Hmm, nothing aye?
I should think that with the kind of swagger you throw around you would be able to rattle off 5 off of the top of your head…or at least come back with a quick “its not my job to disprove your useless kook theories” and furthermore “you know nothing about axiomatic set theory as it was taught to us at Harvard so please get lost”
Hmm, still nothing?
I saw that you were online all that time but have now left. I guess taking 30 seconds to cite one simple example of how .9999… + 1/∞ = 1 leads to contradictions is just too tall an order aye? Not even to solve the 821 message long conundrum and thorn in the side this thread is to you? Perhaps it just too trivial for you to be bothered with? Certainly you can, when you return, and have researched the matter, cite some examples of how the equation above will result in increased global warming, the ultraviolet catastrophe, and a general breakdown of real analysis.
If what you mean by things like .9999… and 1/∞ is the commonly accepted meaning, then .9999… = 1, and 1/∞, if it means anything at all (like the limit of the reciprocal of a quantity that is approaching infinity, under the standard definition of what such a limit is), is 0, and so, non-controversially, 1 + 0 = 1.
If you mean something different by these things, then you’re using your own, non-standard system, in which case the truth or falsity or meaningfulness of your assertion lies within that system.
If you haven’t specified/defined what you mean by .9999… or 1/∞, and you’re just relying on “common sense” or intuition or what we feel in our hearts that they ought to mean, then your assertion has no meaning (or has different meanings to different people) and is not worth arguing about.
Let me walk back my words some. If using the hyperreals were superior to the usual approach, they’d be much more widely adopted.
The NSA approach just swaps one set of baggage (e.g., epsilon-delta definitions of limits) for another set (e.g., the transfer principle and the hyperreals themselves). I don’t feel this provides enough simplification to make the switch. (I’ll admit having a single textbook to choose from doesn’t exactly thrill me either).
Either approach loses sight of what I want a student to get out of calculus: to have a conceptual understanding of what is being taught, to be able to apply this understanding to a wide variety of applications, and to be able to calculate. Expecting students to understand the inner workings and provide rigorous answers during their first exposure to the material is counterproductive, IMO.
In base 10, 1/2 has two decimal representations, but 1/3 has only one:
(1/2) = 0.49999…[sub]10[/sub] = 0.5[sub]10[/sub]
(1/3)= 0.33333…[sub]10[/sub]
This situation is reversed in base 3:
(1/2) = 0.11111…[sub]3[/sub]
(1/3) = 0.02222…[sub]3[/sub] = 0.1[sub]3[/sub]
I hope those claiming that 0.49999…[sub]10[/sub] is some special number distinct from 1/2 will tell us also about 0.02222…[sub]3[/sub], a number which, if “special,” cannot be represented in base 10.
Last month, my contributions to this discussion centered on two ideas.
First, that this whole issue is one of notation. It becomes this huge difficulty for some people only because in their minds, base-10 decimal arithmetic is somehow the right way to do math. It’s not really more right than any other base, it’s just what we’re used to and it works well for a lot of what we need it to do. Convert the problem to base-3 tricimal or to simple fractions and try again. You either have to admit it’s just an artifact of decimal notation, or you need to have some reason to think the alternate notation is somehow not dealing with the same numbers, just from a different point of view.
Second, I object to using infinity in mathematical expressions as if it were just another number. Without some pretty severe limitations put on it, this destroys your math in a hurry. You have to say that ∞ + 1 = ∞. In fact, ∞ + x = ∞ for all x. So, if you accept this kind of math, all numbers are equal to all other numbers. Big problem.
You don’t have to say ∞ + 1 = ∞ if you don’t want to (cf. the system of hypernumbers with a designated infinite element ∞, as I spelt out way back in the ancient mists of the thread (or surreals, or any other such thing)). And even if you do want to, you don’t have to say you can subtract ∞ from both sides to get 1 = 0 if you don’t want to (cf. the cardinals with a designated infinite element ∞, the ordinals with a designated transfinite element ∞, affinely extended numbers, or projectively extended numbers). You get to pick the rules you want to consider, and then see where they take you.
But you are right to say that the whole issue is one of notation.
We have the surplus notation of .9~. Let us use it.
I wouldn’t argue with you on any of that. My phrase, “Without some pretty severe limitations put on it,” was about just this sort of thing.
