How can 0.99999999... be equal to 1?

Question

Let' s take x = 0.99999999...
Then 10x = 9.99999999...
10x -1x = 9.99999999... - 0.99999999...
Therefore 9x = 9
Then x = 1.

But How can 0.99999999... be equal to 1? Though it always approaches 1 it never IS 1. How can you explain This Please?

P.S: Please Don't say rounding off to 1. I am asking about accurate mathematics not rounding-off.

Answer 1

Said recurring decimal formation relates "endlessly recurring mutual reciprocals" of two equal digits numbers ( from number sets either 0...9, or 00...99, or 000...999 or any higher order!)

You will find that ....
1
--- = 0.11111111111111111... and
9

9
--- = 0.99999999999999999... which is "9 times 1/9"
9

Though 9/9 =1 we apply 1/9= 0.11111111111111111...!

So technically nothing is wrong in 1= 0.99999999999999...

similarly reciprocal of 3 is...
1
--- = 0.333333333333333333... and
3

3
--- = 0.999999999999999999
3

Vedic Mathematics makes use of "complementary or mutual reciprocals" to mentally compute "huge number of digits answers" that too without least little truncation error !

An example of two complementary reciprocals are!
1/81=012345679,987654321= 80/81 and..
it is technically (111111111)^2.

If you need (111111111,111111111)^2 answer is... 012345679012345679,987654321987654321

If you need (10^10 times 111111111)^2 answer is...

10^10 times 012345679 (are left digits) and 10^10 times 987654321 are right digits!

Answer is dead accurate!

You can expand said application in too many manners! In fact ancient Indians have done it meticulously!

A second application links to mutual reciprocals and structure of those

You will find 1/17= 0588 2352 9411 7647 recurring (endless times) You will also find that 05882353*17= 100,000,001. You can mentally compute 1/17 through 16/17 by knowing said relation! There are endless mental applications relating mutual reciprocals and their digits! Strangely people hardly use it!

Fact is that, your question touches absolute basics of Vedic mathematics which is now regarded as not-scientific!

Utility makes theories irrelevant!


Regards!

Answer 2

Theoretically, 0.99999999... approaches 1 as a limit, will never reach 1 nor exceed one. However, for all practical purposes (non-theoretical?) it equals one. For example if 0.99999999 is the width of a piece of iron (in meters or centimeters) at some point the error in measurement will be far greater than the next refinement of the number or the refinement will be less than the distance a vibrating iron atom moves. Limits are an essential starting point for the concepts of calculus.

Answer 3

Yeah this is a common mathematics puzzle.

The thing is .9999999.... till infinity and as u have rightly proved we get the answer as one. When you study hihger level calculus you can get the answer to this question.

I have asked someone myself before and they told me they would tell me once i am well versed with the basics of calculus.

It has something to do with like when say x->2 it never is 2 but to find the limit we cancel etc then we substitute x=2.

How ???
well here are some more proofs

We all know that 1/3 (a third) can be written decimally as 0.333333 recurring (i.e. the 3s go on forever). It is also obvious that by multiplying a third by three, we get three thirds (3/3) , which is equal to one.
-
If we look at this same process decimally, take the number 0.33333 recurring and multiply it by 3, and you will see that each 3 in the sequence gets turned into a 9. This gives us .99999 recurring, which, since it is the same as 3/3, is also equal to 1, as explained in the previous paragraph.
-
The reason for this is that as you add more 9s onto the number 0.9 (after the decimal point), it gets closer and closer to 1. Since there are an infinite number of 9s after the point in 0.99999 recurring, the difference between this number and 1 must be infinitely small, and therefore cannot be any greater than 0.
QED.

.999999999~ (recurring) is equal to one because it is .333333~ (1/3) multiplied by 3.


Next one


It's equal to one, and the proff is through the way you convert rational repeating numbers to fractions. if you let x = .999999999~ and let 10*x = 9.9999 repeating then 10 x - x = 9x = 9.99999 repeating - .99999 repeating = 9. SO 9x is 9 and x is equal to one. and since we started out wiht x equaling .9999 repeating we know that .9999 repeating = 1.

This could be done with .999999999~ anbd the result would show that .999999999~ = 1
x = .333...
10x = 3.333...
10x = 3.333...
- x = 0.333...
9x = 3
x = 3/9
x=1/3
.333... = x = 1/3

Ok but the actual and only prrof is calculus. I will tell u as soon as i get to know.

Answer 4

isn't that fact, a million.00000....a million = a million merely like asserting 10 = 13? I recommend, equivalent ability precisely an identical. even nevertheless they are very very close jointly, they are nonetheless 2 diverse numbers. No data would desire to ever exchange that actuality. Math is physically powerful good judgment hence. additionally on your data on the main appropriate, the line 10x = a million.000000....a million is fake 10x = a million.00000... a million (that being one decimal to the left) somewhat, a million.00000... a million isn't a real selection. you desire a definitive like a million x 10^-one thousand multiply it via 10 and you get 10 x 10^-999 it would be less demanding to work out in case you pronounced 9x = 9.0000... 9 the finished equation you made relies upon on that mistake, with the numbers being properly expanded you get a diverse effect.

Answer 5

The situation you pose assumes an infinite number of 9s following the decimal point. That's where the problem occurs. The value of 10x would by definition contain one less decimal place than the value of 1x. But since we're dealing with the notion of infinity, more a concept than a number used to quantify, we can produce a false result, that being ".99999....=1"

Answer 6

anyway the way this was explained to me was in terms of a sequence.
so the sum of n=0 to infinity of .9(.1)^n
because it would be ..9*1 + .9*.1 + 9*.01... to infinity
which is equal to .9999999
but by definitions a sum of a geometric sequnce where the number being raised to the nth power is a/(1-r) where a is the constant and r is the number being raised to a power.
so plug in the numbers and you get .9/.9=1

Answer 7

10*0.9 = 9
9 - 0.9 = 8.1
8.1/9 = 0.9

10*0.99 = 9.9
9.9 - 0.99 = 8.91
8.91/9 = 0.99

10*0.999 = 9.99
9.99 - 0.999 = 8.991
8.991/9 = 0.999

In other words, 9x ≠ 9
9x = 8.999999 .............. 1
because
10*0.99999 ..... 99999 = 9.9999 ..... 999990
The number of 9's in each string must be equal, no matter how long the string. You cannot append a 9 when you multiply by ten. You must append a 0.

No matter how many 9's you string out behind the decimal point, the value only approaches 1. It is not 1.

Answer 8

Take any number beween 0.99999999 and 1.
0.99999999...> any number between 0.99999999 and 1.
Therefore 0.99999999... =1.

Answer 9

Believe it or not, math involves a lot of approximations and rules made up so it all works.

Its like how one third is EXACTLY the same as 0.33 recurring

Answer 10

It's because no matter what, 0.999999... can equal 1 because 1=0.999999....+0.000000...., but since you can't put the 1 on the end of 0.0000.... repetitive, then 1=0.99999.....