If we are given that (x-2)(x-3)=0 and 0.0=0, then we can conclude that both x=2 and x=3 simultaneously. This is because x-2=0 and x-3=0, simultaneously, is consistent with 0.0=0. However, this leads to a contradiction, namely, x=2=3. So, generally we exclude this option while finding roots of an equation and consider that only one of the factors can be zero at a time i.e. all the roots are mutually exclusive. In other words, we consider 0.0 to be not equal to 0.
Now, if we are given that x=0 and asked to find out what x^2 is, then certainly we conclude that x^2=0. It is trivial to observe that this conclusion is made through the following process: x^2=x.x=0.0=0. That is, we need to consider 0.0=0 to make this simple conclusion.
Therefore, while in the first case we have to consider 0.0 not equal to 0 to avoid contradiction, in the second case we have to consider 0.0=0 to reach the conclusion. So, the question arises whether 0.0 is equal to 0 or not. As far as I know, mathematical truths are considered to be universal. However, in the present discussion it appears to me that whether 0.0 is 0 or not, is used as par requirement. Is that legitimate in mathematics?
If we are given that (x-2)(x-3)=0, then at least one of (real) numbers, (x-2) or (x-3), has to be equal to 0 (both can also be equal to 0).
If we are given that (x-2)(x-3)=0, then at least one of (real) numbers, (x-2) or (x-3), has to be equal to 0 (both can also be equal to 0).
Anton Vrdoljak ...thanks for your answer. However, if your suggestion is correct, then it leads to the conclusion x=2=3, because you have x=2 and x=3 as simultaneous true statements. Therefore, I can not accept your answer unless you clarify this dilemma. In fact, in his first proof of the Fundamental Theorem of Algebra, Gauss considered only ``either-or'' as his explanation i.e. either x=2 or x=3 is possible, but not both simultaneously. Boole, in his Laws of Thought did the same while explaining that his logical variables can take values either 1 or 0 (not both simultaneously).
I'm at the bottom of a pit and toss a ball high into the air - well above ground level - and it falls back to my hand. At what time is the ball at ground level (height = 0)? The kinematics equation is: H(t) = .5*g*t^2 +Vi*t + Hi and it's quadratic (just like your example). We set H to 0 and calculate time. We get two answers. Are Math and Physics now illegitimate? No. There really are two times when height equals 0, once while the ball is going up and once again while the ball comes down. As Anton points out, there is no problem with (x-2)(x-3)=0 having two solutions because either factor as 0 makes the product 0. I hope adding a real life example helps make it sensible, Abhishek. Best wishes, Kevin
Kevin Grobman ...thank you very much for your answer. However, I believe you missed the point of my question. The question is not whether there are two roots. The question is whether these two roots are mutually exclusive or both are possible simultaneously. I hope you understand.
Perhaps we are speaking passed each other. I did not mean to convey just that there are two solutions, but those two solutions are sensible and distinct. Let's say you film the ball toss. Two solutions to the equation mean when you look at the film there will be two frames (at those values of t) where the ball really is at height 0. Does that mean these are the same moments in time (same frame in the film)? No. On one frame the ball is on the way up and the other on the way down. It means these moments in time resemble each other in the way modeled by the equation. Stated with Boolean Logic (to address your response to Anton), there are two values of X for which the equation (X-2)(X-3)=0 is true (2 & 3). For any other value for X the equation is false. In no way does this imply 2=3, just like in no way are all moments the same because a ball is at the same height.
@Kevin Grobman, If we could see through the boolean logic then, there is a case which has the condition that both x=2 =3 simultaneously.
Anton Vrdoljak , Kevin Grobman ...if x=2 and x=3 are simultaneously true, then the law of identity (sometimes called Leibniz's law) is violated, which is one of the founding stones of logic in general. You can consult, for example, Chapter III of Alfred Tarski's book: Introduction to logic and to the methodology of deductive sciences, Fourth Edition, in order to understand what is the law of identity.
As far as Grobman's physical examples are concerned, those are not questions of logic and pure mathematics. Doing physics in terms of formal logic has not yet been done. One needs to consider the axioms of geometry, algebra and arithmetic very carefully to write physics in terms of logical jargons, with a special focus on the explanation of whether ``physical dimension'' is ``number'' or ``not number'' (see this link to understand my concernPreprint A logico-linguistic inquiry into the foundations of physics: Part I
).``0 height'' (as Grobman writes) is a meaningless statement, if not blatantly false. For example, Frege wrote on page 11 of his book Foundations of Arithmetic, ``for up to now no one.... has ever seen or touched 0 pebbles''. However, such (logically careless) application of the concept of ``0'' is very useful in doing calculations and experiments (after all we are reaching Mars with such scientific analysis!). The point is that, what is useful is not always logical. Last but not the least, if a really logical analysis of a physical example is used to answer this question, then I shall seek a complete least of the atomic premises of the analysis, which is certainly not given by Grobman. There a clarification need to be given about the distinction between ``number'', ``quantity'' and a clear explanation of ``0 height'' in terms of measurement. Otherwise, such explanations are irrelevant in the context of my question.
