Anaxagoras
Banned
y= 0.9999999
10y=9.9999999
9y=10y - y
9y=9.9999999 - 0.9999999
9y=9
y=1
1=0.9999999
Where did I go wrong?
10y=9.9999999
9y=10y - y
9y=9.9999999 - 0.9999999
9y=9
y=1
1=0.9999999
Where did I go wrong?
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0.9999 = 1, true or false and why?
- Thread starter Anaxagoras
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StopMakingSense
Member
Anaxagoras said:
0.9999999 * 10 is not 9.9999999
BananaBomb
Member
:lol :lol :lol
edit: to the doubters, it can be proved dozens of different ways that .999* = 1
edit: to the doubters, it can be proved dozens of different ways that .999* = 1
Mr. Spinnington
Banned
nerds
Earthstrike
Member
0.9999... is exactly equal to one.
You went wrong nowhere. This is just a consequence of the definition of a repeating decimal as setup by mathematics. A lot of people tend to have a knee jerk reaction to reject for some reason.
A much simply way to think about this is 1/3 + 2/3 = 3/3
1/3 = .333...
2/3 = .6666...
There's several easy ways to rationalize this.
The first is to consider the difference between 0.999... and 1.
You would have 0.000... with a hypothetical 1 as the inifitieth digit. Consequently, you would have a number that is infinitely small. If you have a number that's infinitely small, then it must be the smallest number (absolute value of course). a la zero.
Suffice it to say though, the math will show through.
You went wrong nowhere. This is just a consequence of the definition of a repeating decimal as setup by mathematics. A lot of people tend to have a knee jerk reaction to reject for some reason.
A much simply way to think about this is 1/3 + 2/3 = 3/3
1/3 = .333...
2/3 = .6666...
There's several easy ways to rationalize this.
The first is to consider the difference between 0.999... and 1.
You would have 0.000... with a hypothetical 1 as the inifitieth digit. Consequently, you would have a number that is infinitely small. If you have a number that's infinitely small, then it must be the smallest number (absolute value of course). a la zero.
Suffice it to say though, the math will show through.
StuBurns said:
I believe you meant to say 0.333* x 3 = 0.999*. Because 0.333* x 0.333* = 0.110889.
As for the OP's question, it all depends on the point of view. If you see things technically, then no, 0.999999 does not = 1 because no matter what you do with it, how many 9s there are, there is still an infinitely small difference which keeps shrinking as you add 9s but never completely goes away. In practical use however, 0.999999 = 1.
The fraction argument is a fallacy, because the conversion of 1/3 to decimals relies on a small amount of rounding which introduces a margin of error big enough to make up for the actual difference between both numbers.
Coolio McAwesome
Member
72/9 = 8.0
73/9 = 8.111111
74/9 = 8.222222
75/9 = 8.333333
76/9 = 8.444444
77/9 = 8.555555
78/9 = 8.666666
79/9 = 8.777777
80/9 = 8.888888
Following this same pattern, the next answer should naturally be 8.999999. However:
81/ 9 = 9.0
73/9 = 8.111111
74/9 = 8.222222
75/9 = 8.333333
76/9 = 8.444444
77/9 = 8.555555
78/9 = 8.666666
79/9 = 8.777777
80/9 = 8.888888
Following this same pattern, the next answer should naturally be 8.999999. However:
81/ 9 = 9.0
BananaBomb
Member
Kipz said:
no it's a real number that is equal to one.
RevoDS said:
This is wrong (no offense). .9999... = 1 in theory and in "practice."
The easiest way to think about it is that numbers (real numbers specifically for this problem) are abstract quantities. We represent them with "numerals". So when we write "5," we are writing down a representation of the abstract quantity that is the real number 5. It turns out because of how our number system was set up, the numeral ".999...." and the numeral "1" represent the same real number. Hence, we say .9999... = 1.
Edit: PS, there is no fallacy in the fraction argument.
Haven't seen one of these threads on here in a while. They occurred so often in a short period of time and then suddenly stopped. I thought they were bannable at some point in time.
Anyway, 0.9999 repeating is 1. Numbers are weird, but that's just how it is. You can rationalize it in many ways or you can just acknowledge it as truth. The latter is easier. While there are many ways to prove that 0.99999... = 1, there is no way to confirm the opposite.
I've always thought a better question was this:
If you had a hat with an infinite amount of slips of paper consisting of an infinite amount of numbers, the probability of pulling any random number is zero (1/infinity = 0). In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0). However, if you actually reach down in there and pull out a number, the probability obviously isn't zero because you just pulled out a number. Why is that?
I had my Cal professor explain the answer to this one for me, but I totally forgot what he said haha.
Anyway, 0.9999 repeating is 1. Numbers are weird, but that's just how it is. You can rationalize it in many ways or you can just acknowledge it as truth. The latter is easier. While there are many ways to prove that 0.99999... = 1, there is no way to confirm the opposite.
I've always thought a better question was this:
If you had a hat with an infinite amount of slips of paper consisting of an infinite amount of numbers, the probability of pulling any random number is zero (1/infinity = 0). In fact, the probability of pulling any real subset of numbers is zero (i.e. 10000/infinity = 0). However, if you actually reach down in there and pull out a number, the probability obviously isn't zero because you just pulled out a number. Why is that?
