Mathematical curiosity

vaxer

Moderator Emeritus
This will probably going bore everybody to death :sleep: but?

Normally, you?d think 0.9999? (with an infinit number of 9s) would be smaller then 1 in an inifitly small way. But no, they equate exactly! 1 = 0.9999?

The most simple demonstration is :
c = 0.9999?
10c = 9.9999?
10c ? c = 9.9999? - 0.9999?
9c = 9
c = 1
meaning : 1 = 0.9999?

Funny how things arn?t what they look like. :whip:
 

intergamer

New member
lol...

it's only strange to you because you think infinite decimals are easily defined
the only possible way to define the value of the syntax we use for an infinite decimal is as the limit of an infinite summation. and this summation obviously has limit of 1.
.999... and 1 are not just equal, they are simply two representations for the same number.
 

vaxer

Moderator Emeritus
intergamer said:
lol...

it's only strange to you because you think infinite decimals are easily defined
the only possible way to define the value of the syntax we use for an infinite decimal is as the limit of an infinite summation. and this summation obviously has limit of 1.
.999... and 1 are not just equal, they are simply two representations for the same number.

It actually isn't strange to me, I was just pointing out a commun misconception as the view of this equality can be minde bending.

0.999... can be written as an infinit geometric serie that converges.

0.999... = 9(1/10) + 9(1/10)^2 + 9(1/10)^3 + ...
= (9.(1/10))/(1 - 1/10)
= 1
 

qafir

New member
Vaxer, 0.99999999 never equals 1. But as you look further and further past the decimal point, it begins to approach 1. That's why they invented calculus, largely to handle or describe mathematically what happens as numbers approach each other. Usually this is when a number approaches zero.

For all intents and purposes, the two do approximate each other. But if you were dealing with a micro or macro need for measurement (think sending a probe to venus or counting atoms), you would need accuracy. Treating the two as equal would skew your projects.

Anyway, it's been fifteen years since I picked up a calculus book, so correct me if I'm wrong!!!:D
 

Jay R. Zay

New member
qafir said:
Vaxer, 0.99999999 never equals 1. But as you look further and further past the decimal point, it begins to approach 1. That's why they invented calculus, largely to handle or describe mathematically what happens as numbers approach each other. Usually this is when a number approaches zero.

For all intents and purposes, the two do approximate each other. But if you were dealing with a micro or macro need for measurement (think sending a probe to venus or counting atoms), you would need accuracy. Treating the two as equal would skew your projects.

Anyway, it's been fifteen years since I picked up a calculus book, so correct me if I'm wrong!!!:D

i actually didn't want to post here ever again but i would like to take this opportunity to call you an idiot as i can't stand to see mathematics raped by people who don't understand a thing of them.

numbers don't move. that's what we all should know and this pretty much is the reason why all you say is plain wrong - even if at first glimpse this might seem to be hairsplitting. i am very well aware of what you mean and it's plain wrong.

as we now agree that numbers don't move, they can't approach each other. so the number 0.(9) (meaning 0.9999999... ad infinitum) doesn't approach 1. it either IS 1 or there is a difference between them. a number doesn't change depending on how far you look beyond the decimal point. the number 0.(9) exists as a whole number of infinite decimals and it doesn't give a crap for how far you please to look.

the reason why 0.(9) exists in our world is because decimal numbers exist.

the first step is to divide 1 by 3.

1/3=0.(3)=0.33333...

and, for example, 0.(3)*2=0.(6)

so it seems probable that 0.(3)*3=0.(9), right? yes. but if 0.(9) wouldn't be exactly the same thing as 1, we would get:

1=1/3*3=0.(3)*3=0.(9)≠1

or, more simply: 1≠1

likely? no? ah.

0.(9) is a number that doesn't approach 1. the only thing about 0.(9) that "approaches" 1 is the way you might create this number.

it's an infinite sum.

0.(9)=


(9/10^n)
n=1

now this geometric progression approaches 1 when thinking of it as a slow progress of adding more and more addends to it. but the actual number 0.(9) is only the final result of this progression, not the progress itself. and the result of such a progression always is the limit.

infinitesimal calculus wasn't "invented" because people thought "well, if something is rather close to 1, let's just call it 1, okay?". it was "discovered" because people found out that this limits exist and they can be reached.

there is a popular example for this.

let's say a fast runner (human) and a slow turtle made a race. to make this race more fair, the turtle was given a head start of 100 feet.

now when the runner reaches the point where the turtle is at time zero (let's call it A), the turtle has moved some feet further (although it's moving at a much slower speed) to B. and if the runner reached point B, the turtle got even further to point C. and so on.

so looking at that race this way, it would seem that the turtle would win the race because the runner will never be on the same point as it (or even outrun it). which seems unrealistic but that's what it would look like.

