Hello
I'm sorry if thos has been covered before but I just thought you might be interested in it.
Now, I have been interested in maths, long before I had any real magic experience. And this problem always puzzled me. I have tried to explain it using maths and physics, but I would like to hear the opinions of some magicians. Here goes.

Let X=0.99999...

Therefore 10X=9.99999...

Thus 10X-X= 9.99999-0.99999= 9

Meaning 9X= 9

Therefore X=1
Rather than 0.99999...

My explanation is simply that the concept of a recurring decimal of 9 is idiocy, unless expressed graphically as in an asymptote line.

I appologise that this is not a trick but it left the same dumbfounded look on my math teachers face.

Yes you've just nicely proved that 1=.999999...

Nothing wrong with that.

An infinite series can converge to a finite limit.

Are you familiar with Zeno ?

Jack

1 = 9/9 = 9*(1/9) = 9*0,1111111... = 0,9999999.....

Thanks for the link!

I haven't, but after that article, I should be looking further into it. Would these paradoxes have much application in magic?

Yep!

"Before I can stick the sword through your neck, I must stick it halfway through your neck. Before I can stick the sword halfway through your neck, I must stick it a quarter of the way through your neck. Before I can stick the sword a quarter of the way through your neck, I must stick it an eight of the way through your neck . . ."


(Of course, that's more like Xena's paradox.)

The way I learned it, .999999(repeating) is equal to 1 because there is no number between itself and 1. Therefore they are equal.

Irrational numbers have unique decimal representations.

But many rational numbers have more than one decimal representation.
1 = 1.0 = 1.00 = 1.000 = ... = 1.0000... = .99999999999...
1/2 = .5 = .50 = .500 = ... = .5000000... = .4999999999999...
1/4 = .25 = ... = .2499999999....
1/5 = .2 = ... = .199999999...
1/8 = .125 = ... = .124999999999....
1/10 = .1 = ... = .09999999....

Write a rational number in fraction form, an integer over an integer. Reduce the fraction to lowest form. If the reduced fraction's denominator is divisible by any prime other than 2 or 5, then that rational number has a unique decimal representation. (1/3 only has one decimal representation: .333333...) Otherwise, it has multiple decimal representations.

To explain it generally (in bad english, please excuse):

If you want to turn a decimal periodic number in a fractal number: For every point of the period write a '9'.

Examples:
0,8 8888 ... = 8/9

0,34 3434 ... = 34/99

0,018 018018 ... = 018/999 = 18/999


But what to do if the periodic series starts later?
Then you must subtract the front numbers from the whole number and for every digit from the front number you write a '0'.

Examples:


0,3454545... = (345-3)/990 = 342/990

0,45969696... = (4596 - 45)/9900 = 4551/9900

0,1234567895678956789... = (123456789-1234)/999990000


The other way round:

How can you see, if a fractal turns into a decimal number which ends or in a periodic decimal number or in a periodic decimal number with front digits?

If ouy lokk for the prime-numbers of the lower number of the fractal you have three possibilities:

#) there are only '2's and '5' s --> the decimal number will end

Example 1/50 50 = 2x5x5 1/50 = 0,02 it ends

or 1/8 8 = 2x2x2 1/8 = 0,125 it ends

#) there are no '2' s and no '5's --> it will turn into a peridoic number

Example 1/21 21 = 3x7 1/21 = 0,04761907619 047619....

#) there are both '2's and/or '5's AND other prime numbers --> periodic with front numbers

Example 1/45 45 =5x3x3 1/45 = 0,0 22222222...
1/330 330 = 2x2x5x17 1/340 = 0,00 2941176470588235 29...

Angelo

p.s. Perhaps someone will translate it from my english to real english!