Maths FAQ: 0/0, Infinity/Infinity, What is Infinity, 0.999..=/=1

[shmoe]

OK, didn't know calculators could do that; I don't own one.

Some will, I think some will also say 1.25!=error. I'm not sure about mine, I can't find it and I don't think it has batteries even if I could. Maple will deal with them though (it also has built in gamma of course).

 

[mathwonk]

by the way I am puzzled as to why my definition of .9999..., as the smallest number not smaller than any of the finite approximations, i.e. as the least upper bound of the sequence {.9, .99,.999,.9999,...} of rationals, was considered only "intuitive" and not rigorous.

 

[matt grime]

Is that how it came across? Sorry, I'll correct it.

 

[matt grime]

reading back i don't see that i said your description isn't rigorous. could you elaborate for me?

 

[mathwonk]

i was referring to the remark:

"I think there should be an intuitive version and a rigorous version establishing the notion that 0.9...=1. The intuitive version was posted by mathwonk and the rigorous verison from Mr. Grime.: by phoenixthoth.

but i shouldn't have said anything. he probably meant my 1 = 3(1/3) = 3(.3333...)

= .9999... argument.

 

[mathwonk]

Just to play devil's advocate, isn't any textbook in math essentially a list of FAQ's with answers? I suspect people who ask these FAQ's just do not read.

 

[mathwonk]

on the other hand, what about adding some frequently ignored principles, like "we will not do your homework for you", "please do not double post", "try to be polite (restrict use of demands all in caps, refrain from repeating the same post if it at first proves unappealing to us ),...

 

[matt grime]

OK, thought you were referring to me. The link to tim gower's page shows how to make the reals from decimals.

I can't think of any textbook I came acros that showed why 0.99..=1, or said what infinity 'is'. The topics I have in mind are those that use "intuitively" concepts without defining the concepts. And these questions get asked a lot, and the same people post the same replies, so it'd be good to have the replies in one place before the question gets asked. Perhaps.

 

[mathwonk]

an idea i put elsewhere is to answer the FAQ: 'what is a "good" textbook in ...'? we could list a bunch of recommended books, especially free online ones, and tutorial websites in one standard place.

as to infinite decimals as reals, they are treated that way in the appendix to spivak's calculus book. I taught from that one summer to a bunch of bright high schoolers on an NSF sponsored project. I also wrote notes proving all usual properties including how to add multiply etc... infinite decimals, and prove archimedean property, completeness etc... Two of the students even wrote a rap song about real numbers, unfortunately not preserved. Of course like all my stuff the notes are simply mouldering on a desktop computer but not online.

I first learned about infinite decimals and the possible ambiguity of their use in representing reals such as 1 = .999... from the calculus book of Courant vol 1, pages 8-10, some 40 odd years ago in freshman calculus. I recall being very impressed at the clarity of his explanation at the time, as if "scales had been lifted from my eyes".

 

[Gokul43201]

Is this idea going somewhere or has it already fallen by the wayside ?

Mentors ?

Also, the conversation between matt and a math beginner here is revealing :

https://www.physicsforums.com/showthread.php?t=43825

Beginners seem to find the language daunting.

This should be perfectly clear : (I quote from matt) "plus there is only one ordering of the empty set - an ordering of a set of n elements being all the ways of picking the n objects out in turn (without replacement). the set with no elements in is the empty set, there is only one way to pick out elements to get a set with no elements in it." yet it clearly isn't.

Perhaps the use of a little verbosity, and a sprinkling of examples may get the point across to a beginner ?

 

[mathwonk]

i agree with Gokul that beginners find math language daunting. if you have ever read one of my posts you will note i seldom use standard math language in talking to beginners, or even to non beginners, but try to use everyday common sense language.

i also agree with Gokul that beginners find examples more helpful than theory, but admit i am frequently too lazy to provide them.

I disagree that it should be perfectly clear to anyone that the empty set has only one ordering. it was not clear to me for example. it seems to boil down to the rather odd fact (for beginners) that the empty set has exactly one subset.

 

[Gokul43201]

I disagree that it should be perfectly clear to anyone that the empty set has only one ordering. it was not clear to me for example. it seems to boil down to the rather odd fact (for beginners) that the empty set has exactly one subset.

Okay. Perhaps it is more intuititve that C(n,0) = 1 (or is it not) ? Could that be used instead to explain why 0!=1 works ?