BUT with infinitesimal calculus, you will notice one thing: that the locations of the runner and the turtle converge at one point. and this is the very point on which the runner has outrun the turtle. so doesn't the runner reach that point? isn't he able to outrun the turtle?

of course he is. because the limit of this progression can be reached. THIS is what infinitesimal calculus tells us. and it doesn't tell us that we can forget about minor differences between numbers because nobody will look so closely anyway.

and what can you learn out of this? as you probably won't understand mathematics much better than you did before reading this, you could at least learn that an opinion about something always is as good or as worthless as the informations behind it. in other words: if you don't know much about something, better refrain from indoctrinating people about it.
 

Finn

Moderator
Staff member
Jay R. Zay said:
i would like to take this opportunity to call you an idiot
I don't know much about math either, but I do know that you rarely would miss such an opportunity. ;)
 

qafir

New member
Dear J-Lo,

I almost called you Jello, but given the language/cultural divide between us, I wasn?t sure you?d understand that I was implying you had gelatin for brains. So I?ll leave it with a pop reference you can probably manage.

Where do I begin? First off, I?d like to say you?re spectacularly unique here at this forum. I?ve read hundreds and hundreds of posts, and yours stand out as the most painful to read. I mean, when someone takes so much time to be so very wrong, where do the rest of us even begin???

So I?ll start with the niceties. I understand your perspective. English is not your primary language, and it is unlikely you would ever have read a Calculus text in English. So rather than lambasting you for that, let me clear up a few vocabulary problems first.

Ok?mathematical modeling: We use graphs to model functions and equations. It helps us analyze problems on paper that are vastly too complex to analyze in real world terms. Calculus, actually, was invented specifically to describe the arc a projectile follows after launching it. Like many scientific advances, Calculus was inspired by the need for better warfare.

The simplest mathematical model is a one-dimensional coordinate system called (in English, at any rate), a number line. It?s a straight line. Zero is assigned somewhere along it arbitrarily, and unit distances (-3,-2,-1,0,1,2,3) are picked and written along this line. One the line, meaning in the model, numbers approach. You approach zero. Or you approach a different point along the number line.

You are partially correct. Numbers are what they are. But in a function, the function ?approaches? certain solutions, including the limit.

So please do forgive me for using English in my post. But had you actually read the post instead of rushing in limp-brained to insult someone, you would have noticed that my first two sentences were, ?Vaxer, 0.99999999 never equals 1. But as you look further and further past the decimal point, it begins to approach 1.? I can see how your ESL take on things would misinterpret that to mean the numbers jumped into a Ferrari and zoomed along a freeway. Again, I can only apologize for my native tongue and forgive you for never having studied math in an English classroom.

But think about what I said again. I know it hurts your brain. You hate reading what others post before insulting them. But try. Pretty please?

I said that as you look further and further past the decimal point, the number approaches 1. The number 0.9 (think 90%) is a lot closer to 1 than the number 0.99 (think 99%). So, as I indicated, the further past the decimal point (on a number line) you go, the more the decimal approaches 1.

Does that clarify your confusion? I hope so. Let me know if it doesn?t, and I?ll see about hiring you a tutor. I hear that many twelve year olds in your country can understand this much. I?m sure they work cheap.

You wrote,
numbers don't move. that's what we all should know and this pretty much is the reason why all you say is plain wrong - even if at first glimpse this might seem to be hairsplitting. i am very well aware of what you mean and it's plain wrong.

as we now agree that numbers don't move, they can't approach each other. so the number 0.(9) (meaning 0.9999999... ad infinitum) doesn't approach 1. it either IS 1 or there is a difference between them. a number doesn't change depending on how far you look beyond the decimal point. the number 0.(9) exists as a whole number of infinite decimals and it doesn't give a crap for how far you please to look.

Do you think, now, that you might want to recant that position? Pure mathematics is different from the real world. But as we can all probably agree, Calculus was created for, and continues to exist for, measuring real-world situations too complex to measure without a simplified model. But again, I?m talking about the difference between 0.9, 0.99, and 0.9999999999999999999. Depending on where you choose to cut off your decimal places, the number does actually get a lot closer to 1 the further along the number line you go.

It perplexes me that you chose to include a bastardized version of Zeno?s Paradox (the turtle story). It doesn?t really support your point at all.

