x = 0.9999999999999999.............
10*x = 9.9999999999999999.............
-------------------
9*x = 9
=> x=1 ??
but x = 0.99999999...............
can we always say that 0.999999999999999999.... is equal to 1
I think it should be 0.999999999999999999.... is nearly equal to 1
please explain
2007-03-28 03:39:52 · 6 answers · asked by Life sucks . but u gonna love it 2 in Science & Mathematics ➔ Mathematics
0.999… + y = 1
find y ?
wow really math gets interesting when we go deep into 0 and infinity
2007-03-28 03:59:45 · update #1
No.
It is *exactly* equal to 1.
You have offered an algebraic proof.
Here is one using fractions:
1/3 = 0.333...
3 * 1/3 = 0.999...
1=0.999...
Other proofs are discussed here:
http://en.wikipedia.org/wiki/0.999...
2007-03-28 03:45:35 · answer #1 · answered by Jerry P 6 · 6⤊ 1⤋
This gets into the concept of "infinity". When you say that a number repeats "infinitely", then you have to take a mental leap. This is what you have proven by saying that 0.9999... = 1
It's the same as saying 0.33333... = 1/3
x = 0.3333333
10x = 3.3333333
-------------------------
9x = 3
x = 3/9 = 1/3
If you terminate the number 0.99999999... at any decimal place, even the 10,000,000,000,000,000th decimal place, THEN you can say that it "approximates" 1.
2007-03-28 10:45:17 · answer #2 · answered by Dave 6 · 1⤊ 0⤋
Any mathematician of the conventional kind will argue that 0.999... (recurring) equals 1 exactly....
Some will even go out of their way to call those who believe otherwise idiots....
But what they aren't mentioning is that it requires you automatically accept the axiomatic definition of "real numbers".... including the claim that all fractions can be represented in decimal form. If you believe that 0.333... = 1/3 ... then you must accept that 0.999... = 1.
On the other hand, if... like me... you do not believe that all fractions can be precisely expressed in decimal form but only approximated to a degree of maximum closeness... then it quite clearly does not.
Likewise, my belief of what "infinity" actually represents is very different to what mathematicians consider "infinity" to be, and therein lies the problem.
But then as I have noticed, if you go near a mathematician with the notion that you disbelieve their axioms.... they will react to it much like a christian will to the denial of god..... It is quite humourous.....
Break anything down far enough and you come to realise that to some extent it all relies on blindly believing the unproven.
2007-03-28 11:02:08 · answer #3 · answered by Nihilist Templar 4 · 1⤊ 3⤋
Well, .999999999999999 is not exactly equal to 1, but putting a bar over the 9s is a way to express 1. The bar means "repeats to infinity".
Similarly, .66666 with a bar over it is a way to express the exact quantity of 2/3.
2007-03-28 10:47:24 · answer #4 · answered by tinal22 2 · 1⤊ 4⤋
.999... is the limit of a sum of an infinite series.
lim (n -> inf) sum (k = 1..n) (9 / 10^k)
That limit is 1.
2007-03-28 10:55:50 · answer #5 · answered by ? 4 · 0⤊ 2⤋
9*x=9 only on your calculator. it rounds up to one because it cannot display all of the digits.
2007-03-28 10:43:59 · answer #6 · answered by tonikat 2 · 0⤊ 3⤋