It’s not trivial to note that all of the many people who have entered this thread to dispute that 0.9999~ = 1 have consistently ignored the existence of posts like these. Your math, IOW, is so irrelevant to their understanding of math that it is not even a topic for comment. That’s why I posit that they don’t want an answer from math at all, but from arithmetic.
Oh, as noted at the bottom, that post’s not for the people who are entering this thread to dispute that 0.9999… = 1. I’m not posting for them now. That topic’s been well-covered, so I’ll talk about anything else interesting that comes up with anyone else who brings it up. Anyone who still cares about the the nominal subject of the thread can go read everything that’s already been written about it (or, more likely, ignore it and post in write-only mode).
Oh, and:
Typo corrected in bold
Associativity of addition u + (v + w) = (u + v) + w
Commutativity of addition u + v = v + u
Identity element of addition There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.
Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v
Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F.
Distributivity of scalar multiplication with respect to vector addition a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv
Real numbers are a subset of vectors, so what’s true for vectors is also true for reals.
Those are just the axioms for a commutative ring with unity. (They hold for R, but don’t characterize it.) There is no first-order characterization of R; a common set of axioms is that it’s the unique ordered, complete field.
Your explanation here is undoubtedly the right one. It’s trivial to find anything you’d ever want to find, including all the proofs and disproofs in this 24-page thread, online (even just the relevant Wikipedia page is more than enough), but some people would rather argue their pet theory on a message board. Oh well.
Also, I feel a bit left out for never having gotten any of those handwritten messages from cranks asking me to validate their pet theory (usually along the lines of, “I have this amazing theory that proves Fermat/Einstein/Feynman/whoever wrong, but I just need a bit of help with the math.”) I’ve gotten some email to that effect, but there’s something to be said for finding a packet of crazy in your departmental mailbox.
Here’s an axiomatization that works, within the background of first-order Boolean logic with equality (I’ll write it in prose, rather than formulaically, but if you’d prefer the latter, that can be supplied):
[ul]
[li]There is a sort of thing called “reals”.[/li]
[li]Addition: You can + reals together, producing a real as output; this is associative, commutative, and has an identity element called 0.[/li]
[li]Negation: For every real x, there is also a real -x such that x + (-x) = 0.[/li]
[li]Multiplication: You can * reals together, producing a real as output; this is associative, commutative, and has an identity element called 1. Furthermore, * distributes over +, in the sense that x * 0 = 0 and x * (y + z) = x * y + x * z.[/li]
[li]Order: Some reals are called “semipositive”; the semipositives include 0 and 1, but not -1, and are closed under + and *. For every real x, either x or -x is semipositive; the only case where both are semipositive is when x = 0.[/li]
Finally,
[li]Intermediate value property: Any term you can build up from variables in the above language (of +, 0, -, *, and 1) is called a “polynomial” in those variables. Given a polynomial P(x) in one variable, if there exists a real y such that P(y) is semipositive, and there exists a real z such that -P(z) is semipositive, then there exists a real w such that P(w) = 0.[/li][/ul]
We can also define subtraction, >, etc., as definitional shorthand from the above in a straightforward way. Basically, feel free to use all the basic properties of real arithmetic which you’d expect from your first algebra course, and then add the intermediate value property for polynomials on top of that.
The above structure is called a “real-closed field”; note that, if we were to formalize this in first-order logic with only a single sort of reals, the intermediate value property would have to be an axiom schema with infinitely many instances rather than single axiom.
It’s true, as Itself notes, that these axioms do not pin down a unique model, but that is always the way in first-order logic, for any theory with an infinite model. They do, however, have the completeness property that any first-order statement in their language is either provably true or provably false from them.
Whoops, a subtle mistake to fix here: instead of demanding P be a polynomial in one variable as defined here, I should let P be the result of specializing an n+1 variable polynomial with any n choices of reals to fix its first n arguments to.
Thus, for example, we will have that the intermediate value property is applicable to P(x) = a * x * x + b * x + c for any choice of a, b, and c, not merely integer choices.
On another note, I axiomatized ordering above in terms of “semipositive” (aka, “>= 0”), for minor personal aesthetic reasons, but you could change the axiomatization to be in terms of the more familiar word “positive” (aka, “> 0”) instead, if you prefer; just make the appropriate changes to the Order section, and replace “semipositive” by “positive” in the Intermediate Value section.
