Does 0.9999999... = 1?

[stemann]

If 0.9^ did not equal 1, then there would be a number in between 0.9^ and 1, which there isn't.

 

[michel]

This is the first I have seen of this thread; I can't see any justification for saying that 0.999999999 =1 - it isn't.

Just as a question, Fluffy, what would you do with "

" ? Historically, mathematicians have tried to find the answer (which, of course, you know, there isn't one); I was always taught to use the fraction value as opposed to the decimal one.

I agree with what others have said; once 0.999999999=1, what about 0.99999998 ?........

 

[ch'ang]

Well after rethinking my rash retort I feel i must retract it. Mostly because of this 1/3=.3'
1/3x3=.3'x3
.3'x3=3/3
.9'=1

Since numbers can written an infinite number times .9' just happens to be another way to write 1. Thanks for clearing this up for me guys.

P.S. I blame my misunderstanding of this concept on my math teach who routinely denounces the evils of decimals XD.

 

[Willamena]

What do you mean by "not possible"? It is not possible for me to split this coke can into 83 equal parts but that does not mean it is impossible. Just because I lack the ability to split something an infinite amount of times does not make it impossible.

I can't = impossible.

That's exactly what it means.

 

[ch'ang]

If 0.9^ did not equal 1, then there would be a number in between 0.9^ and 1, which there isn't.

Quoted for truth

 

[Willamena]

Do you have a reference for that definition?

The dictionary.

Get a cake and cut it into thirds. You have just produced 3 infinitely recurring decimals in the real world (0.3^). To say that infinitely recurring decimals do not exist would be to say that mathematics is incorrect in the division of 1 by 3 and that therefore the process of division (and as a consequence multiplication, addition and subtraction) are all false.

No, I have produced three pieces of cake. Three is not an infinite regression.

And no one said they didn't exist.

 

[SoyLeche]

This is the first I have seen of this thread; I can't see any justification for saying that 0.999999999 =1 - it isn't.

Just as a question, Fluffy, what would you do with "

" ? Historically, mathematicians have tried to find the answer (which, of course, you know, there isn't one); I was always taught to use the fraction value as opposed to the decimal one.

I agree with what others have said; once 0.999999999=1, what about 0.99999998 ?........

A couple of problems here:

We are not talking about 0.999999999. We are talking about the number 0.9999... where the 9s continue infinately.

As far as I know there is not fraction value of pi. It's an irrational number.

0.99999998 is an entirely different number than 0.9^

 

[Fluffy]

Ugh...I hate math. Ok, correct me if I'm wrong, but I seem to remember that some graphs of certain equations have lines that "infinitely approach" a number, say 1. The line gets closer and closer to 1 (on either axis), but never actually reaches it. Isn't this basically the same thing?

What you are describing are known as asymptotes. They are usually first encountered when shown the graph of y=1/x. http://en.wikipedia.org/wiki/Asymptote See the second figure from the top.

This is a similar idea because it is generally stated by mathematicians that at infinity, the line will touch the axis. It is similar because it deals with a concept of infinity that developed due to the internal inconsistencies of the previous concept and this version of infinity is the same as is being used here. However, beyond that, the similarities end.

This is the first I have seen of this thread; I can't see any justification for saying that 0.999999999 =1 - it isn't.

Hey michel, I am uncertain if you are approximising when you say "0.9999999" but I just want to make sure that you understand that I am not talking about 0.9999999 but "0." followed by an infinite number of 9s (for which the mathematical notation is 0.9^)

I agree with what others have said; once 0.999999999=1, what about 0.99999998 ?........

Keeping the above in mind, it is impossible to have 0.9^8 because this would involve placing an 8 at the end of an infinite sequence of 9s. "at the end of infinity" is similar to ideas such as "outside of space" and "before time". It is certainly not a concept that is mathematically supported. Infinity has no end at which the 8 may be placed.