In the Paradox, yes, there is a point where the runner and the turtle?s points converge. And yes, that is the ?limit.? But there is a fundamental error in Zeno?s reasoning. It assumes an infinite amount of time is used to cover a distance divided into infinite numbers of segments. So Zeno?s setup was flawed from the beginning. I?m not sure how that particular image furthered our discussion here, but then again, most of your writing is flawed. Perhaps I shouldn?t be so shocked.

Oh, I just looked at your post again. I thought I should clarify. In English, Calculus is referred to as Integral, not infinitesimal.

But back to my original meaning in my previous post: One never equals 0.999999999?But in certain functions (abstract mathematical functions and graphs), it is important to know what happens really really close to the point on the graph where .999999 would either become 1 or would be undefined.

In simple real world situations, such as determining how much gasoline is consumed per hour at different speeds and elevations, treating 0.99 and 1.0 as equal might not be a big problem. But if you?re trying to land a $3 billion probe on a different planet, you?d better not round your numbers off to the second decimal place.

And Jay, for goodness sake, could you please stop posing as a smart man? You may be bright, but reading your posts over the last month, I can?t say they reflect that very well. You are so quick to jump in and attack others that you often forget to read through your own posts to make sure they are logical, ordered, and true.

IN this last case, you moved so quickly that you forgot to read my first sentence. Then you called me an idiot and turned around and asserted the very same thing. Remember, writing a lot of words poorly is never a substitute for clear thinking.

Also, for everyone else, I did actually write, ?Anyway, it's been fifteen years since I picked up a calculus book, so correct me if I'm wrong!!!? I was just trying to join in on a discussion, and I was humble enough to admit my own limits. Wow, I didn?t even need Calculus for that!
 

intergamer

New member
qafir said:
Vaxer, 0.99999999 never equals 1. But as you look further and further past the decimal point, it begins to approach 1. That's why they invented calculus, largely to handle or describe mathematically what happens as numbers approach each other. Usually this is when a number approaches zero.

For all intents and purposes, the two do approximate each other. But if you were dealing with a micro or macro need for measurement (think sending a probe to venus or counting atoms), you would need accuracy. Treating the two as equal would skew your projects.

Anyway, it's been fifteen years since I picked up a calculus book, so correct me if I'm wrong!!!:D


0.9999.. repeating doesn't approach 1, it EQUALS 1. they are the same number, exactly, entirely, and thoroughly.
 

vaxer

Moderator Emeritus
Ok... I never thought that a thread about math could turn into a flame war.
One that doesn't understand insn't necesarily stupid, by all means, keep that in minde people!

qafir, I think that there's somthing you havn't understood. When I wrote 0.9999... or even 0.(9) it means there is an INFINIT number of 9s. So when you say that 0.9 (90%) is 0.1 away from 1 it's true, but 0.9999... (100%) is 0 away from 1. There's a difference between 0.999, 0.999999 and 0.99999999 but not between 1 and 0.999...
 

qafir

New member
vaxer said:
This will probably going bore everybody to death :sleep: but…

Normally, you’d think 0.9999… (with an infinit number of 9s) would be smaller then 1 in an inifitly small way. But no, they equate exactly! 1 = 0.9999…

The most simple demonstration is :
c = 0.9999…
10c = 9.9999…
10c – c = 9.9999… - 0.9999…
9c = 9
c = 1
meaning : 1 = 0.9999…

Funny how things arn’t what they look like. :whip:

Intergamer and vaxer...look at the original thread post.

10c [ meaning 10 x 0.9999999] = 9.9999999.....I agree.
10c-c = [9.99999...-0.999999...] = 9
9c = [9 x 0.99999...] = 8.99999... (not 9)
c= (8.999999... / 9) = 0.9999999

There is no point in that equation where .9999999 = 1 It only does so if you round off to the first decimal place. But we're not talking about rounding.

So the only point in your scenario when .99999... = 1 is if you round off.

And CH: I agree. Can't we just have a discussion without people jumping in and throwing out words like idiot. It's childish, I know, but I'm feeling a lot like rubber surrounded by gallons of glue.
 

vaxer

Moderator Emeritus
qafir said:
10c-c = [9.99999...-0.999999...] = 9
9c = [9 x 0.99999...] = 8.99999... (not 9)

by writing this you are saying that 10c-c != 9c meaning that 1 != 1
So you got something wrong (9 x 0.9999... != 8.9999... it's a misconception).

to save you time 1 = 0.9999... is a very well know result (a little suprising at first), so whatever you try to demonstrate it won't be correct.

