I ask you to postulate a number that is between 0.999… and 1.0. You might say, 0.99999 (five nines.) I can then respond with 0.999999 (six nines) and thus have put another number that is closer to 1.0 than your number was.
No matter what number you postulate, between 0.999… and 1.0, I can always provide another number that is closer to 1.0. You can’t win the game; I always can.
Since you can never win the game, the victory goes to me. This is, in essence, the definition of “limit.” No matter how small an epsilon you offer, I can always come up with a delta that is still smaller. The point is that no number you can ever offer will serve the purpose of rebutting the claim that 0.999… equals 1.0
Sure if we calculate Pi or 1/3 we get an infinite number.
But what I am asking is where do mathematicians draw a line between +1 to anything you say (which is finite and a number) and infinity which is not a number?
My reason, if it helps, was a discussion on another message board about approaching light speed in a rocket ship.
Ignoring the practical limits of rocket ships you can go arbitrarily closer to the speed of light forever. You can approach infinity with no limit.
So when do we say we’ve turned the corner? Or can I truly approach the speed of light forever (without ever quite reaching it)? Seems here is my 0.999…=1 is an issue.
Must be different learning styles. At the very least it shows that you primary school math textbooks, calculator and computer believe also concur that 0.999… equals 1. Getting people to believe that fact alone is the major hurdle in the argument. The finer points about the properties of infinities, the fundamental concepts behind them and proofs are of secondary concern unless you are an academic.
I was around for several rounds of the 0.999… = 1 wars and any concept that is easy to demonstrate to anyone is a valuable tool to have at your disposal. Most people don’t care about math theory much at all but there are a surprising number of them that will get angry when you tell them the two are the same.
I think that perhaps Whack-a-Mole may simply be failing to take mathematical notation into account. W-a-M, when you write the number “0.999…”, the periods don’t just mean “we were too lazy to write the full number of 9s”. It means “there are infinitely many 9s.” That’s the line you’re looking for.
0.9 does not equal 1 - correct
0.99 does not equal 1 - correct
0.999 does not equal 1 - correct 0.999…does equal 1, the three periods tell you that there’s not just three, or four, or fifty 9s there. They go on forever. That’s what tells you it’s infinite, and is equal to 1. You may also have seen this number written with a small horizontal line above the 9.
Now I’m really confused. You’re getting 0.999999999 there because a calculator rounds off the initial division from a theoretically infinite decimal to an approximate result, finite decimal accuracy.
Just to be clear - I do realize that you’re trying to grant intuition, rather than claim a rigorous proof. I just don’t get it at all.
It seems to me all you’ve shown is that 1 equals approximately 0.9999999 (finite, non-recurring) when the calculator does some rounding along the way.
Yes and that is what would happen because of other weird effects that additional question introduces. This, more complicated question, is in the realm of theoretical physics and not just math. You introduced relativity and time dilation now as well which complicates the overall question greatly. I am a lay reader of theoretical physics but I don’t pretend to be competent at it just like 99.999% of the people in the world. I do know that you can approach the speed of light but never reach it no matter how much energy you have so that is the simple answer.
This is what the formal definition of a limit (in this case, the limit of a sequence) is designed to nail down, without any reference to “infinity.” Basically, once you tell me how close you need to be to 1, I can always tell you “how many 0.999… you need.”
Seems my issue is infinities that have no end (which would seem definitional) and some that, in theory, could end (e.g. +1 forever).
For instance Pi is infinite. 1/3 is infinite. +1 forever however is sort of infinite. We could stop at any point and say that is a finite number. Can’t do that for Pi or 1/3.
Maybe this is the breakdown. I always knew even as an elementary school student that my calculator wasn’t rounding at all just because it had a screen of a certain size unless I told it to. You can only see some of the numbers but they will go on forever if you want to work more of them out in the most boring homework exercise ever or waste your money on the world’s biggest calculator. The displayed digits (0.333…) are only part of the result and the full thing literally goes on forever.
I learned the repeating digit notation (i.e.; 0.333…) in elementary school and we knew what it meant at least superficially. That is a basic start in understanding an infinite series even if it is just at a superficial level.