Just as a question, Fluffy, what would you do with "

" ? Historically, mathematicians have tried to find the answer (which, of course, you know, there isn't one); I was always taught to use the fraction value as opposed to the decimal one.

Pi is one of the mathematical nuggets that I have not yet studied in depth. I understand it to be an irrational number (so it cannot be accurately written as a fraction). However, it is also a transcendental number and so it is part of an entire realm of mathematics I have yet to study. I am uncertain how you would go about calculating a transcendental number (the definition of one is sufficiently complex so I will not post it here) but as far as I am aware, it is not possible to write them without approximising.

However, the decimal to a sufficient degree will always be more accurate than the fraction. The fraction is useful for "simple" calculations in which relatively imprecise answers are sufficient (eg understanding the use and impact of Pi does not require an accurate output)

The dictionary.

Thank you but I was specifically after a reference that stated that 0.9^ is "a number approaching 1 that can never quite get there". Apologies for my skepticism but I do not believe that you will be able to find a respectable mathematician who will endorse that statement.

You should also know that an "infinite regress", as you have defined it, involves convergent series and that it is well documented that when you sum and infinitely long series made up of convergent terms, you will get a finite number. http://en.wikipedia.org/wiki/Geometric_series#Infinite_geometric_series

No, I have produced three pieces of cake. Three is not an infinite regression.

And a piece of cake is a fraction of the entire cake. It is 1/3 of the entire cake and 1/3 is an infinitely recurring decimal. That you have 3 pieces of cake makes sense since 3 x 1/3 = 3/3 = 1 and you have 1 cake.

I can't = impossible.

That's exactly what it means.

I agree but "I can't fly" does not equate to "flight is impossible" which appears to be analagous to what is being argued in that particular case.

 

[Ryan2065]

So, what's the fraction for pi?

Er, we don't use the decimal either... we use that fun symbol for pi =p

If you want to get technical about pi this site lists some of the fun equations of pi:
http://www.geom.uiuc.edu/~huberty/math5337/groupe/expresspi.html

If you use the decimal your results are going to be off.... same with using the decimal for 1/3 rather than just 1/3. It is usually only acceptable to use the decimal form for fractions with exact answers (ie 1/2 = .5) but even then most don't.

I am very surprised. How do these mathematicians deal with numbers that cannot be written in the form m/n (that is to say that cannot be written as a fraction)? Only rational numbers can be written as fractions whilst all irrational numbers have to be noted using decimals (You can use surds for some but I don't think all irrational numbers can be written as a surd either).

There is always a form for them that does not use decimals. Wether you are talking about i or e... Theres some fun square roots that are irrational...

Well I guess I've never read anything that says all irrational numbers have a representation other than decimal form but I've never used or seen any. My studies are not number theory though... My work involves knot theory. Mathematicians don't tend to like decimals so I would imagine if they found a number that had to be expressed as a decimal they would label it with a letter rather than keep it in decimal form

 

[Ryan2065]

Pi is one of the mathematical nuggets that I have not yet studied in depth. I understand it to be an irrational number (so it cannot be accurately written as a fraction). However, it is also a transcendental number and so it is part of an entire realm of mathematics I have yet to study. I am uncertain how you would go about calculating a transcendental number (the definition of one is sufficiently complex so I will not post it here) but as far as I am aware, it is not possible to write them without approximising.

heh, see the link from my above post... Its possible to write pi in an exact form... its just not the exact decimal form. The link above gives all the exact forms of pi...

However, the decimal to a sufficient degree will always be more accurate than the fraction. The fraction is useful for "simple" calculations in which relatively imprecise answers are sufficient (eg understanding the use and impact of Pi does not require an accurate output)

I hope you said this backwards... The fraction (ie the exact number) is always more accurate and most of the time easier to deal with than the decimal =p Espically now since computers are calculating more and more things for mathematicans... They can't compute a number after it is so big in decimal form (it ends up rounding at the end) so you alot of times do not get exact results. If you run the same equation on 3 different computers and they all get different answers its off to the pencil and paper method.

