 0 multiplied by infinity is indeed indeterminate. So your arguement here is wrong, to put it simply.

ReplyDeleteJustice Katju , please do not put an effort to explain those who are not willing to understand you on logical grounds , rather its more of an emotional hue and cry . You cannot reason with fanatics who are self righteous in their own eyes . Division by 0 in arithmetic is an undefined number , you are correct on that .

ReplyDelete@ mathematic intellectuals , Infinity is not a number . http://scienceblogs.com/goodmath/2008/10/13/infinity-is-not-a-number/

As for division by 0 in calculus please read more on calculus and division by 0 and refer this paradigm :

f'(x) = lim (h->0) (f(x+h) - f(x))/h

So in the sense that we are taking the limit to zero this looks like division by zero, but look at what happens with this definition, for a common function like f(x)=x2

we have

lim(h->0) ((x+h)2-x2)/h

lim(h->0) (x2+2xh+h2-x2 )/h

lim(h->0)(2x+h)

now we can take the limit to get x'=2x

which is the deriveative of x2, calculus does not make divide by zero possible, it simply takes the limit to zero, and uses careful mathematcal practices to yield meaningful results.

What we have in the above two comments is an example for bad reasoning based not on logic but fallacies .

Dear Lucius,

DeleteGet it right. It is neither a question of what mathematics one understands or how great you are at reasoning! Justice Katju's judgement has affected an individual, probably in a negative way. It is a known fact that his understanding is wrong.

Is this the right way of delivering justice? If you think it is, you probably need to justify to yourself whether you are blinded by your own bias!

Since you bring in your reasoning, I hope you go back to your math and refer to something called Complex Analysis.

DeleteI dont know whether you have an idea or not, there are branches of mathematics, where there is no negative scale, which means there are only right-sided limits, where limit of a/x as x tends to 0 and a not equal to 0 exists and is defined as infinity.

Problem with people like you is "you run away from issues like this one" citing something which you do not fully understand!

It is up to Shri Katju now to let us know what he would like to do. May be tender an apology to the poor lecturer? He has set a precedent. It is bad and needs to countered.

Have you even read the judgement ? And please don't forget the Judge cannot deliver a judgement on his whims , he is not above the spirit of law . There is rule of law not rule of judge , the order be it any must be based on reasoning or merits and satisfy the procedure and ingredients required by law . Judge or any other body is not supreme but law is supreme .

DeleteJustice Katju havn't even mentioned the judgement how do you it has affected him in a negative way ? I don't suppose any judge can base his judgement on such a question which Justice Katju posed . So please , do not talk about what you do not know . Read the judgement if you can find it , find the flaws and point it out . Usage or words such as "probably" affected will not get anyone anywhere .

You say its a known fact his understanding is wrong ? Angst and hate against somebody does not make him stupid , I think you are biased , you are not even willing to see his point . when he says "Indian society is communal " where is he wrong ? its quite true , I live in UP Muslims and Hindus are very communal . When he says " Indian societies are superstitious " And "90% of them are fools" how do you prove him wrong ? Most of the people in India thinks there is an invisible man in Sky who watches their every action and will punish them for bad deeds ( I am speaking about your god whosoever you believe) I have studied theology for 10 years and I know that it has caused more damage than any other doctrine . Not to mention your god is " God of Gaps" he will always be hiding in lacunae in science or something which science doesn't know yet .

And I am not biased , because I disagreed with him when he backed Jaipur protest against Salman Rushdie and banning of "Satanic verses " I don't think beliefs are to be respected when they are entirely absurd and lead to violence , that is my opinion and I am entitled to think differently .

When Justice Katju said " 90% of Indians are fools " I wasn't offended in the least but concurred that view , I am not communal , neither I value caste system , I don't believe in superstition , I don't believe in religions ( after extensive studies of all major religions of world) , I think critically challenge my views and strive to improve myself and have great deal of interest in science and I am an avid reader of science , history and law . I simple don't fit his criteria of "fool" , that is why i am not offended .

I am with Justice Katju for doing what is right , if he goes communal next day and orders genocide , I will not be with him . I realize he is a man of critical and scientific thinking and he calls spade a spade something which Indian society isn't used to , he wants to remove evils of Indian society and has logical ( not fanatic) nationalism in him , that is why I support him not because of any other reason .

I really suggest you to look at his things objectively , you are being too subject .

