is 0,99999 the same as 1?

Question

This is a mathematical statement: 0,9999999999999 (inifitely) = 9/9 = 1.

Do you agree, yes or no and why?

@ highschoolmathpreparation...
"They are the same. Theres a lot of infinity trickery going on there but I present a proof that hopefully limits this:
x = 0.9999999999999... (*)
10 x = 9.9999999999... (**)
Subtract (**) and (*):
10x - x = 9.99999999... - 0.999999999999...
9x = 9
x = 1
BUT x = 0.9999999999 ... by (*)
Hence 1 = 0.999999999999999... as required"

um... if (*) = 0,9999999.... = 1
10x - x = 9.99999999... - 0.999999999999...
9x = 9 isn't correct anymore.
Because it would mean that 10x-x = 9.99999... - 0.999999... = 9.99999... - 1 = 9
but!
9.99999... - 0.999999... = indeed 9
but 9.99999... - 1 = 8,9999... damnit!
That's what I mean, 1 and 0,9999.. are DIFFERENT! you can't replace 0,9999... with 1 in every math question! cuz
1/9 = 0,1111...
and 1/9 x 9 ≠ 0,999999....!
This is why we invented FRACTIONS]
0,111 with billions of 1s after the comma ISNT the same as 0,11111......
if we wud do it precisely, I wud say that
0,11111111.... ≠ 1/9!
pfff

Answer 1

OK. I have a PhD in Mathematics. From Harvard, no less. And if you wonder whether I'm making that up, try Googling on my name.

.9999.... (infinitely repeated) is defined as the LIMIT of the following sequence:

.9
.99
.999
.9999
and so on.

That is to say, it is the limit as n-->infinity of 1 - 1/(10^n)

That limit is 1.

Hence, .999 .... (infinitely) = 1.

9/9, of course, also equals 1.

It's that simple.

If you don't know what the "limit" of a sequence is, you should either study up and learn that, or else take this fact on faith in the mean time until you do.

Answer 2

They are the same. Theres a lot of infinity trickery going on there but I present a proof that hopefully limits this:
x = 0.9999999999999... (*)
10 x = 9.9999999999... (**)
Subtract (**) and (*):
10x - x = 9.99999999... - 0.999999999999...
9x = 9
x = 1
BUT x = 0.9999999999 ... by (*)
Hence 1 = 0.999999999999999... as required

Hope this helps!

What surprises me most is the number of people that think this isn't true!!! What is happen with our mathematics education!!! Sure this requires calculus to formally prove but everyone should have seen this proof at some point!!! This is a basic proof... I don't know I'm disheartened by the number of incorrect answers and arguments.

To the asker:
Yes 8.999999999999... = 9
Similarily 7.99999999999999.. = 8 by the exact same proof they're all equally valid.

And you are foolish to believe that fractions came about to solve this problem. Fractions came about to solve the problem of inverting integers - we wanted for ever integer a a 'number' b so that a*b = 1 With the rational numbers - or fractions, we now have that. While not the original intent - we have nor formalized the notion to agree with this.

Remember - we can create infinite decimals WITHOUT fractional representation:
0.1234567891011121314151617...
This is an infinite decimal thats not repeating (I believe its trancedental as wlel for those who are good with algebra) So fractions clearly were not created to "take care of" 0.111111111111... because it doesn't even take care of all cases of infinite decimals.

Once again you need to take calculus to fully understand this - any time you are dealing with infinity you really should be using some calculus to give a formal argument - technically all the arguments presented here are incorrect because they are not fully rigourous in that they follow from axioms or their off spring. Yes I also agree that this proof is not formal BUT it gives the correct idea and insight without needing a BMath degree to figure out.

So no you can't take "The billionth" digit of an INFINITELY repeating number - thats against the rules of infinity - you cannot take a finite part and ever expect to get the whole. So your argument is fallacious.

Yes you can replace 0.9999999999999... with 1 in EVERY single math question but again you can't do much with this - 0.9999999999999... * 8 is impossible to work with. however 8*1 is okay and easy - in my argument 10 works nicely since we can just more the decimal place - it really only in fact works with powers of 10 because again the math is tricky - like I said infinity is a VERY tricky concept - if you like this concept take some abstract math courses with calculus to fully appreciate this.

