recurring fractions

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tiredmum

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I'm having a maths crisis. I'm converting recurring decimals into fractions and have 0.0009 where 9 is recurring. I keep following the the method in the text book and getting the answer 1/1000. but 1/1000 is 0.001. I know 0.0009 recurring is near as dammit 0.001 - but this is maths! It's meant to be precise. Can someone please explain?Where am I going wrong?
 
tiredmum said:
I'm having a maths crisis. I'm converting recurring decimals into fractions and have 0.0009 where 9 is recurring. I keep following the the method in the text book and getting the answer 1/1000. but 1/1000 is 0.001. I know 0.0009 recurring is near as dammit 0.001 - but this is maths! It's meant to be precise. Can someone please explain?Where am I going wrong?
Click to expand...
By definition:

0.999999... = 1​

When you start calculus - you will be presented with "reason". Now just "assume" it is true by definition.
 
tiredmum said:
I'm having a maths crisis. I'm converting recurring decimals into fractions and have 0.0009 where 9 is recurring. I keep following the the method in the text book and getting the answer 1/1000. but 1/1000 is 0.001. I know 0.0009 recurring is near as dammit 0.001 - but this is maths! It's meant to be precise. Can someone please explain?Where am I going wrong?
Click to expand...
I can't explain where you are going wrong because you haven't shown what you've done. Can you show your working?
 
tiredmum said:
I'm having a maths crisis. I'm converting recurring decimals into fractions and have 0.0009 where 9 is recurring. I keep following the the method in the text book and getting the answer 1/1000. but 1/1000 is 0.001. I know 0.0009 recurring is near as dammit 0.001 - but this is maths! It's meant to be precise. Can someone please explain?Where am I going wrong?
Click to expand...
So you have
x= 0.000999999....
1000x= 0.9999....
10000x= 9.99999....

Subtracting, 9000x= 9 so \(\displaystyle x= \frac{9}{9000}= \frac{1}{1000}\).

"0.009 recurring" is NOT "as near as dammit 0.001", It is exactly \(\displaystyle \frac{1}{1000}= 0.001\).

Yes, mathematics is exact- if you do it right!
 
tiredmum said:
I'm having a maths crisis. I'm converting recurring decimals into fractions and have 0.0009 where 9 is recurring. I keep following the the method in the text book and getting the answer 1/1000. but 1/1000 is 0.001. I know 0.0009 recurring is near as dammit 0.001 - but this is maths! It's meant to be precise. Can someone please explain?Where am I going wrong?
Click to expand...
If .0009999... is not exactly .001, then please give me a number in between the two numbers.
 
Jomo said:
If .0009999... is not exactly .001, then please give me a number in between the two numbers.
Click to expand...
still confused here. how can two different numbers equal each other? When I picture 0.0009 recurring it's a string of 9 floating to infinity, like a tired mountain climber trying to get to the top of the mountain and only getting a little closer each time, but never quite making it - there is always another 9....
 
HallsofIvy said:
So you have
x= 0.000999999....
1000x= 0.9999....
10000x= 9.99999....

Subtracting, 9000x= 9 so \(\displaystyle x= \frac{9}{9000}= \frac{1}{1000}\).

"0.009 recurring" is NOT "as near as dammit 0.001", It is exactly \(\displaystyle \frac{1}{1000}= 0.001\).

Yes, mathematics is exact- if you do it right!
Click to expand...
That is how I was going it, and that is the answer I got. I think I'm just having an existential crisis over the answer
 
tiredmum said:
still confused here. how can two different numbers equal each other?
Click to expand...
Two "different numbers" cannot equal each other! But there can be two (or more) ways to represent the same number.

When I picture 0.0009 recurring it's a string of 9 floating to infinity, like a tired mountain climber trying to get to the top of the mountain and only getting a little closer each time, but never quite making it - there is always another 9....
Click to expand...
Well, that's just weird!
 
tiredmum said:
still confused here. how can two different numbers equal each other? When I picture 0.0009 recurring it's a string of 9 floating to infinity, like a tired mountain climber trying to get to the top of the mountain and only getting a little closer each time, but never quite making it - there is always another 9....
Click to expand...
Of course if you insist that they are two different numbers then you will never agree that they are equal. Are you saying that since they look different then they must be different? So you do not agree that .5 = 1/2? How about 2/3=4/6?

Please divide 1 by 3. You will get that 1/3 = .33333... . I think you'll buy that one.

Now multiply both numbers by 3. You'll get 1 = .9999.... . They are equal!
 
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tiredmum said:
still confused here. how can two different numbers equal each other? When I picture 0.0009 recurring it's a string of 9 floating to infinity, like a tired mountain climber trying to get to the top of the mountain and only getting a little closer each time, but never quite making it - there is always another 9....
Click to expand...
Your mistake is saying that the climber is not getting there! That is not true!

So you claim that .999.... will get close to 1 but never make it. Wonderful! Then give me a number between .9999.... and 1.

We can play the game differently. Tell me how close to 1 .9999.... can get? Maybe you'll say that it will never get within .0000001 of 1. I can show you that you are wrong by finding a string of 9 (like .99999999) that is within .0000001 of 1 and .999........> .99999999 !!
 
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