Originally posted by Cheshire Cat Why does this work though? Obviously they are two different numbers. And what is your abbreviation rec.?
1 - 0.9rec = 0.0000...infinite times...00001
Given the 'infinite' number of preceading zeroes the difference is zero, as we never actually get to the '1' at the end.
So as the difference is zero, the numbers are the same, regardless of how they are written.
Its just a quirk of using the term 'infinite' which is a bit of a brain twister.
Originally posted by Toe 1 - 0.9rec = 0.0000...infinite times...00001
Given the 'infinite' number of preceading zeroes the difference is zero, as we never actually get to the '1' at the end.
So as the difference is zero, the numbers are the same, regardless of how they are written.
Its just a quirk of using the term 'infinite' which is a bit of a brain twister.
What a horrible notation.
0.0000...infinite times...00001, means that there exist a position (the position of the 1) beyond infinity. Which one might that be? 🙂
Originally posted by genius http://www.redhotpawn.com/board/showthread.php?id=5557 is the other thread with this problem. (started by moi!...😏)
I checked out that thread, thanks. Unfortunately, it does not show why this works. I like the idea about the 1 at the end of the recurring zeros. Of course, if that were to work, that would mean that you could conceivably have a 9 at the end of the reccurring nines. Which means that .9 rec. is not equal to one. Even though the "proof" still works.
Originally posted by Cheshire Cat I checked out that thread, thanks. Unfortunately, it does not show why this works. I like the idea about the 1 at the end of the recurring zeros. Of course, if that were to work, that would mean that you could conceivably have a 9 at the end of the reccurring nines. Which means that .9 rec. is not equal to one. Even though the "proof" still works.
well, there isn't a one at the end of the 0's, cause there is no end of the 0's...
Look at the row of numbers 1, -1/2, 1/2, -1/3, 1/3, -1/4, ...
Now look at the sum S. The second and third term add to 0, as do the 4th and 5th, 6th and 7th, and so on. So S = 1.
Now place bracets in S; (1-1/2) + (1/2-1/3) + (1/3-1/4) +...
1/3 - 1/4 = 4-3/4*3 = 1/4*3, and in the same way 1/4 - 1/5 = 1/4*5 etc.
So S= 1/1*2 +1/2*3 + 1/3*4 + ... = 1
Now, 1/2*3 = -1/2 + 2/3 and 1/3*4 = -2/3 + 3/4 and so on. Thus S becomes S= 1/2 -1/2 + 2/3 -2/3 +3/4 - 3/4 +....=0 because all the subsequent numbers add up to 0.
With similar operation you can get S > 1 and S < 0 as well...
Originally posted by Cheshire Cat Why does this work though? Obviously they are two different numbers. And what is your abbreviation rec.?
look at the difference 1 - 0.99999999...
It's 0,00000.... wich is the same as 0. It isn't "infinite 0's and then a 1", but it's "infinite 0's". There isn't anything beyond infinity.
As for the first problem. the Square root isn't the opposite of squaring. The square root of something is always positive, while negative numbers also have squares...
Originally posted by TheMaster37 look at the difference 1 - 0.99999999...
It's 0,00000.... wich is the same as 0. It isn't "infinite 0's and then a 1", but it's "infinite 0's". There isn't anything beyond infinity.
Whether it is equal to 0, or not, does not answer the question. They are two different numbers and therefore not equal. The "proof" shows that they are equal and therefore must be wrong. The question is what is wrong with the "proof"?
As for the first problem. the square root isn't the opposite of squaring. The square root of something is always positive, while negative numbers also have squares...
This is just not true. The square root of a number always has a positite and a negative. eg. The square root of four is either 2 or -2. Likewise, the square root of -4 is the same as the square root of 4 times the square root of -1 therefore it equals either 2i or -2i.
Originally posted by Cheshire Cat Whether it is equal to 0, or not, does not answer the question. They are two different numbers and therefore not equal. The "proof" shows that they are equal and therefore must be wrong. The question is what is wrong with the "proof"?
They are not two different numbers. They are two ways of writing the same number, just as 1.0 is the same as 1.000 and 1/2 is the same as 3/6 or 0.5.
0.11rec means 0.1111... which means "not 0.1, nor 0.11, nor 0.111, but the number which you are approximating more and more closely as you increase the number of digits in the representation". There is only one number that fits that description, and that is 1/9. In the same way, 0.9rec just means 1
The definition of a recurring decimal means that any number represented by a non recurring decimal can also be written as a recurring decimal that ends in 9rec. For instance, 0.25 can also be represented as 0.2499rec
Whether it is equal to 0, or not, does not answer the question. They are two different numbers and therefore not equal. The "proof" shows that they are equal and therefore must be wrong. The question is what is wrong with the "proof"?
Every proof given in the thread "0.9 rec =1" was valid. If 1-0.9rec = 0 then 1=0.9rec. Your logic here is specious. If the proof says they are equal and is valid, then they are equal. 0.9rec = 9/9 =1.
This is just not true. The square root of a number always has a positite and a negative. eg. The square root of four is either 2 or -2. Likewise, the square root of -4 is the same as the square root of 4 times the square root of -1 therefore it equals either 2i or -2i.
The square root of a number is the principle value of the "square root" operation....the positive one, by definition. This is a convention to give even powers unique inverses.
[b]
This is just not true. The square root of a number always has a positite and a negative. eg. The square root of four is either 2 or -2. Likewise, the square root of -4 is the same as the square root of 4 times the square root of -1 therefore it equals either 2i or -2i.
The square roo of a number is the principle value of the "square root" operation....the positive one, by definition. This is a convention to give even powers unique inverses. [/b]
Originally posted by Cheshire Cat Originally posted by royalchicken [b]
[b]
This is just not true. The square root of a number always has a positite and a negative. eg. The square root of four is either 2 or -2. Likewise, the square root of -4 is the same as the square root of 4 times the square root of -1 therefore it equals either 2i or -2i.
The square roo of ...[text shortened]... is a convention to give even powers unique inverses. [/b]
A convention set in place by who?[/b]
Mathematics is just a bunch of conventions set in place by interested parties. This convention makes the concept useful and eliminates silly paradoxes.
Originally posted by royalchicken Mathematics is just a bunch of conventions set in place by interested parties. This convention makes the concept useful and eliminates silly paradoxes.
I am afraid that I do not see any "silly paradoxes". Whether people use this convention, or not, does not make the negative square root invalid. Perhaps I'm wrong....more stuff to look into. 🙄
Originally posted by Cheshire Cat I am afraid that I do not see any "silly paradoxes". Whether people use this convention, or not, does not make the negative square root invalid. Perhaps I'm wrong....more stuff to look into. 🙄
Look. (-x)^2 = x^2. But sqrt(x^2) = x. This way f(x) = sqrt(x) can preserve its status as a function. You made a silly paradox based on this in the first post, which richjohnson kindly pointed out.
Originally posted by royalchicken Look. (-x)^2 = x^2. But sqrt(x^2) = x. This way f(x) = sqrt(x) can preserve its status as a function. You made a silly paradox based on this in the first post, which richjohnson kindly pointed out.
28 Aug '03 11:21
1=2
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