It just seems that most of the people using infinity in expressions in their arguments are not specifying limitations like you are, and in fact seem to think such rules are not needed, because they think their logic is intuitive. If their intuition leaves out these sorts of explicit rules, you get nonsense like I pointed out.
Ah, alright. I can’t argue with you not arguing with me. 
In some ways, we already have. Just not as part of the real numbers. Which is what most of this thread is about.
Sure you can.
1/3 = 0.33333…
1 - 1/3 = 1.00000… - 0.33333… = 0.66666…
1/7 = 0.142857 (that should be an overbar, not an underline, anyway the point is that it’s 0.142857142857142857 and so on).
1 - 1/7 = 1.000000… - 0.142857 = 0.857142
My comment was a throw away remark (let us throw it away).
I do not understand this comment. In what sense is .9~, in standard usage, not part of the real numbers?
It is until you claim that it should have a meaning distinct from 1.0~, which you seemed intent on in the comment I actually replied to.
Valmont314, you want an example of how .9999… + 1/∞ = 1 leads to contradictions? O.K., here’s one:
Start with .9999… + 1/∞ = 1.
So therefore 9.9999… + 10/∞ = 10 if we multiply both equation by 10.
Subtracting this first equation from the first, we see that 9 + 9/∞ = 9.
Now if 1/∞ means something, then 9/∞ means something. If .9999… is not the same as 1, then 9 is not the same as 9. There’s your contradiction.
Also you claim that when we calculate a limit, we’re supposed to use our intuition. The problem is that our intuition is often wrong. What’s the limit of this series?:
(1/1) + (1/2) + (1/3) + (1/4) + (1/5) + (1/6) + (1/7) + (1/8) + . . .
Your intuition might tell you that it has some limit. After all, if you calculate the sum of a lot of terms, you’ll see that the total grows more and more slowly. So you might think that it has a limit.
But that’s wrong. The series has no limit. Break up the series as follows:
((1/1)) + ((1/2)) + ((1/3) + (1/4)) + ((1/5) + (1/6) + (1/7) + (1/8)) + . . .
In other words, put one term in the first group, one term in the second group, two terms in the third group, four terms in the fourth group, eight terms in the fifth group, sixteen terms in the sixth group, thirty-two terms in the seventh group, etc. All the terms in the first group are larger than 1, all the terms in the second group are larger than 1/2, all the terms in the third group are larger than 1/4, all the terms in the fourth group are larger than 1/8, all the terms in the fifth group are larger than 1/16, all the terms in the sixth group are larger than 1/32, all the terms in the seventh group are larger than 1/64, etc. So the sum of the series is larger than this series:
(1 * 1) + (1 * (1/2)) + (2 * (1/4)) + (4 * (1/8)) + (8 * (1/16)) + (16 * (1/32)) + (32 * (1/64)) + . . .
Each of the terms in that series is larger than 1/2. So the sum of the original series grows without limit. For any given positive number, there is a length of the series so that the sum up to that length is greater than that positive number.
One could define “.9~” to mean 2; that is a real.
Perhaps you meant that if we wanted to have a special meaning for the relevant base 2 expression, we would already have distinguished “.9~” from “1”. Well perhaps so, but strictly speaking, we need not, and if we haven’t we have surplus notation, because we have “.9~” and “1”; ergo, it is not impossible to represent the base 2 expression (while keeping in-tact almost all of the standard notations meaning).
Yes. By all means let us throw your original comment away and throw this one away, too. That’s the nicest, gentlest thing I can think of right now.
Can you think of anything to say that shows I have said something false?
Suppose you define .9999… as being equal to 2.
Then you start with the equation .9999… = 2.
You multiply it by 10 to get 9.9999… = 20.
When you subtract the first equation from the second one, you get 9 = 18, which is clearly not true.
So if you define .9999… as being equal to 2, you quickly reach a contradiction, which means that it’s not a good idea to choose that definition. Saying that mathematics is a matter of sticking to your definitions doesn’t mean that any set of definitions that you choose would be equally good. Some of them are self-contradictory. Some of them produce a system that has no contradictions but which is useless for doing mathematics.
The idea is to define “.9~” as being substitutable wherever “2” the number occurs, not “2” the numeral. So “9.9~” is not equivalent to “9.2”, for instance. (All I want to do is map one term to another; not other terms that can be built up from that term.)
Now, granted, you cannot express multiplication identically, but that doesn’t mean things need to get mucked up. Just take whatever your rule is, append, “if .9~ is a term, substitute 2”.