 

[Chrono]

i agree with Gokul that beginners find math language daunting. if you have ever read one of my posts you will note i seldom use standard math language in talking to beginners, or even to non beginners, but try to use everyday common sense language.

i also agree with Gokul that beginners find examples more helpful than theory, but admit i am frequently too lazy to provide them.

I agree with all this, also. A friend of mine uses examples and nothing else when he studys math. Not only can the math language be daunting but it can just be downright confusing and that alone can turn people off to it.

 

[Hurkyl]

I've been quite distracted as of late, I haven't had the time to properly follow the thread.

 

[matt grime]

Is this idea going somewhere or has it already fallen by the wayside ?

Mentors ?

Also, the conversation between matt and a math beginner here is revealing :

https://www.physicsforums.com/showthread.php?t=43825

Beginners seem to find the language daunting.

This should be perfectly clear : (I quote from matt) "plus there is only one ordering of the empty set - an ordering of a set of n elements being all the ways of picking the n objects out in turn (without replacement). the set with no elements in is the empty set, there is only one way to pick out elements to get a set with no elements in it." yet it clearly isn't.

Perhaps the use of a little verbosity, and a sprinkling of examples may get the point across to a beginner ?

I agree, this was off the cuff nonsense really. however, there is exactly one set of cardinality 0 in the power set of the empty set, which is what I was trying to get across.

That there is one ordering of the empty set isn't obvious, and for that matter it could be argued that it is infact an axiom that it has one ordering since it is difficult to define orderings of sets with no elements.

 

[matt grime]

Okay. Perhaps it is more intuititve that C(n,0) = 1 (or is it not) ? Could that be used instead to explain why 0!=1 works ?

That only works if we *have* to define n choose zero as n!/n!0!, there's no reason to suppose that we aren;t' just making a special case here either. However, I for one find the consitency argument reasonably compelling, and we could just say that it can be justified by thinking about empty orderings if one so chooses.

 

[matt grime]

That other thread with the poster asking aobut 0! was interesting in that it revealed they were asking someone to prove a fact about 0! when they had seemingly had no explanation of what n! describes (orders of sets) personally I don't think that n! shoud be defined as the number of was of ordering a set. It's just a recursive function that happens to be used to indicate the orderings of a set (for n non-zero), otherwise what's to stop aleph-0! being a factorial?

 

[mathwonk]

i decided the number of orderings equaled the number of permutations, i.e. the number of graphs of bijections between the empty set and itself, and since those graphs are subsets of the empty product set, there is at most one of them, the empty graph. now that is a bijection by default i suppose.

i have always reminded people the reason we make conventions is to make our lives easier, so if setting 0! = 1, makes more formulas hold then I'm all for it.

 

[matt grime]

The problem there (and I don't think it ought to be a problem, but I can foresee it being one) is trying to explain how you can have a funtion from the empty set to itself and that there is exactly one of them (because it's the initial object in the category set anyone?)

 

[mathwonk]

As usual, (the graph of) a function from the empty set to itself is by definition a subset of the product of the empty set with itself, satisfying the properties that define a function, i.e. every element of the empty set occurs exactly once as first entry in some pair of the subset.

The only subset of that empty product is the empty graph which does satisfy the function property for the empty domain. so there is exactly one such function.

Of course my point was that it was not obvious how to explain this to a member of the general public.

 

[matt grime]

Ah! That's something that ought to get a mention: the definitoin of a function. Mind you I'd probably only get on my high horse and contradict teachers who set silly and unmathematical questions such as: what is the largest domain of the function 1/(x^2-2)

 

[Zurtex]

It all looks good to me, I think a general FAQ will save a lot of time. Although it will need to include as many methods as possible on how to prove something and demonstrate them clearly enough for anyone to understand (remember that a lot of people who claim that 0.999... does not equal 1, will also probably have little understanding of mathematical terms beyond early years of high school).

 

[Chrono]

It all looks good to me, I think a general FAQ will save a lot of time. Although it will need to include as many methods as possible on how to prove something and demonstrate them clearly enough for anyone to understand

I can't wait for it! I'm already getting excited.

 

[mathwonk]

I once taught a prep class for potential elementary school teachers in which we discussed why .9999... = 1, in our general discussion of what numbers mean and how you represent them and add them and so on, including "carrying" and "borrowing". One of them got so excited about it she tried to convey it to her peers at her next job. I think they shunned her though.

 

[Chrono]

I think they shunned her though.

Now, why would they do such a thing like that?

 

[Hurkyl]

Bumping so I don't lose this again!

 

[Gokul43201]

Rebumping ... in light of parallel discussion here.