0 or 1 ??

what is the value of $[0.9999999999999999...]$ ??

here $[x]$ represents the greatest integer function.

#GreatestIntegerFunction(Floor)

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The thing that most people have confusion about is the infinite number of $9$ s after the decimal.They tend to think that $.999999...$ approaches $1$ but never quite reaches $1$ . First thing to realize is that $.999...$ does not only reach one but it is exactly equal to $1$ ,and when I say exactly,I really mean it.Now,computing the above expression is a simple matter of substituting $1$ in place of $.99999...$ .

If you still aren't contented,then think about it this way.What do we mean when we say that two real numbers are different?It means that there is at least one more real number between them(In fact there are an infinite number of them). Now answer this:Can you find any other real number between $1$ and $.9999....$ ?If you can't ,the only conclusion is that they are equal.

Rahul Saha - 6 years, 4 months ago

The Value = 1

let X = 0.999999999999999999 ......

then 10 X = 9.999999999999999 ......

subtract (10 X - X) = 9 X = 9 =====> X =1

Ossama Ismail - 6 years, 4 months ago

yes the answer is one. one of example why 0,9999..........=1. look 10/3= 3,33333333333333..... then 10= 3times3.333333333....................... 10=9.9999999999999999. next 10= 9+0.99999999..............................subtract 10-9=0.9999999......... so 1=0.999999999999........

Ivan dennys Opit - 6 years, 4 months ago

The value of the expression will be zero and not one. Since the value of $X$ which is given approaches the value of one but it really does not become equal to it and will always remain less than it and thus $[X] = 0$ .

Sudeep Salgia - 6 years, 4 months ago

The value is $1$ .

This is basically asking the greatest integer that is less than or equal to the limit of $1-{10}^{-n}$ as $n$ approaches infinity. Since the limit is $1$ , so is the final answer. The order matters.

Mursalin Habib - 6 years, 4 months ago

madhav srirangan - 6 years, 4 months ago

What I think is that the main question here is that whether 0.9999999...... is equal to 1 or not.and that I think it is.Therefore the answer should be 1.

Aman Jain - 6 years, 4 months ago

agreed.

sakshi taparia - 6 years, 4 months ago

AS THIS IS AN INFINITE SUM I.E 0+9/10+9/100...................... SO ON AND SO FOURTH, WE SEE THAT ITS VALUE APPROCHES 1. WELL I DONT UNDERSTAND THE STIGMA/CONUNDRUM HERE. THIS FITS RATHER PERFECTLY INTO THE DEFINITION OF GIF THAT GIF OF ALL NUMBERS LYING BETWEEN 2 INTEGERS IS EQUAL TO THE INTEGER LESS THAN OR EQUAL TO IT. HERE 0.9999999999999999999999........... CAN BE TREATED AS THE LAST REAL NUMBER B/W 0 AND 1 AS IT NEVER REACHES ZERO. SOME WILL ARGUE THAT TO MAINTAIN CONTINUITY WE WILL REGARD IT AS ONE BUT THINK AGAIN.

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS. HENCE IT IS ZERO.

Arnav Jain - 6 years ago

Is 0.9999999999999999999.... an integer?

Archit Boobna - 6 years, 3 months ago

Yes, it is.

Mursalin Habib - 6 years, 3 months ago

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS. HENCE IT IS ZERO.

Arnav Jain - 6 years ago

I don't know much about continuity but I think an infinite geometric series as defined as the limit of the geometric series as it approaches infinity. Thus $.999\cdots =1$ follows by applying the definition.

Btw, what do you mean by the last real number between two numbers.? If there aren't an infinite number of real numbers between $a$ and $b$ , then $a=b$ . A "last real number" can't exist in an open interval like $(0,1)$ .

Rahul Saha - 5 years, 12 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$