@robjohn You commented in your answer to the question "How to prove this particular limit as x->0 (sin x ) / x = 1" "understood, to be proven later". In any definition, there will be things that are left as understood. & a little bit of hand-waving can make believable... These are excellent examples of a recurring theme in mathematics found from the earliest elementary school days all the way up through Calculus and beyond...
@Skullpatrol That doesn't mean the proof doesn't exist; it just means that for this specific case, detouring into a complete proof breaks the flow of the issue in question.
I can't take this much longer. I thought I had reached the necessary points in FA to be admitted to the exam and now I found out that it's actually going to be a close shave and that I have to get full marks in the next 3 assignments.
@WillihamTotland Again, I agree a several mile long proof is unrealistic, but try to think of the earliest example of this when we were all told division by zero is impossible, and presented with the definition of division based on multiplication, but were never given the proof that zero times any number is zero.
In any definition, there will be things that are left as understood. & a little bit of hand-waving can make believable...
but as you said detouring into a complete proof breaks the flow of the issue in question.
Ten different letters of an alphabet are given. 2 of these letters followed by 2 digits are used to number the products of a company. In how many ways can the products be numbered? Answer of mine is (10p2)*(10p2) = 2025 but right answer is (10*9)(10*9)=8100 by principle of counting , why is that we can't use direct formula in this (10p2)*(10p2) ?
@robjohn Ten different letters of an alphabet are given. 2 of these letters followed by 2 digits are used to number the products of a company. In how many ways can the products be numbered? Answer of mine is (10p2)*(10p2) = 2025 but right answer is (10*9)(10*9)=8100 by principle of counting , why is that we can't use direct formula in this (10p2)*(10p2) ?
@robjohn you've already answered that , thanks , I was doing the wrong formula
@robjohn Would you like to elaborate on your comment to the question "How to prove this particular limit as x->0 (sin x ) / x = 1" "understood, to be proven later". In any definition, there will be things that are left as understood. & a little bit of hand-waving can make believable...?
@AsafKaragila " boring and bothersome" is how most people describe mathematics in general, what I'm trying to discuss here is why they feel that way or more precisely what is it about mathematics that gives it a bad rap.
These are excellent examples of a recurring theme in mathematics found from the earliest elementary school days all the way up through Calculus and beyond...
To provide excruciating detail at an elementary level can detract from the process of understanding the idea being explained. The purpose of a proof is to convince the reader of the truth of that which is being proved.
@robjohn If you find this topic to be ill-defined for a mathematical chat please say so and I will stop due to the risk of being called a troll, etc...
But as far as the area or arclength being proportional to the angle of the arc/wedge is one of those details that may require calculus and limits to show rigorously, but which is obvious to most readers.
@WillihamTotland As I said, it may require calculus to prove rigorously. However, one must be careful that the same thing being proven was not used in developing the calculus being used.
@robjohn Is "left as obvious" the same thing as circularly defined within a very small circle? Such as a numerical expression represents a particular number & A particular number is what is represented by a numerical expression.
@Skullpatrol not circularly defined, but something that the reader should have no difficulty accepting as true, but whose rigorous proof would take the discussion far afield and almost off-topic.
What a particular reader "should have no difficulty accepting as true" depends on the reader.
@robjohn What do you think of a numerical expression represents a particular number & A particular number is what is represented by a numerical expression?
I guess it would depend on the expression and context. sum 1/k^2 is commonly accepted as pi^2/6 but depending on the purpose of the expression its proof is appropriate.
Could I make one comment on a "starred" comment made by QED on the side panel on the right? "how do we get people to stop thinking of "infinity" as a number"
@WillihamTotland This is a tough one to not "confuse practical considerations with mysticism."
If we take away the reflexive property of equality from infinity and say infinity=/=infinity then it can not be a real number and people will stop thinking of infinity as a number.
For there are different orders of infinity that are certainly not comparable to each other
We don't need to go to the cardinality for reasons to say that \infty \not= \infty
saying that \infty=\infty leads us into shaky ground when computing with \infty, and as JM points out, we have to be very careful how we define computations involving \infty
Also, the idea of \infty and NaN in computer science bleeds back into pure math and obscures what \infty is in math.
Oops ... I'm sorry I asked such a loaded question. I humbly apologize for all this. The question started off as: how do we get people to stop thinking of "infinity" as a number? And I said take away some of the properties of Real numbers that people use with it such as the reflexive property of equality. Then people can not say infinity=infinity and must say infinity=/=infinity ... that should stop them from thinking of infinity as a Real number, in my opinion.
@Skullpatrol By THAT I mean that it is obvious that you lack whatsoever mathematical education. While it is clear that you want to understand, it also seems that you focus too much attention on the wrong parts of mathematics. If you want to understand these things study logic, study set theory, study category theory.
@Skullpatrol And yeah, I was trying to change the subject. I humbly apologize sir. I am a dust at your feet, please forgive me.
@RajeshD how do we get people to stop thinking of "infinity" as a number? And I said take away some of the properties of Real numbers that people use with it such as the reflexive property of equality. Then people can not say infinity=infinity and must say infinity=/=infinity ... that should stop them from thinking of infinity as a Real number, in my opinion.
Either way, I cannot be a part of a global conspiracy about changing the way people think. I am Jewish by ethnicity and this will automatically create a negative bias towards both Judaism and the cause of the conspiracy.
So as long as I am present in the channel you cannot ask people this, or I will be held a part of it.
The question is a pet peeve. What's worse is your inexplicable rattle afterwards which contain pseudointellectual ideas which are neither math nor philosophy and were long tossed into the garbage bin of pop science and pop philosophy to stew in the drip of mindless drones all over the internet.
I have an acceptable answer. The problem is that a person limited so much by his own mind cannot grasp the fact that none of that matters. None of the questions which you have vomit through the keyboard onto this chatroom have any place in mathematics and hardly even so in mathematical philosophy. They contribute nothing to mathematics and simply waste time of both people dealing with this sort of question.
This puppy is entirely useless. I wanted a Great Dane because I wanted people to stay off me and change sidewalks when they see us coming but she's so cute, gorgeous and well-behaved that everyone wants to touch her.
@AsafKaragila So you don't like BO either! This reminds me of the discussion we had the other day. I've been wondering if a guy who doesn't use deodorant is more likely to be single. And the other way around. So, Asaf and Jonas are not single which makes it likely that they shower and use deodorant.