Propositions are true or false; not so for variables, logical symbols, etc. Height of 0 is neither true nor false; it's a location within a frame of reference to map an equation to reality. I designed the example to be intuitive and like yours. It wasn't meant as Pure Mathematics, though Classical Mechanics of Physics was developed within an axiomatic system by Isaac Newton in Philosophiae Naturalis Principia Mathematica. Returning to (X-2)(X-3)=0, rather than saying X is "simultaneously" 2 and 3, a more precise statement is the set of values for X in which the equation is true is 2 and 3. That is, don't say "X=2 & X=3" instead say " X ∈ {2,3} " No contradiction. I felt I could be helpful by adding an intuitive, concrete example. Unfortunately I feel it's best I step away now; I can not think of additional ways to help you see how you're mistakenly concluding 2=3.
Kevin Grobman ...thank you again for your reply. Here goes mine, corresponding to each of your statements.
``Propositions are true or false; not so for variables, logical symbols, etc.'':
I agree but I could not understand why this statement is written. Anyways, thanks for rewinding.
``Height of 0 is neither true nor false; it's .... reality.'':
Then there was no point of giving such example to answer a question of logic. Logic can be done with statements which are either true or false. Also, I do not understand the meaning of the phrase ``height of 0''!!
``I designed the example to be intuitive and like yours.'':
I am confused because Boolean algebra in different from Brouwer's Intuitionistic logic (Heyting algebra). The law of the excluded middle gets questioned if you bring in `intuition'. Further my question does not involve any intuition and therefore, your example was certainly not like mine. My question deals with the way in which we come to conclusions from a simple quadratic equation about its roots and the logical connections among those conclusions.
``It wasn't meant as Pure Mathematics, though Classical Mechanics of Physics was developed within an axiomatic system by Isaac Newton in Philosophiae Naturalis Principia Mathematica.'' :
When I read Newton's principia, I do not find any logical analysis with a strict attachment to the axioms and definitions i.e. by considering them as formal statements. Rather those were informal reasonable and useful analyses. Mach (Science of Mechanics) and later Bridgman (The Logic of Modern Physics) would call the logic of physics as ``operational'' rather than formal. For example, one can immediately see statements like ``velocity=2'' in Principia, which is formally illogical because there is no unit mentioned on the right hand side. It is a basic query that comes to mind regarding the unit with which that velocity is measured because the number ``2'' depends on the unit of measurement. However, if the reader accepts that there is some unmentioned unit is such statement and carry on with such analysis to do some meaningful calculations and practical applications, then it becomes a truth by practice. This certainly does not mean the the statement ``velocity=2'' is a logical statement because measurement of a quantity is done by another standard quantity (unit) of the same type (physical dimension) which generate some number. This is the way measurement was given meaning by Einstein among many others e.g. see The Meaning of Relativity by Einstein if you are honestly interested. But the physics literature is plagued with such illogical statements where the measuring units are not mentioned while doing calculations and it is accepted like that. I shall not waste more time in discussing this issue as it is not immediately relevant in the context of my question.
``Returning to (X-2)(X-3)=0, rather than saying X is "simultaneously" 2 and 3, a more precise statement is the set of values for X in which the equation is true is 2 and 3.'
That is, don't say "X=2 & X=3" instead say " X ∈ {2,3} " No contradiction.'':
There is no contradiction because this is not an answer to my question. Everybody knows that there is a set of two roots (2,3) and you have just written it with formal notation. Your statement actually means that ``do not ask the question'' and just accept that there are two solutions without worrying about the logical connection between the two. I have no problem if you are satisfied with your understanding. It is a personal choice.
``I felt I could be helpful by adding an intuitive, concrete example.'': Thank you for trying to help me. I am glad.
``Unfortunately I feel it's best I step away now; '': As you wish.
``I can not think of additional ways to help you see how you're mistakenly concluding 2=3. '': Then, please do not waste your precious time anymore to find additional ways. As far as ``mistakes'' are concerned, I have made many in my life and it is fun. For example, I put a dot on the paper and say that ``it has no parts and therefore it is a point'' because Euclid said so. Then I realize that I can not see the dot if it has no extension; so it must have parts but not describable in terms of geometry. But this mistake is not considered as ``mistake'' because everybody makes this mistake to do geometric analysis. So, it is just a useful lie. Therefore, even if I have now mistakenly concluded 2=3 (as you wrote), I shall try to make it useful. Let me keep my belief as you keep yours.
Regards
Abhishek
The question and most of the answers seem to be crazy for me.
More interestingly, the question of division by zero is also investigated
some well-known mathematicians.
Whenever, someone is starting with an axiom system for the set
R(+, . ,
Wrong assumptions lead to wrong results.
Correctly ,we must say ,If we are given that (x-2)(x-3)=0
x-2=0 OR x-3=0.
This leads to
x=2 OR x=3,
BUT not x={2,3} in the same time, that is
0.0=0, then we can conclude that both x=2 and x=3 simultaneously.
I think it clear.