 

[SoyLeche]

Er, we don't use the decimal either... we use that fun symbol for pi =p

Sure, but that fun symbol is absolutely worthless anywhere outside of an equation. I would bet that anyone that needs to use pi (say, an engineer) will use the decimal version rather than a fractional version - knowing full well that it is an approximation. We know pi out to enough decimal points to fit within the tollerance of most any use that we need.

 

[SoyLeche]

Its possible to write pi in an exact form... its just not the exact decimal form. The link above gives all the exact forms of pi...

I didn't see a single "exact form" of pi there. They all had elipses (...).

I don't believe it is possible to write down exactly what pi is equal to, no matter what form you want to try.

 

[Ryan2065]

Sure, but that fun symbol is absolutely worthless anywhere outside of an equation.

I am a mathematician... I care about exact answers...

I would bet that anyone that needs to use pi (say, an engineer) will use the decimal version rather than a fractional version - knowing full well that it is an approximation.

engineers do not care about exact answers... they care about answers close enough to what they need so whatever they make won't break.

We know pi out to enough decimal points to fit within the tollerance of most any use that we need.

But I can't see any mathematician using anything other than the symbol for pi even if we find the decimal form (that is if we find something that is repeating). It would be too long to write out.

 

[Ryan2065]

I didn't see a single "exact form" of pi there. They all had elipses (...).

I don't believe it is possible to write down exactly what pi is equal to, no matter what form you want to try.

Sure we know the exact form... If we didn't then there would be many different versions (Everyones version of what pi should be) and the actual number wouldn't be able to be calculated.
Here is a nice looking equation for pi

Again, if it was not possible to write pi exactly in any form then we would not be able to compute it.

 

[Willamena]

Thank you but I was specifically after a reference that stated that 0.9^ is "a number approaching 1 that can never quite get there".

But that's what having no initiating term means --the infinite series "0.9^" will never be complete and "1" is complete; therefore, 0.9^ can only ever approach 1.

And a piece of cake is a fraction of the entire cake. It is 1/3 of the entire cake and 1/3 is an infinitely recurring decimal. That you have 3 pieces of cake makes sense since 3 x 1/3 = 3/3 = 1 and you have 1 cake.

But I do not hold an "infinitely recurring decimal" in my hand, I hold one piece of cake in my hand. A whole piece. The infinite series that you profess to represent something "real" is in fact entirely conceptual.

 

[SoyLeche]

Sure we know the exact form... If we didn't then there would be many different versions (Everyones version of what pi should be) and the actual number wouldn't be able to be calculated.
Here is a nice looking equation for pi

Again, if it was not possible to write pi exactly in any form then we would not be able to compute it.

You haven't written what pi is equal to though. See those ...'s there? That's no more an exact representation than 3.14... is.

 

[Ryan2065]

You haven't written what pi is equal to though. See those ...'s there? That's no more an exact representation than 3.14... is.

That is what pi is equal to... That pattern continues on forever... Its an exact representation =p

I'm a mathematician so representating numbers in an infinite series is just as exact as a symbol. (ie .3333333333333^ = 1/3) In computations its better to use 1/3 but that does not take away the fact that .3333^ is 1/3 and as long as you note that .33333 goes on forever it is an exact representation.

 

[SoyLeche]

Again, if it was not possible to write pi exactly in any form then we would not be able to compute it.

Last I checked we aren't able to compute it. We've got pretty darn close approximations, but not exact.

 

[SoyLeche]

That is what pi is equal to... That pattern continues on forever...

Which cannot be written. That's what I said

 

[Ryan2065]

Last I checked we aren't able to compute it. We've got pretty darn close approximations, but not exact.

The approximations are in decimal form... There are many forms of a number.

Which cannot be written. That's what I said

It cannot be written in decimal form, but again, there are other forms of a number.