Anyway thank you for proving that my thread is useful... (you must hate these three dots by now!)
 

qafir

New member
I do hate those three dots...

But I'm confused. I still don't see how your post proves that 1= 0.999... I know at some point it makes sense to round it off, but I did the math, and if you divide c (0.99999...) by 9, you still get another infinitely repeating decimal. It only works out to 1 if you round off.
 

Indy Parise

New member
qafir said:
I do hate those three dots...

But I'm confused. I still don't see how your post proves that 1= 0.999... I know at some point it makes sense to round it off, but I did the math, and if you divide c (0.99999...) by 9, you still get another infinitely repeating decimal. It only works out to 1 if you round off.
This has absolutely nothing to do with rounding at all. All it is saying is it's face value

0.99999999999.................=1
if you do the math, it is simple enough and i'm only a freshman in highschool, you will get that answer any way you slice it unless you make a miscalculation. What you are saying about the further down the line you go, the closer you get to 1, but again that is rounding. We are talking about an unrounded number, it just happens to be neverending, or in some other cases is just too long to even fathom writing down (think 100 pages of continuing 9s.) but this is not the case here. The number is just 0.999999999999999.... continued on forever, unrounded. Not cut off at any spot. Just continuous 9s.


BTW.Don't you think it is a bit hypocritical to say "correct me if I'm wrong" and then when somebody does correct you, you tell them you were right and they are wrong?
 

qafir

New member
IndyP,

The number 0.9999... with an infinity of nines NEVER becomes 1 on any linear number line. The more decimal places you view, the "closer" the number becomes to 1, but it is never equal to 1. The reason you believe it does is because it would be highly difficult to use an infinitely repeating decimal in any calculation. So for convenience, in math, you say that the answer is 1. But that's an approximation. By definition, it IS rounded off.

Consider the similar case of Pi. Pi is used in simple every day math and treated as pi = 3.142. If you take into consideration a few more decimal places, you get pi = 3.14159265. And if you networked every single computer on the planet or that will ever exist, you'd never be able to calculate pi entirely, because it's an infinite repeating decimal. Yet in your highschool math class, when you need to figure out the area of a circle, don't you use 3.142? It's not because pi = 3.142. That's a rounded number. Pi acutally equals the infinite chain of decimals. But for every day use, we have to draw the line somewhere. You pick how many decimals you're willing to use.

It's the same with the 0.999...repeating number. It never equals 1. But in most applications, you can treat it AS 1, because they're close enough. Ask you teacher. Be specific. Ask if it actually equals in every way the number 1, or if we just treat it as 1 because it simplifies the math.

If you want to calculate something beyond the normal, like distances measured in multiple lightyears, then those extra decimals are important. That's why it takes a ruler to measure a piece of wood for a door frame and a supercomputer to figure out how to land a few tons of metal and rubber on the surface of mars. We simplify things when they don't matter. But that doesn't mean the simplification is entirely accurate. Just good enough for the job at hand.

By the way, I didn't mind Jello's perspective on the math. But he began his post by calling me an idiot. Keep it respectful, and we can have an interesting discussion. Otherwise, you deserve what you get. And I'm not convinced at all that he was right. This thread still only works if you choose to arbitrarily round an infinite repeating decimal off to 1.

Just for clarification, will someone please explain the following:

if c=0.999999999 followed by an infinite number of nines
and 9c = 9 x 0.999999999999...
how does 9c not = 8.999999999999.....?
I understand how the math works if you do the trick, but any equation is tested by inserting the original values and seeing if it reaches an accurate answer.
 
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intergamer

New member
qafir said:
IndyP,

The number 0.9999... with an infinity of nines NEVER becomes 1 on any linear number line. The more decimal places you view, the "closer" the number becomes to 1, but it is never equal to 1. The reason you believe it does is because it would be highly difficult to use an infinitely repeating decimal in any calculation. So for convenience, in math, you say that the answer is 1. But that's an approximation. By definition, it IS rounded off.

Consider the similar case of Pi. Pi is used in simple every day math and treated as pi = 3.142. If you take into consideration a few more decimal places, you get pi = 3.14159265. And if you networked every single computer on the planet or that will ever exist, you'd never be able to calculate pi entirely, because it's an infinite repeating decimal. Yet in your highschool math class, when you need to figure out the area of a circle, don't you use 3.142? It's not because pi = 3.142. That's a rounded number. Pi acutally equals the infinite chain of decimals. But for every day use, we have to draw the line somewhere. You pick how many decimals you're willing to use.