*subjective

DeleteAnd I think you forgot the most basic rule of mathematics , Division by 0 is impermissible and its undefined . infinity isn't a number and it is the general rule of mathematics that division by 0 is undefined not infinity . Please , read some books on it or research on internet . You havn't presented any rebuttal you just claimed

" I dont know whether you have an idea or not, there are branches of mathematics, where there is no negative scale, which means there are only right-sided limits, where limit of a/x as x tends to 0 and a not equal to 0 exists and is defined as infinity."

This is a weasel argument , name the branch and cite the relevant rules

Coming on topic at hand what about the general rule ? Refer general rules of arithmetic and other mathematical branches , most have the same general rule . I suggest you consult some very experts in this field and by expert I mean real experts who know what they are talking about .

When Justice Katju said , the number is undefined he was correct . I say so because it concurs the general rule of mathematics and please read the Judgement before criticizing it . I welcome a healthy debate and I don't run from issues . Your argument has no gravity in it please speak from an objective and logical point of view not subject just because you think something is correct , doesn't mean its correct .

It is possible to argue with those who know and those who do not know. It is impossible to argue with those who think they know.

DeleteShri Katju knows what he did was wrong. I rest my case.

@Codename V "It is possible to argue with those who know and those who do not know. It is impossible to argue with those who think they know. "

DeleteI really concur with your view .

this explanation is based on misunderstandings of some basic mathematical definitions. Firstly, undefined and indeterminate are not the same thing. In maths there is a proper definiton of indeterminate forms,which includes 0/0 and infinity/infinity (1/0 , 10/0 etc are not included in that definition) http://en.wikipedia.org/wiki/Indeterminate_form

ReplyDeleteso saying that 1/0 is indeterminate is wrong on the face of it. Now, coming to undefined and infinity, i am baffled by the stupidity shown by people across the board on this issue. If we agree that infinity is not a number and rather a notation that says that the particular result is ridiculously large then how can we differentiate b/w saying that something is undefined or infinity. if infinity is not the same as undefined, what is it?

Infinity and Indeterminate are two distinct and different

Deleteconcepts, they can not be used interchangeably. Infinity

(+ve infinity) is bigger than any finite number, it,

therefore, can not be equal to any number; Indeterminate,

on the other hand, is equal to more than one finite

number.

To say that 1/0 is Indeterminate is not correct because it

is 0/0 which is Indeterminate. We will see how.

We know 1 X 0 = 0 which gives 0/0 = 1, also 2 X 0 = 0

giving 0/0 = 2. So, 0/0 = 1, also 0/0 = 2, depending on

our starting equation. Therefore, 0/0 is Indeterminate.

If someone can show 1/0 to be equal to more than one

number, then, and only then, it would be settled that 1/0

is Indeterminate.

exactly

DeleteJustice Katju,

ReplyDeleteWhy you insist on boring us to death (plus waste miles of forests each time your rant is per force published by newspapers)? Why not be content and enjoy your twin incomes - judicial pension and salary (?) from PC (non)job - and leave rest of us alone?

Dear Gopi,

DeleteYou are epitome of idiosyncratic person. It's clear that a substandard person like you prefers to quantify everything in terms of money (look at your cheap way of mentioning pension + salary). If you find his articles boring, don't read it. Third class people like you should not comment and waste others time.

Take a chill-pill.

Couldn't have said better

DeleteTo Anonyms,

DeleteKab tak aap dono Baby Katju, Papa Katju ko defend karte rahogey "Anonymously"....He is capable enough to answer all these critisim oh his own.

Kind Sir (Justice Katju) , you can see the rebuttals presented by respected people , they imply by their assertiveness that they know maths better than anyone else and yet are completely oblivious to general rule of mathematics , and quote wikipedia which can be edited by almost anyone and is not a reliable source of information unless corroborated by reliable citation .

ReplyDeletePeople are baffled by stupidity that " division by 0 = undefined number and not infinity " again action oblivious to general rule of mathematics . After careful analysis , they have one thing in common that is logical fallacies :

Argumentum ad hominem

Argumentum Ad verecundium

and Strawman .

However for sake of clarity infinity is a value a mathematical concept which cannot be expressed in finite digits or numbers .

On the other hand, undefined means that no value has been associated with the variable so far.

Final conclusion : I implore you not to address such pitiful display of angst and contempt against you .

Dear Lucius,

DeleteI follow Justice Katju's articles / comments very closely and 95% time I am in agreement with his views. And I've been avidly following his blogs as well.

You have rightly pointed out that people (whom I call as semi-literate) give him suggestions on Maths, Science, Religion and above all Law. It is the hallmark of the ignorant to assume they know more than others, even in fields they have only second-hand information.

And many Indians simply don't accept their knowledge to be incorrect, because they are the classic Hindi phrase, "Koowe Ka Mendak".

