(In case my last paragraph gives the wrong impression, I don’t particularly have any problem with anything you’ve done in trying to bring posters such as 7777777 to understanding, The Hamster King. I’m just defending my choice to engage in the more sympathetic exercises I’ve engaged in, on top of all the other posts.)
Even your own cite points out the problem in this idea. Personally, I consider it a proof by contradiction:
- Let’s say …99999 = E
- Multiply E by 10 to get 10E, which would be …999990.
- 10E - E = 9, clearly. One could also say E - 10E = 9.
- The first equation resolves to E = 1. The second to E = -1. Both these results follow from the premises.
The problem is treating an infinite value (in this case symbolized by E) like a conventional algebraic variable. It’s the same as defining B = 0, slipping a “now divide both sides by B” into the sequence, and getting a nonsensical or contradictory result.
James Tanton clearly recognizes this.
No, you’ve made a sign error. 10E - E = -9, clearly, just as expected. We can make good sense of …999 = -1, if we want to, as I’ve noted above; for example, using the method outlined in this post to interpret infinitary decimal notation into the real numbers, or in terms of 10-adic numbers, or various other things.
Which isn’t to contradict that it is also frequently useful to instead think of …999 as a positively infinite value, as you say.
I agreed with pretty much your entire post, including the quote above. Unfortunately, I’m not sure this is a case of someone willing to explore but cowed by more sophisticated and cynical individuals, I think it is the opposite. Self-motivated exploration is not a quality usually associated with wanton stubborness to engage.
Oh, I agree. 7777777 is a lost cause. But it’s the general principle of the thing, for the sake of anyone else reading the thread.
This goes back to what I said earlier. People who don’t like the fact that 0.9999… = 1 want everything to be arithmetic. Your argument is conceptually equivalent. You are insisting that numbers are arithmetic and anything not learned in high school is meaningless, useless, or just plain insulting to normal people.
Math is math. It includes arithmetic but that’s like saying that a human includes nostril hair. Nostril hair is useful but is such a ridiculously trivial part of being human that inflating its importance is laughable.
If you’re a layman, then here’s what you do when the topic is math. Shut up and listen. Maybe even learn.
You say the same thing more explicitly in your next post so I’ll just concentrate on a different aspect of it.
I’m under the impression that most of Chomsky wrote on linguistics has been discredited to the point where only a few diehards adhere to his theories, similar to the fate of Freud.
Whether humans have an innate sense of counting numbers is debatable. Its truth doesn’t matter because - again - arithmetic is not math.
Not at ALL what I said
Note that I actually specifically said that are “useful”
Also not what I said. I said an innate ability to learn math, and a species-wide tendency to develop it (note that numerous societies independently figured out the same mathematical principles). I also pointed to a proven innate sense of geometry that was proven in a study
You clearly weren’t getting what I was writing, so feel free to go back, read it carefully, don’t add anything into it or put your own spin on it and instead actually try to listen and absorb it, and then maybe we can talk
Nope. If E is …999999 and 10E is …999990 :
*E*: ........99999999
10*E*: - ........99999990
==============
9
Why not? They both go infinitely to the left, after all. Similarly:
10*E*: ........99999990
*E*: - ........99999999
==============
-9
…or 9, if we assume E can be treated like a conventional algebraic variable. My point is that it cannot.
What is your argument that 10E - E = 9 by treating E like a conventional algebraic variable? I’m not seeing it.
(I assure you, you can treat E like a conventional algebraic variable in a consistent manner which determines it to be -1 and not 1. I’ve noted several natural contexts in which such reasoning applies.)
I don’t think he made a sign error. If I had to guess, I’d say he thinks 10*E is like E with the same number of 9s but now with a zero at the end and thus bigger than E.
But an infinite number of 9s shifted left with a zero at the end is still an infinite number of 9s. And so 10E and E both have the same infinite number of digits. But since 10E has a zero in the tens place rather than a 9 it is smaller than E, leading to the -1 result. You never get out to infinity where 10E has 1 more digit than E.
Which is non-intuitive for sure if I’m understanding that line of reasoning correctly. Apologies if I’ve gotten it wrong and muddied the waters even more and hopefully you’ll set me straight if so.
Because you’re giving E a coefficient and treating it as if it was meaningful. It’s like asking “what’s 10 times ∞?” The answer is “∞”, not “10∞”.