Moreover, by the definition given in calculus, you can wasily prove that 0.1111111111111... = 1/9 but again it seems like few people reading this post have any idea of calculus or even as to the ACTUAL reason why this is true. So please - if you want pick up your favourite abstract math book, look up the chapter on decimal representations and there you'll get your answer fully. Since I doubt many people will a) be ambitious enough to do so or b) Won't understand enough math to do so - this answer remains as the correct answer.

As required.

Yes Kyle G. You have the right idea - I am currently obtaining a degree in Pure Mathematics degree - glad to see someone else has some good mathematical insight and understanding of calculus (Please note: to those reading calculus is a subsection of Real Analysis - but since most people don't really know what Real analysis is - and don't understand how hard it may be I have used the word calculus instead because people have an idea that calculus is not easy). By the way clever epsilon argument in disguise (for all epsilon greater than 0, you can append enough 9's so that 0.999...9 > 1-epsilon) - I'll remember that next time I have to explain this to someone.

Keira D. The integers and the natural numbers are the same infinity - there exists a one to one correspondence between them: Use |-> to represent maps to
0 |-> 0
positive integer n |-> 2n
negative integer n |-> 2|n| -1
So they have the same number of elements.

Just to Expand on Curt's point below,

Here is the formal proof:

1) Define a decimal exmansion 0.a_1a_2a_3... as
0.a_1a_2a_3 = limit(sum(a_k/10^k, k=1..n, n=infinity)
(using standard maple notation there)
Okay now we know what a decimal expansion is. This is the definition there is no disputing this. Any good mathematics text book has this. Next we have the following lemma:
2) Sum((a-1)/a^k,k=1..n) = 1 - 1/a^n
Proof: Expanding the left side we get:
1 - 1/a + 1/a - 1/a^2 + ... +1/a^(n-1) - 1/a^n
Which is a telescoping series - thus eliminating the middle terms we get the right hand side as required.
3) We claim:
0.9999999999... = Sum((a-1)/a^k,k=1..infinity) = 1
when of course a = 10.
Proof: We know (again a =10 here but I do not want to change the notation:
limit(Sum((a-1)/a^k,k=1..n), n = infinity) = limit(1 - 1/a^n, n = infinity) by lemma 2 and taking limits to infinity on both sides. Now the RHS is
limit(1 - 1/a^n, n = infinity)
= limit(1 , n = infinity) - limit(1/a^n, n = infinity) (Using limit rules)
= 1 - 0 (By the Archemedian property or however else you'd like to see it)
=1
BUT the LHS is
limit(Sum((a-1)/a^k,k=1..n), n = infinity)
= limit(Sum((9/10^k,k=1..n), n = infinity)
= 9/10 + 9/100 + 9/1000 +....
= 0.9999999999999....
So 1 = 0.999999999999999... as required

There now you have the 100% irrefutable calculus proof - if you do not accept this you do not accept mathematics. This is fully rigourous and I never want to hear anything else of the sort that this is not true.

lol this is amazing how much bad mathematics is on this one post... phenomenal... Our poor education system...

Answer 3

will if the number is .999 Repeating then it is equal to 1. here is why
1/9=.111 Repeating
2/9=.222 Repeating
3/9=.333 Repeating
4/9=.444 Repeating
5/9=.555 Repeating
6/9=.666 Repeating
7/9=.777 Repeating
8/9=.888 Repeating
9/9= .999 Repeating also a number divided by it's slef is one so that means that .999 Repeating is exactly equal to 1. So yes I do agree that they are equal.

Answer 4

OK, let me settle this for all the clear disagreement and idiocy in these responses. I have a degree in abstract mathematics.

Yes, .999999 repeating infinitely equals 1.

The reason is going to get into a subject called "real analysis," but I'll explain a little.

Two numbers are equal if there is no number in between them. That is one definition of equality. For example, is 1.0000 equal to 1? Yes, because there is no number in between 1.0000 and 1.

Now let's look at .99999 repeating. Is there a number greater than that but less than 1?

No. .8? No, .8 is less than .9.
.99? No, because .99 < .9999999 repeating
.999998? No, because that is < .9999999999
.99999999999? No, because that is < .9999999999999999999

There is literally NO NUMBER that exists between .9 repeating and 1. Therefore, they are equal.

There are many other explanations one can give, but this is one of the simplest ones. I could invoke a real proof involving things called "Cauchy sequences," but someone actually disagreeing with what I've said above would never understand Cauchy sequences anyways.