To sum up, fools will continue to take objections to points, based on emotions and indirectly-acquired knowledge. By voicing their frivolous opinion, they further prove their stupidity.

I enjoyed your views / arguments as much as I enjoyed reading Justice Katju's blog.

Cheers!

Thank you brother :)

Delete@lucius .. using the terms 'infinity' and 'undefined' interchangeably can be pardoned, but using 'undefined' and 'indeterminate' interchangeably can't. Wouldn't you agree?

DeleteDear Lucius,

DeleteI'm choosing to remain anonymous, purely because of my job. I'm an Asst. Professor at an American University, in the Economics Dept. I don't want to make my personal views public and give an impression of being biased towards certain economic policies.

I too consider myself educated, not because of my PhD, but because of my critical thinking and exhaustive reading on Science, Religion, Economics and some Law. So, it was nice to express someone expressing their scientific views.

@Sid: You are basing your argument on personal opinions. Learn to base your arguments on facts, which most people don't. Think about it: Who question Justice Katju's views or ideas? Generally people with no real education or repute. How often do you find, say, Amartya Sen criticizing Justice Katju? You will never, because accomplished people do realize that Justice Katju is a scientific man and goes by logic. So, I wouldn't agree with your point and highly recommend you to change yourself and attain better levels of knowledge.

Cheers!

@Sid: Well, this Wiki link might help you understand things better: http://en.wikipedia.org/wiki/Division_by_zero

DeleteI humbly agree with your decision to remain Anonymous as protecting identity on internet is very important . My own name is a literal roman translation of my real name and I use it as a pseudonym . I understand the responsibilities of your job and work . I must say your credentials are quite impressive to say the least . :)

DeleteThanks Lucius, for your kind words. Over the past decade if one thing has become increasingly clear to me is the fact that when people investigate, free of bias and do their fair bit of research, their opinions match. However, people who think based on the information fed to them, they will never make sense. I'm glad our opinions are coinciding on most instances.

DeleteCheers!

I still think sir that 1/0 is a very large number but it would be a number; as you proposed correctly with using limit; that we call infinite; but intermediate is something that you can not determine at all, we generally saw 0/0 is indeterminate; as making 0 fraction of 0 is you can not do at all.

ReplyDeleteSo what I mean is, infinite is a very large number that you can never reach; while indeterminate is something that you can not determine at first place.

DeleteDear Mayank,

Delete1/0 is not a large number, but non-existent. Hence, Justice Katju's point that it is undefined is absolutely right.

To give you a point to think on... consider this: Ask yourself a question, 'who is questioning Justice Katju's views?' Not even a single well reputed person, because they know he is right. On the other hand, those criticizing him are sem-literate idiots (sorry for harsh language, but they deserve it). And do understand, he has been a Supreme Court Judge, who will have no hesitation in refraining from making remarks, if the subject is not known to him.

Cheers!

@anonymous ..'Justice Katju's point that it is undefined is absolutely right.' u say. Now if u had actually read the article, u would know that this wasn't his point. He said that 1/0 is 'indeterminate'. So apart from your ignorance of the concept of infinity, your unsubstantiated support for Justice Katju is also based on a non-reading of the disputed article. You keep focusing on 'who' is questioning Justice Katju instead of focusing on the facts of the matter.

Delete@Sid: So, you still hold by your views. You claim, 'infinite' is more closer to 'undefined' than 'indeterminate' is? Did I understand you point correctly? I don't want to talk Maths to you, as I hold a PhD is Economics from a US University and have done my share of Maths.So, let's argue the English language semantics. There is no such thing as 'more aptly interchangeable and less aptly interchangeable terms' in English language. And using the language principles if you are able to 'determine' what 'infinity' is, you win. Else, you will not lose, but educate yourself. I highly recommend you to do so.

DeleteAnd if you are not certain (unless you don't want to fit into Bertrand Russell's words, "The problem with the world is fools and fanatics are so sure of themselves and wiser people are full of doubts") try to avoid words like 'absolutely' in your arguments.

Take a chill-pill!

Infinity and Indeterminate are two distinct and different

ReplyDeleteconcepts, they can not be used interchangeably. Infinity

(+ve infinity) is bigger than any finite number, it,

therefore, can not be equal to any number; Indeterminate,

on the other hand, is equal to more than one finite

number.

To say that 1/0 is Indeterminate is not correct because it

is 0/0 which is Indeterminate. We will see how.

We know 1 X 0 = 0 which gives 0/0 = 1, also 2 X 0 = 0

giving 0/0 = 2. So, 0/0 = 1, also 0/0 = 2, depending on

our starting equation. Therefore, 0/0 is Indeterminate.

