I laugh so far ... Task: I have 2 numbers: A = 1 B = 0.9999999999999 ... = 0 (9) Question: Are these two numbers equal or not?

Not sure what the notation in B is. Is the number that begins with .999 a repeating decimal? What does the = 0 (9) represent. If you are wondering if 1 and 0.99 (repeating decimal) are equal, the answer is no.

That's higher math than I ever learned. Doesn't make intuitive sense, but I'll take your (and Wiki's) word for it.

I didn't want to wade through the Wiki. I'd look at it as B = .999999... (forever) 10 X B = 9.999999... (forever) Now, subtract B from 10 X B. (10 X B) - B = 9 X B = 9.99999... - .99999... = 9 Finally: B = (9 X B)/9 = 1

Whatever wiki says or anyone else, they are not equal, if you plug 1 into an equation you get different results than if you plug in .99999. Repeating numbers are theoretical, they have no practical use because eventually in the real world you must cut them off for the sake of space or time.

There's an inherent problem with that, most mathematicians haven't recognised. (Similar to the Infinite Hotel Paradox) There are different versions of infinity, you can't modify infinity without changing it. 10 X 9.99999... is still going to have an effective zero place holder on the end, even though the 9s go on forever. (Yes, that sounds paradoxical) In the Infinite Hotel Paradox there's an inconsistency, where the concept is attempting to treat hotel guests and rooms fundamentally different from each other. So, in conclusion, although I recognise many mathematicians would disagree, my opinion is still that 0.999999... does not equal 1.

That's why mathematicians prefer rigorous proof, and don't rely on individual opinion or gut feel. Here's the formal proof: https://en.wikipedia.org/wiki/0.999...#Formal_proof

During discussions like these, people often forget that mathematics is based on a set of rules and definitions collected and/or created by humans. As a mathematician, you essentially claim that this and that is true without regard for anything (especially nature), describe as precisely as possible what you mean, and look what else you can derive from a preferably small set of basic rules. "Standard math" just turns out to be really useful in many practical applications, like physics, where I'm coming from. That doesn't mean that you can't create your own kind of math that you prefer. Unfortunately, you're unlikely to find something that contradicts "standard math" in a big way, while also being more useful in the real world. I think that's the reason almost everyone calls "standard math", just "math". I am unaware of any good alternatives. It's proven to work and there will be more and more extensions, but there seems to be no need for a complete overhaul at this point and possibly ever. For you to agree that 0.999... = 1, you need to accept and understand some basic axioms, definitions, real numbers, algebra, and limits, and show that all of this is free of contradictions. This has been done. The most contentious issue here are probably limits. Does it really make sense to claim something like, the limit of 1/n as n goes to infinity is equal to 0? You can look up how a limit is really defined with those deltas and epsilons and try to derive consequences from that definition, like 0.999... = 1. While limits have many practical uses (physicists can't achieve much without calculus), you can still refuse to use them for completely arbitrary reasons. That's essentially what the 0.999... is not equal to 1 crowd is doing. They're rejecting certain certain axioms and/or definitions from "standard math". They can do that, of course, but they're usually unaware of how much useful math they really lose when they're doing so. I would say that losing calculus that way, for example, is totally unacceptable.

Great points. In the natural sciences and engineering all those 9's after the first 3 or 4 are undoubtedly garbage. It's important to know how many of the endless streams of digits our calculators emit have any meaning at all.

Hardly, 1 is a usable concept whereas .9999 is a theoretical They are similar not the same. I can add 1+1 and get 2, you cannot do the same with your .999999...

Yes I can. .9999...+.9999...= 2, every time, everywhere. That is because 1 and 0.999.... are the same number. You are simply wrong, but you are invited to attempt any example or proof to demonstrate otherwise. In fact, demonstrating that 0.9999....+0.9999...= 2 is one of the easiest ways to prove to students that 0.9999...= 1. Do you understand why? I can show you. It's actually a fun example, as students inadvertently fall 4ss backwards into proving 0.9999...= 1, using this example.

Not true, since .99999 is infinite, you would still be doing the equation, in fact you haven't even gotten to the addition part...

We don't have to to perform that infinite equarion, when we understand a principle that always holds. We don't have to perform the infinite iterations possible to know that (pi)r^2= the area of the circle, as we can prove this using a proof. So you are really barking up the wrong tree, there. When adding 0.9999... to itself, we don't have to perform math forever. We can simply agree that each place after the decimal will be 9. (This is a very simple proof, actually). In short summary, you will always get 18 and carry the 1, for any corresponding decimal place. So, 0.9999...+0.9999...= 1.9999..., by simple aritmetic and proof. Still with me? But, oops! You have just proven that (0.9999....+0.9999...)=(1 + 0.9999...), therefore proving 0.9999...=1. Sorry dude, you are just not going to to will your incorrect claims to suddenly become correct. 0.9999...=1 not just by arithmetic, but even by definition.

Of course you have to perform the infinite equation, because the second you cut it off you change the results. They are not the same, one is actual and practical, the other is theoretic and unusable...

At no point do we have to cut it off. That is an error to think so. We know exactly what will appear in every decimal place, by simple aritmetic proof. Just as we can define 1 to equal 1.000..., with a zero in every decimal place. We don't have to perform arithmetic for eternity to arrive at 1.000...+1.000...=2.000... We can simply show this by arithmetic proof. No, they are no different.

Show us the work, in a manner we can see and understand. You can't do that without cutting it off. You are correct, you don't have to do anything special with whole numbers, not so once you add that decimal point and claim infinite number chains...

Which you only know by intuition, not by arithmetic proof. What work do you need to see? And don't you find it a bit presumptuous on your part to demand even more detail from me, when you have not presented a single example of ANY equation that yields a different result, when 0.9999...is plugged in instead of 1? Do you have an example of a proof that shows they are not equal? ANYTHING? You can claim pi=8 all day, but nobody should believe you, until you prove it. And again...which work, exactly, do you need to see? Please be specific. The only part of all this that has not been shown by simple, arithmetic proof is that 0.9999...+0.9999..= 1.9999.... Is this the work to which you are referring? Again, please be specific.

The people who do not understand this are suffering from, essentially, a version of one of Zeno's paradoxes (which have been discredited and resolved by our understanding of the sums of infinite sequences). Essentially, one of them describes a paradox of movement. Shoot an arrow at a target, and it must first travel halfway to the target. It must then travel half the remaining distance first. And so on and so forth, with the conclusion that the arrow can never reach its target, as it must always travel half the remaining distance first. The paradox arises in the fact that the arrow always does, in fact, reach its target. The greatest thinkers at the time were all but paralyzed by this thought experiment. How does mathematics deal with such a contrived paradox of human intuition? Simple...sums of infinite series. The sum of the numbers 10*(1/2)^x, for all integers x from 1 to infinity, is 10. The arrow reaches its target 10 feet away. Every time. There is no paradox. It does not take an infinite amount of time for this to occur. Just as it does not take an infinite amount of time to arrive at this sum.