One final proof that someone else already mentioned here is:

Let .999999999 repeating = x
10x = 9.999999 repeating
10x-x=9.9999 repeating - .99999 repeating
9x = 9
x=1

But x=.9 repeating.
Therefore, 1=.9 repeating.

Answer 5

if the 9's extend indefinitely then yes, it is the same as 1. There are two ways of looking at this:

1) The difference between this number and 1 is infinitesimally small, therefore there is no difference.

2) 1/10 of this number is 0.0999999999999... again continuing infinitely. The difference between these two numbers, 9/10 of the original number, is exactly 0.9. Since 9/10 of the number is 9, the number must be 1.

Answer 6

9/9ths does not convert to .999999 ad infinitum. It converts to simply 1. The logic is based on a faulty understanding of infinte repeating decimals. 1/3 becomes .33333 ad infinitum, and 2/3 becomes .66666 ad infinitum. One might think if 1/3 + 2/3 = 1 then .33333 ad infinitum + .66666.....= .9999999....= 1. .999999....... is approximately one, but is not one, not the same thing at all BY THE LAWS OF ORDINARY ARITHMETIC, but in the higher Mathematics is is the functional equivalent of ONE. Higher forms of Mathematics sometimes use different assumptions and axioms of the lower forms. Riemanian Geometry was created when Riemann decided to change a few basic axioms and premises and got a system equally valid, but vastly different! This makes one wonder if ALL systems are equally valid, or if in fact NO system is valid at all, and all we do is play number games and imagine we have proved something when in reality we have proved nothing. It was Godel who proved that you can't prove anything WITHIN a system, using the system itself. I'm sure if Godel had taken further time, he'd have realized that his own proof was invalid by his own law! He sure couldn't have proven it by some outside system!

If I had $999,999,999.99 in the bank American dollars, for all practical purposes I'm a Billionaire, but to be technical I'm a penny short of a Billionaire. But how short is .999999..... from one? To answer it with anything would be to invoke the infinitessimal numbers that Newton envisioned (smaller than the smallest possible decimal but still greater than 0). Later mathematicians canned the concept of infinitessimals as it only clouded the issue and helped nothing. But then you find yourself in a paradox. You cannot give as an answer the amount that .999999.....is deficient, that if you added it, it would become 1, therefore, .99999999 is equal to one!

Want to experience something interesting about calculators and the logic by which they operate?

Take a large number like 398439 and if you have the scientific kind you can take the square root. Keep taking the square root of the answer over and over and eventually the calculator will spit out 1. The number keeps getting closer and closer to 1 like 1.0000000593 and therefore by the laws of Calculus if the the number of operations are infinite, taking the square root of the square root of a positive integer an infinite number of times will equal 1. The poor calculator can't handle infinity, so it simply spits out 1 and rather quickly at that! Try it and see.

Mathematicians hate infinity too, and wherever possible try to get rid of it! Infinity like division by zero can really screw up your calculations! Is infinity a true number or just a mere concept? Various systems have been devised to allow one to add, subtract, and multiply or divide Infinity or with infinity but they don't resemble ordinary rules of mathematics at all! And even worse! Not all infinities are EQUAL, there are orders of infinity embedded within infinity!
The set of all integers "{....-6, -5,-4,-3,-2,-1, 0, 1, 2 ,3,4, 5, 6....} HAS to be larger than the set of all positive integers alone {0, 1, 2, 3, 4, 5, 6, 7, 8.....} because it has MORE members. Lewis Cantor created the concept of the Transfinite number or orders of infinity. If the second set was Alef nul, the first and larger set would be Alef prime as it contains the first set within it!!

Answer 7

Yes it is exactly the same as 1.

Look at it this way, no matter how tiny of number you imagine (call it delta), the difference between the infinite-nines and 1 is less than the delta.

So the difference becomes zero in the limit and infinite-nines is equal to 1.

Answer 8

1/3 = 0.3333333333333333333333333333.....
3 * 1/3 = 3 * 0.33333333333333333333333....
3/3 = 0.999999999999999999999999999....
1 = 0.99999999999999999999999999...

Answer 9

hehe
it is. I asked the same question a few times earlier.
Thing is, half the sillies on this site don't even know the diff. between .999... and .999. They don't know what a repeating decimal is!

Answer 10

yes

0.9999 ..= 1

multiplying with 10

9.9999.. =10

let0.999.. be x

9+x=10

x=10-9

x =1

ie, .999999.....is 1