If someone can show 1/0 to be equal to more than one

number, then, and only then, it would be settled that 1/0

is Indeterminate.

D Balasubramaniam, Director of LVPEI - L V Prasad Eye Institute - Hyderabad, on the op-ed in question:

ReplyDeletehttp://www.thehindu.com/sci-tech/science/article3889304.ece

"As Shakespeare said: this judge doth protest too much, methinks".

I do not know if Justice Katju maintains this blog, but if he does, I hope he reads this one.

This is also for the anonymous economist commentator who holds a degree from a US university (as if that is the greatest qualifier) and the other over-enthusiastic commentator who brings in his life in UP and his secular value system for a question purely related to mathematical sciences. Before these two over-enthusiastic folks start their inanities about D.Bala, may be they should - just for a moment - go to the Hindu website and read D.Bala's articles in the past (he has been writing for The Hindu for like years now!)

Sir,

ReplyDeleteZero divided by zero has many answers.

0/0 = 1 (any number divided by the same number = 1)

0/0 = 0 (Zero divided by any number = 0)

0/0 = infinitive (any number divided by 0 is endless which is named as infinitive)

As such 0/0 is indeterminate

But

1/0 has only one answer and it is infinitive

Example 1 If X = 5 and A = 4, Find the smallest of the following three.

I) 2(X + A) – 8DIVIDED BY X + A+1

II) 8(X - A) – 8 DIVIDED BY X + A –1

III) 7(X – A) – 6 DIVIDED BY X – A –1

Answer : II) is the smallest as its value is Zero; Value of I) is one and III) is infinity ( a biggest number)

Example 2

Step

1..Let a = 1 and b = 1

2..Then a square = ab

3..asquare - bsquare = ab - b square

4..(a + b ) (a - b) = b ( a - b)

5..a + b = b

6..2 = 1

Step 4 is not correct as (a – b) – 0, and cancellation of 0 on both sides - dividing by 0 in balancing numbers in an equation is impermissible and such action will result in absurdity. Further cancellation of square in both sides of an equation is permissible only in the case of algebraic symbols and the same is impermissible in respect of constant numbers in view of double roots (+ 0r -).. There is vast difference between indeterminate (unable to decide right one) and impermissible (no entry for deciding). It is not impossible for a justice to consult experts in Mathematics.

Instead of generously accepting his fault he has entered for justification in vain. Raise a question totally irrelevant and insulting an innocent university professor as unfit on the wrong premises is untolerable. No one is infallible. A honourable man will always be first to accept his fault and seek pardon, He will avoid irrelevant questions hear calmly, bring out truth and decide rightly: he will not jump into conclusions in haste. The honourable judge may introspect for self corrections.

G.Kannapiran, Chennai

Respected Sir.

ReplyDeleteI read your arguments as well as this post regarding infinity. This is what I have to say on this matter because this opinion has not been exactly expressed.

1. As you rightly said, division by 0 is not permissible, and hence 1/0 is neither infinity, nor indeterminate it is simply "Not a number"

2. Indeterminate is something which is a number but whose value cannot be determined. But it has a value.

3. When we fall in these corner cases of indeterminate numbers we have to take the limits on parts that is causing the contention. In the case of function 1/x when x becomes 0 we have the problem. So we have to take the limit of x tending to 0.

4. In order to ensure that the limit exists we have to approach the limit from both sides of the number line. And if the answer matches, then we say that the limit exists.

So

Lim ( x -> 0+ ) 1/x tends to positive infinity

Lim (x->0-) 1/x tends to negative infinity.

Hence

Lim (x->0) 1/x is indeterminate (as the value doesn't converge on approaching the limit on both sides of the number line.)

The only "field" in which the question of asking what is 1/0 makes sense is the field of complex numbers. And here the answer is indubitably what is called "Complex infinity" or more simply, "Infinity". In more technical terminology, 0 is a first prder pole of the function 1/x, and the limit as x goes to 0 (as a complex number) of 1/x is infinity. The idea of treating Complex Infinity as a number in its own right leads to some very beautiful theory (with a wide range of applications), see for example the notion of a Riemann sphere.

DeleteIn the instance of this mathematical question, Justice Katju was very clearly wrong (and the poor mathematics teach was completely right). My only hope is that the distinguished jurist that Justice Katju is, he hopefully did not let this silly question come into play in his judgement of the teacher's case.

Another important point: in any reasonable university around the world, an undergraduate student who answers "indeterminate" without any further explanation to the question of what is 1/0 would unfailingly get a failing grade for his/her effort.

Delete