You can get an answer if you decide on certain conventions and apply them consistently, but not necessarily meaningfully.
Well, great, I’m confident you can get a whole bunch of answers that seem perfectly logical. Somewhere along the way, though, there’s going to be an assumption made about E that may not quite be supportable.
10E is like E with the same number of 9s but now with a zero at the end. And an infinite number 9s shifted left with a zero at the end is still an infinite number of 9s. And since 10E has a zero in the (minor correction) units place rather than a 9, you do get, upon subtraction, 10E - E = -9, or E - 10E = 9, both amounting to E = -1. You do (on this account of things) never get out to infinity where 10E has 1 more digit than E.
All of that is true. And all of that leads to E = -1. I’m pointing out that Bryan Ekers was mistaken in supposing there to be a clear algebraic derivation of E = 1. There is not; there are well-studied mathematical contexts in which all the usual rules of algebra (not dealing with ordering) apply, such that E unambiguously equals -1.
Yes, the non-intuitive part is that this clashes with usual rules of ordering like “Adding positive values makes you bigger, not smaller” and “-1 is smaller than 0, not bigger”. So all of this can only be interpreted in a context where we either weaken those rules, or give up on the idea of ordering entirely. [Just as when we consider a numeric system in which there is a square root of -1, we do not impose the usual ordering structure upon it]
Let’s go back to this. Why do you say “clearly” here? What makes it clear that 10E - E is 9?
You later say:
That part is clear. Digit-by-digit subtraction gives us E - 10E = …0009.
But I am confused as to why you considered the first part clear. What is most clear is that 10 - E = -9, by digit-by-digit subtraction, as you yourself demonstrated in a subsequent post.
Yes meaningfully. I’ve spelt out such meanings! There’s nothing unmeaningful about, e.g., the 10-adics; it’s just a meaning you are unfamiliar with.
If E - 10E = 9 then E = -1
If 10E - E = -9 then E=-1
Not what you wrote in your previous post where you said E could resolve to 1 or -1.
Typo corrected in bold. Anyway, sich_hinaufwinden has said more clearly and succinctly what I’ve been trying to point out.
Still, let me point it out again:
Why do you say “or 9”, when you just finished showing me that the calculation actually comes out to -9? What justifies flipping the sign?
Because if we were going to treat E like a conventional algebraic variable that happens to have 9 as a final digit:
10E is E shifted one digit to the left and has 0 as a final digit.
10E - E = 9.
Of course, E being infinite in length, this doesn’t work. Now, I’m currently reading the wiki page on p-adic numbering systems, a mathematical concept I was not previously aware of, and I’m willing to entertain the idea that once I get a handle on how it works, I could end up saying E = -1 in a p-adic system. I’m pretty sure the qualifier will be necessary.
Hey, no argument here. My initial comment was directed at 7777777, who (as far as I could tell) was citing James Tanton who wrote that …9999 = -1 while using conventional arithmetic and pointing out the limitations of doing so.
No, 10E - E = negative 9. You carried out the calculation yourself, above. In the units place, 10E has a 0 and E has a 9, so we get 0 - 9 = negative 9 in the units place. In every other place, 10E and E have the same digit 9, so we get no contribution. Thus, we ultimately get negative 9.
Let me put it this way: What is 99990 - 99999?
It doesn’t matter because you’re using numbers of finite size, abandoning the essential concept of E, namely that it is of infinite size.
I’m cheerfully prepared to admit that there are approaches to this issue outside my current sphere of knowledge, but I’m not about to get worked up about it.
Why would you say 0 - 9 = negative 9, 90 - 99 = negative 9, 990 - 999 = negative 9, etc., but applying the same treatment to …990 - …999 would give us positive 9?
I agree it is an essential fact about E that it is of infinite size. But regardless of that essential fact, I don’t see why you would look at a subtraction where the first value has a units place of 0 and second value has a units place of 9, and conclude that this means the result “clearly” gets a contribution of +9 in the units place rather than -9.
You say this is what conventional algebraic rules would demand, but the conventional rules are that 0 - 9 = -9, not 0 - 9 = +9.
I’m glad. However, I still feel you would benefit from re-examining your initial argument that 10E - E clearly equals 9. Without further explanation for how you came to that conclusion, this still seems to me a sign error; a calculational slip, like the child who mistakenly supposes 42 - 18 = 36.