There are many wrong assumptions like this. For example
if x=y, then
x^2-y^2=yx-y^2
(x-y)(x+y)=y(x-y)
x+y=y
2y=y
2=1, this is wrong !!!!
``Wrong assumptions lead to wrong results.'': Nice beginning!
``Correctly ,we must say ,If we are given that (x-2)(x-3)=0
x-2=0 OR x-3=0. This leads to x=2 OR x=3, BUT not x={2,3} in the same time, that is 0.0=0, then we can conclude that both x=2 and x=3 simultaneously. I think it clear. '':
No, it is not clear. Please clarify the following: Do you mean EITHER x-2=0 OR x-3=0 (i.e. the roots are mutually exclusive)? It seems so because you have excluded the case ``both x=2 and x=3 simultaneously'' which is equivalent to REJECTING 0.0=0 (equivalent to ACCEPTING 0.0 not equal to 0) as you have admitted in writing. In case, you REJECT 0.0=0 (which I assume you have done), can you please answer the following question: Given x=0, what is x^2 (i.e. x.x)?
``There are many wrong assumptions like this. For example if x=y, then
x^2-y^2=yx-y^2
(x-y)(x+y)=y(x-y)
x+y=y
2y=y
2=1, this is wrong !!!!''
In this example you have made a mistake to get to this wrong result. While going from the second to the third step, you have cancelled (x-y) from both sides. However, this is not possible because x-y=0 and 0/0 is indeterminate. This is not at all similar to what I have asked. If you do wrong calculation and arrive at a conclusion which you call wrong, then that is tautology.
Yes!!! (x-2)(x-3) = 0, inplies x = 2 OOORRR x = 3. Noooottt!!!! x = 2 and x = 3,
Because logically is wrong!!!!
@Pedro, I don't understand why the word or is emphasised . As per my understanding , a quadratic equation must possess exactly 2 roots so this equation (x-2)(x-3)=0 contains two distinct roots which are x=2 and x=3. For better visualisation , one can easily plot that equation to find the roots where the function cuts the x-axis. It does not mean that x=2=3. Do I miss any necessary information?
Abhishek Majhi The field of real numbers is an integral domain where a*b = 0 implies either a = 0 or b = 0 or a=b=0. the equation you mentioned follows the case either a = 0 or b = 0. if you take (x-2)(x-2) = 0 here 0*0 = 0 for x=2.
Pedro Mateus ...Can you please confirm whether you are REJECTING 0.0=0 or ACCEPTING 0.0=0? If x=2=3 is wrong and if we do not want to do wrong mathematics, then we should not accept this and that can only be done by REJECTING 0.0=0. Isn't it?
@Gourab Kumar Sar...do you want to mean that 0.0=0 is sometimes true and sometimes false depending on the context or the problem at hand?
To all:
Borrowing from Munkres: In ordinary everyday English, the word “or” is ambiguous. Sometimes the statement “P or Q” means “P or Q, or both” and sometimes it means “P or Q, but not both.” Usually one decides from the context which meaning is intended.
In mathematics, one cannot tolerate such ambiguity. One has to pick just one meaning and stick with it, or confusion will reign. Accordingly, mathematicians have agreed that they will use the word “or” in the first sense, so that the statement “P or Q” always means “P or Q, or both.” If one means “P or Q, but not both,” then one has to include the phrase “but not both” explicitly.
What is beeng said is that: Logic and good sense has to rein when people do Mathematics. Is it reasonable to write x = 2 = 3? Mathematicaly no. Logicaly no. Humanly no. Although it comes from (x - 2)(x-3) = 0, it is not accepltable to write down x = 2 = 3. In this case you need to discern. x = 2 Or x = 3, Or in this case is exclusive, not inclusive. This type of OR is common in Programming. In these contexts you can't mix things. Othewise you get into that kind of absurdity x = 2 =3.Rationally wrong.
The polynomial p(x)=(x-2)(x-3) has two roots over the reals. They are 2 and 3. That says nothing about 0*0. x=2 is one root. That means that (x-2) divides p(x) or q(x)=p(x)/(x-2) = (x-3) is in the ring of polynomials over the reals. The root of q(x) is equal to 3. Hence both (x-2) and (x-3) divides the original polynomial p.
BTW 0*0 is zero. As pointed out in an integral domain (no divisors of zero - a*b = 0 implies either a or b must be zero) 0* any other element is zero. However, 0/0 is not defined since 0 is not a unit (that is 0 has not multiplicity inverse). In the reals all elements except zero are units.
The original assumption made that p(x)=0 implies that x = both 2 and 3 is wrong since p(2)=(2-2)*(-1)=0 and similarly for p(3) = 0 and p(x) = 0 if either x = 2 or x =3.
Now if one considers the polynomial of two variables p(x,y)=(x-2)(y-3) then it has one solution in the affine space R^2, that is the point (x,y)=(2,3).
either of x-2 or x-3 is equal to 0 but not both at the same time.
The equation f(x)=0 may have no solutions. It may have one, but it may have more. This is not surprising and contradictory to anything.
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