I'm posting one puzzle, riddle, math, or statistical problem a day. Try to answer each one and post your answers in the comments section. I'll post the answer the next day. Even if you have the same answer as someone else, feel free to put up your answer, too!
Tuesday, July 31, 2007
What's Wrong With That?
The following is what seems to be a mathematical proof that ten equals 9.999999.... What's wrong with it?
a = 9.999999... 10a = 99.999999... 10a - a = 90 9a = 90 a = 10
The problem comes when you multiply by 10 and subtract. Since 9.999999... is not a rational number (that is, you can't express it in closed form with a fraction, like you could with, say 1.111111... = 10/9), we can only express it to some degree of accuracy (some number of significant digits, that is). Then, when you multiply by 10, the level of precision you chose is moved to the next level. If you do it this way, the problem resolves itself. For example:
a = 9.999 10a = 99.99 10a - a = 89.991 9a = 89.991 a = 9.999
The same principle applies for any number of significant figures you choose.
this can also be displayed in the following pattern 1/9= .111111... 2/9= .222222... 3/9= .333333... ... 8/9= .888888... follwoing this logic, 9/9= .999999... , where the fraction 9/9 also = 1
I used the definition of geometric sums, a/(1-r). I still say that level of precision argument I pointed out is sort of the crux of the trick in your proof, but apparently 9.99999.. = 10 in our decimal system for some reason. Good post, Mike.
Yeah. I was a bit worried when I saw Abe's post, but other peoples' took me back to reality and reminded me that, yes, this is the same logic as the 0.999... = 1 proof
could the blog author please come forward and let us know whether this was a trick question or whether they just screwed up on this one?
Hello DJ Lower/kkairos, I am the blog author and I believe it is a trick question. I've been wrong enough in the past not to be completely certain about that.
It definitely is a trick question! That is one of the extraordinary and magnificient features of real numbers. You can't tell which is the closest smaller number than 10, since 9,999... = 10.
Abe's response is the correct. The principal difference is the type of number: rationale or non-rationale. You can't mix orange and lemons :):):) When we say 1/3=0.333333..., that is an non-rationale aproximation. The correct is 1/3. The difference is minimum (correct) but there is a difference. So, when you're talk about "aproximation" the equations are correct: "The rationale approximation of the non-rationale 0.999999... is 1"
we can prove that 9.99999...... is a rational number and all other numbers in this format (repeated form) are rational numbers a = 9.999999... 10a = 99.999999... 10a - a = 90 9a = 90 a = 10 by Zafarullah khan
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The problem comes when you multiply by 10 and subtract. Since 9.999999... is not a rational number (that is, you can't express it in closed form with a fraction, like you could with, say 1.111111... = 10/9), we can only express it to some degree of accuracy (some number of significant digits, that is). Then, when you multiply by 10, the level of precision you chose is moved to the next level. If you do it this way, the problem resolves itself. For example:
ReplyDeletea = 9.999
10a = 99.99
10a - a = 89.991
9a = 89.991
a = 9.999
The same principle applies for any number of significant figures you choose.
I always thought that this was a legitimate proof, and so the 1 in 89.991 was just a condition of not extending the ...999's.
ReplyDeleteIf this proof is invalid, what does that say about the following:
1/3 + 2/3 = 1
1/3 = .33333...
2/3 = .66666...
.33333... + .66666... = .99999...
Therefore .99999... = 1
?
this can also be displayed in the following pattern
ReplyDelete1/9= .111111...
2/9= .222222...
3/9= .333333...
...
8/9= .888888...
follwoing this logic,
9/9= .999999... , where the fraction 9/9 also = 1
I'm going to side with anonymous and nick on this one. There's (I believe) nothing wrong with the equations. 9.999... is equal to 10.
ReplyDeleteIt was a trick question.
I researched it, and it looks like you're right.
ReplyDelete9.99999... is an infinite geometric sum:
9*(1/10)^0 + 9*(1/10)^1 + 9*(1/10)^2 = 9 / (1 - 1/10) = 10.
I used the definition of geometric sums, a/(1-r). I still say that level of precision argument I pointed out is sort of the crux of the trick in your proof, but apparently 9.99999.. = 10 in our decimal system for some reason. Good post, Mike.
Yeah. I was a bit worried when I saw Abe's post, but other peoples' took me back to reality and reminded me that, yes, this is the same logic as the 0.999... = 1 proof
ReplyDeletecould the blog author please come forward and let us know whether this was a trick question or whether they just screwed up on this one?
Hello DJ Lower/kkairos, I am the blog author and I believe it is a trick question. I've been wrong enough in the past not to be completely certain about that.
ReplyDeleteIt definitely is a trick question!
ReplyDeleteThat is one of the extraordinary and magnificient features of real numbers. You can't tell which is the closest smaller number than 10, since 9,999... = 10.
sorry, 9.999...=10 :)
ReplyDeleteI am Finnish and we use "," as the decimal separator
Abe's response is the correct. The principal difference is the type of number: rationale or non-rationale. You can't mix orange and lemons :):):)
ReplyDeleteWhen we say 1/3=0.333333..., that is an non-rationale aproximation. The correct is 1/3. The difference is minimum (correct) but there is a difference.
So, when you're talk about "aproximation" the equations are correct: "The rationale approximation of the non-rationale 0.999999... is 1"
we can prove that 9.99999...... is a rational number and all other numbers in this format (repeated form) are rational numbers
ReplyDeletea = 9.999999...
10a = 99.999999...
10a - a = 90
9a = 90
a = 10
by Zafarullah khan