1 = 1.000000000000000000…
1 ≠ 0.999999999999999999…
because every decimal is different as you can see
furthermore
71/7 = 1
but
70.142857 = 0.999999
70.142857142857 = 0.999999999999
70.142857142857…= 0.999999999999…
therefore
1/7 ≠ 0.142857142857…
Thank you for that. We might as well just close the thread, based on your brilliant insight.
But what do you think about people who say that 1/2 = 2/4? Clearly they are wrong, because every fraction “is different as you can see”.
Thank you for your thanks.
The case with 0.5 is different than 0.99999999…because clearly there
is not the problem with infinitely many decimals.
1/2 = 2/4 = 0.5 = 0.50000000000000…
as you can see, not the same problem as before.
The problem exists only with fractions like 1/9, 1/7, 1/6,1/3 etc because they have unending decimal representation.
The same problem does not happen with fractions like 1/8,1/5,1/4,1/2 etc.
As you wrote, exactly the same problem as before with infinite decimals.
As you also acknowledge, decimal notation is only a representation, originally meant for integers. It is imperfect for rational numbers, because it produces unintuitive identities like .9999… = 1.000… . That’s why you better use the fractional notation unless your application uses approximations anyway.
I’m sure this has been said before upthread. I feel more like educating than like reading at the moment though.
I would refer you to an earlier post in this thread.
Wrong.
The 0s at the end of the decimal representation 0.50000…can be ignored. The 9s
at the end of 0.99999999… can’t be ignored.
Can you refer me to an earlier post where it has been proved that
1 ≠ 0.999999999999999…
Oh well, I mean to stay out of the thread but this is one of those things that I can’t ignore.
This HAS been mentioned upthread. In fact, variations on this precise misunderstanding have been made since the very first few posts. No need to dig through a ton of posts to get there.
Basically: prove any statements you make. People have shown in a variety of ways and with varying degrees of rigor that the two are the same.
What you have done is simply made a claim. In math, we prove our claims if we wish to use them.
So, the first challenge: if two real numbers are not the same, you can compute a difference between them. What is the difference between 1 and 0.99999…? And show your work. I’m not being (entirely) snarky. Mathematicians don’t accept things without proof, so showing your work is equivalent to making your point.
Very firm. Very ignorant, but very firm.
And may I also demand you proving that 1=0.99999999…
Because in math, we prove our claims.
And don’t prove your claim with various tricks, for example that because
1/3 = 0.33333333333… therefore 3*0.33333333333…=1.
The above “proof” is not valid because
31/3 = 1
but
30.333333333…= 0.999999999…
and you just can’t write that these two identities are the same, it would constitute
a claim without a proof.
The difference between 1 and 0.99999… is infinitesimal. Its value cannot be
written be down because it is a variable, but it is not equal to 0, therefore
it can’t be ignored.
You mean beyond the numerous proofs in this thread?
Well, here’s one very early in the thread.
If you’re not going to bother even trying and won’t accept logical reasoning, there’s really nowhere to go.
No such thing. Point out which real number is called “infinitesimal”. Reading back through this thread (wow it’s old and pointlessly long), I see that I myself have made this very point before.
The numbers above, both 1 and 0.999…, are real numbers. Their difference, as the reals are closed under addition, is also a real number. “Infinitesimal” or whatever you choose to call it is NOT a real number. If both are real numbers, the difference has to have a real number value.
Again, what is the difference? Only a real number is an acceptable answer to that question. So, which real number corresponds to the difference?
Beyond that question, to get meta for a moment, this line of reasoning shows one of those cases where language can make a difference in thought.
I can describe, using words, a red laser that emits light at 810nm. The problem: 810nm is actually green. I can play make believe games with this red laser that emits in the green portion of the spectrum, but it won’t make any difference to the actual world. I can use words to describe something that can’t really exist, except maybe to people who are red/green colorblind.
The same thing is true for this “real number that has no set value but is imperceptibly bigger than 0” business. We can use words to describe a situation that is nonsensical in context. In this particular context of real numbers, such a number doesn’t exist and further using it makes no sense.
The challenge becomes to recognize when such a situation arises in our thinking.
ETA: And harking back to my earlier post, the point still stands. You can create this “infinitesimal” value or whatever and add it to the reals. But we won’t be talking about the standard real numbers anymore. So, we run right back into the same problem I mentioned a couple years back. Under your new, fancy modified reals, maybe the two values are different. But that still makes no difference on the standard reals that are the original subject under discussion. Using this modified real number line to make statements about the standard real number line is a big no-no mathematically.
It’s been proven dozens of times in this thread. If you don’t accept those proofs then there’s not much we can do to help you.