The answer is easy to say and hard to deeply understand.
That “line” as you put it is drawn just before the smallest infinity. How many is that? More than every number which is not infinity. In fact it’s infinitely more than the biggest number smaller than infinity. (loosely stated).
There is no process of “+1” as you put it which can cross the boundary from “biggest finite number” to “smallest infinite number”. Those two kinds of numbers simply don’t connect. Period, Amen.
The reason they don’t is the words “biggest finite number” are totally meaningless. There is no such thing. Period, Amen.
Take any finite number. Take the biggest example you can imagine. You *can *add one to that number. And you therefore get a new “biggest finite number I’ve imagined so far.” And you can carry this process out forever without ever crossing over to infinity. You just keep creating bigger and bigger finite numbers forever.
Infinity is something that lives beyond the edge of that never endable process. In fact it (Aleph null, the “smallest form” of infinity for the experts) is *defined * (small arm wave) as the thing that lives beyond the end of that never ending process.
Once you can accept all this at face value without asking why (or why not) you’re ready for the next step.
I’m not sure what you mean. It would be more precise to say that pi and 1/3 have infinite decimal expansions. But if you “stop at any point”—for instance, if you write 0.333 and stop—you do indeed have a finite number, one that is close to but not equal to 1/3.
Now I’m even more confused. How do you think calculators work? You realize that they do round off results to finite decimal accuracy, right? The calculator does not store the intermediate result as “one third”, it stores it as a finite decimal that does not exactly equal one third.
I mean, as long as I can add a “9” to the list it is finite but there is a special class where adding another “9” to the list is meaningless because the list is endless.
So I have two sets. My finite (albeit big) list and my infinite list and the two sets are not comparable in a meaningful sense.
Ran out of time on a good edit of my last paragraphs. Try this instead:
Infinity is something that lives beyond the edge of that never endable process. In other words, it exists out beyond the edge of something which has no edge.
In fact it (Aleph null, the “smallest” form of infinity for the experts) is *defined *(very small arm wave) as the thing that lives beyond the end of that never ending integer +1 process. There are other bigger nastier infinities lurking infinitely farther out beyond that one.
0.999 with 3 nines is not infinite. 0.999… with an infinite number of nines is not infinite. It’s 1.
The decimal expansion of 0.999 has 3 or 4 digits depending on how you count. The decimal expansion of 0.999… has infinite digits. An infinite length decimal expansion is not infinity.
This is darn good. It’s effectively a restatement of what I was posting while you were posting. You’re in the right ballpark now.
Dude! You just blew my mind. It is true though. Some infinities are larger than other. For example, the range of integers less than 0 is infinite but smaller than the infinity of all positive and negative integers.
Besides, 1/3 is not “infinite” – i.e., does not necessarily have an infinite expansion.
You just wrote it using three symbols. 1/3. That’s the explicit notation.
Pi doesn’t have the same kind of explicit notation because it’s non-repeating and can’t be simplified. But “1/3” carries all of the information that “0.333…” does.
In a previous thread, it was noted that one could re-define numbers to be of the form “a + ub” where “u” is some extremely small number, like 10^-googol. At this point, numbers would have properties somewhat akin to complex numbers.
This isn’t how numbers are defined now, but if you really insist on wanting “something” to exist between 0.999… and 1.0, you actually can, so long as you’re willing to accept all the consequences. (One biggie is having to accept that u*u = 0.)
(Well, why not? Non-Euclidean geometry, non-Peano numbers. Life is fun!)
I am a Systems Analyst and developer by profession. I know how calculators work and did even as a child.
The calculator is a red herring anyway and isn’t important. That is just the lazy way to demonstrate it quickly. You can also do long division by hand and quickly see that you will be in for a very long night, week, year or infinity if you ever want to get to the end of 1 divided by 3. It is perfectly obvious after a couple of minutes what the results are and what will always be. Now multiply it back.
Like I said earlier, even primary school kids know about infinitely repeating decimals. It isn’t meant to be a rigorous proof but it is a good hands on demonstration of the concept for those that are inclined to learn concepts through their own experimentation.