I had my Cal professor explain the answer to this one for me, but I totally forgot what he said haha.
It's a featured article on Wikipedia.
Earthstrike
Member
RevoDS said:
There's not rounding in the fractional representation of 1/3. Also, answers to mathematical questions do not depend on "points of view".
Plain and simple mathematics is the evaluation of the set of consequences and relations amongst a given set of definitions. By the definitions in our number system 0.99999 repeating is exactly equal to one.
You're argument contains a fallacy however because you say "there is still an inifinitely small difference that never goes away". Actually it HAS to go away by definition.
Proof: Lets say there is some infinitely small difference that is greater than zero. Divide that by 10. It is now smaller, meaning it wasn't an infinitely small difference. The only infinitely small difference is zero.
Mr. Spinnington
Banned
sorry, math nerds
No, it doens't depend on your point of view. They are different representations of the same number. Technically, practically, and in every way.RevoDS said:
"4/4" looks different to "1", but people easily accept that these are different representations of an identical number. The exact same deal applies to 0.999... = 1
Earthstrike said:
By "infinitely small", I meant that it was equal to 1x10^-n, where n is the amount of 9s after the 0. As the 0.9999.... grows ad infinitum towards 1 without ever actually reaching it, the difference between 1 and that number shrinks at the same rate.
There IS rounding, for the very same reason there is a 0.0000....1 difference between 0.9999....9 and 1
In practice, we simply assume 0.999999 = 1 for convenience and because the 0.000001 difference is mathematically insignificant. But it is still there no matter what, thus 0.999999 and 1 are not the same number.
Assuming 0.999... is 1 is the very same thing as assuming 99 is 100, only on a smaller scale.
99/100 =9.9/10
9.9/10=0.99/1
You are correct that 0.9999...9 does not equal 1. However, we aren't talking about 0.9999...9.RevoDS said:
Or, to put it another way,
0.9999...9 != 1
0.9999...9 != 0.9999...
Specious reasoning. You are offering one pair in a convergence series. The correct observation is that the difference between the 0.9...9 series and 1 approaches 0 as the series is extended.RevoDS said:
This magical 0.000...1 number doesn't exist. You can't have an infinite amount of 0's and then add a 1 at the end, because if you can add a 1 at the end then you obviously could have added more 0's instead hence it wasn't an infinite amount. Therefore, as you say, the difference between 0.999... and 1 is 0.000... which is simply another way of writing 0.RevoDS said:
But there is no 0.000000.........1 difference (for the proper 1-0.999...= argument). No matter how far you go, that tailing one doesn't exist. And because there is no difference between the numbers, they are the same.RevoDS said:
The entire mathematic world agrees that they are equal to each other, and not just due to "mathematical insignificance" like you claim.
No it doesn't, because the second it stops being an infinite series it stops being equal.
ratcliffja
Member
tokkun said:
This is the one I always use. It seems weird, but it's actually true. The entire concept of repeating decimals is to express a ratio or fraction in decimal form. Repeating numbers can always be expressed as a number out of a series of 9s. For instance, .123123123... = 123/999.
Earthstrike
Member
RevoDS said:
You do not make assumptions in math. We're not assuming 0.999... is equal to one. We have proven it with nothing more than addition, subtraction, division, multiplication, and algebraic substitution.
Math is not felt out or worked out by analogy to thought processes. You have the rules defined within the system and the definitions of the system. With these you can demonstrate the 0.999... = 1, ergo, 0.999... is exactly equal to one.
Handycrap101
Banned
no imo
BananaBomb
Member
Zaptruder said:
What does the smallest margin even mean? There isn't such a thing. How do you represent the number one less than infinity?
And to the guy above me, unless I am in grievous error, 3! and 9! are both real numbers...
The "smallest margin" is 0. In the real number system there is no number that is the "smallest possible size but larger than 0". Number systems that include such numbers (called infinitesimals) include hyperreal numbers; however, for analysis purposes these are considered non-standard.Zaptruder said:
Earthstrike said:
There are assumptions made in math, but they are fundamental assumptions.Wikipedia said:
I'd guess:Zaptruder said:
y = 1 - ε
Code:
12.8
+ 6.1
-----
18.9
The sum of each column can be taken to find
the total sum.
0.333333...
+ 0.333333...
+ 0.333333...
-------------
0.999999...
BananaBomb
Member
Since I feel like there is more controversy in this thread than is acceptable, let's just seal the issue with a simple (and mathematically sound) proof.
Sum of a geometric series with a = 9 and r = 1/10 is 1. End of story.
Sum of a geometric series with a = 9 and r = 1/10 is 1. End of story.
ColtraineGF
Member
My oh my, never expected to see this argument at GAF. Eh.
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
^This link holds all the arguments you can shake a stick at, and common refutations to the fact and whatnot.
http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html
^This link holds all the arguments you can shake a stick at, and common refutations to the fact and whatnot.
There probably is, but I'm not a mathmetician.ccbfan said:
If it's just the multiplication bit you don't like, try...
Otherwise, there are a whole bunch of the proofs at http://en.wikipedia.org/wiki/0.999...#Proofs don't use decimal representation of a fractions at all.
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