It's the same with the 0.999...repeating number. It never equals 1. But in most applications, you can treat it AS 1, because they're close enough. Ask you teacher. Be specific. Ask if it actually equals in every way the number 1, or if we just treat it as 1 because it simplifies the math.

If you want to calculate something beyond the normal, like distances measured in multiple lightyears, then those extra decimals are important. That's why it takes a ruler to measure a piece of wood for a door frame and a supercomputer to figure out how to land a few tons of metal and rubber on the surface of mars. We simplify things when they don't matter. But that doesn't mean the simplification is entirely accurate. Just good enough for the job at hand.

By the way, I didn't mind Jello's perspective on the math. But he began his post by calling me an idiot. Keep it respectful, and we can have an interesting discussion. Otherwise, you deserve what you get. And I'm not convinced at all that he was right. This thread still only works if you choose to arbitrarily round an infinite repeating decimal off to 1.

Just for clarification, will someone please explain the following:

if c=0.999999999 followed by an infinite number of nines
and 9c = 9 x 0.999999999999...
how does 9c not = 8.999999999999.....?
I understand how the math works if you do the trick, but any equation is tested by inserting the original values and seeing if it reaches an accurate answer.

qafir, 9c does equal 8.9999...
however, 8.9999...=9 by definition. They are the same number.
I am not arguing that 0.999... = 1, I am telling you. :)
 

Indy Parise

New member
intergamer said:
qafir, 9c does equal 8.9999...
however, 8.9999...=9 by definition. They are the same number.
I am not arguing that 0.999... = 1, I am telling you. :)
exactly, see. This is what we are all saying. It is a rather obtuse and hard to understand concept that two seperate numbers like that can be one and the same. It took me a while when my teacher said it (last year my teacher went a little overboard, though it does help me now in High School.) but once you do the math, you figure out that it is true. We are not saying that 0.99999999..... is rounded to 1, we know that already.
 

Gustav

New member
I didn't read the whole thread, but I'm with Jay on this. I don't think 0.999999... could ever be equal to 1.


The most simple demonstration is :
c = 0.9999?
10c = 9.9999?
10c ? c = 9.9999? - 0.9999?
9c = 9
c = 1
meaning : 1 = 0.9999?


I read this and it just doesn't make sense to me.

What is this? 10c ? c = 9.9999? - 0.9999?

If you take it away from the 9.99999... then you also have to take it away from the 10, which would make it 9.11111....

Jay, what do you mean you never wanted to post here again? Surely you didn't mean on this board. There must be a misunderstanding. What did you really mean?
 

intergamer

New member
Gustav said:
I didn't read the whole thread, but I'm with Jay on this. I don't think 0.999999... could ever be equal to 1.
It's essentially a definition. 0.999... equals 1 regardless of what your opinion. The true nature behind why this is true is obviously over your head, so we use little demonstrations like:

The most simple demonstration is :
c = 0.9999…
10c = 9.9999…
10c – c = 9.9999… - 0.9999…
9c = 9
c = 1
meaning : 1 = 0.9999…


I read this and it just doesn't make sense to me.

What is this? 10c – c = 9.9999… - 0.9999…

if c = 0.999..., then surely you believe that 10c = 9.999...?
and then you believe that 10c-c = 9. It's almost begging the question to "prove" anything with this technique, because 0.999...=1 by mathematical definition. 0.999... is not some kind of process, it is a concept involving an infinite number of 9s. Infinite, not almost infinite. But this is just an example to help the weak-minded believe. Since 0.999...=1 follows directly from the definition of the decimal system, then strictly speaking believing that you can do floating arithmetic like 9.999... - 0.999 = 9, is more of a stretch than believing 0.999...=1.

If you take it away from the 9.99999... then you also have to take it away from the 10, which would make it 9.11111....
If you subtract 0.9999... from 10, you get 9. 0.999... and 1 are representations for the same number, just as (2+1), (1+2), and 3, are representations for the same number.

Think about this. Do you believe that if you added 9's infinitely, that the numbers you get "converge" to 1? Then that's all you need to believe. The common syntax for writing down numbers in our decimal system works so that we define numbers after the decimal as negative powers of two. So 0.23 = 2*10^-1 + 3*10^-2. 0.999... is an infinite summation in this respect. Infinite summations can be added. It adds to 1. But even this begs the question, because it's more simple than that. By the definition of our metric space over the real numbers, two numbers are equal if and only if there are no numbers in between. There are no numbers in between 0.999... and 1. It's even more simple than that, but for our purposes, simple may in fact mean more difficult to understand.

This is so trivial..
 
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