interesting fractions

Cool goIITian

Posted on
5 Jul 2007 18:45:24 IST

Posts: 30

5 Jul 2007 18:45:24 IST

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interesting fractions

Mathematics , Algebra , academic

Cool Fraction Problem
A friend of mine sent this to me today, see what you think of it:

The decimal representation of 10/81 is cool:

0.123456790 123456790 123456790 123456790 and so on forever.

But the missing 8 is a bit of a flaw. The easiest way to remedy this is to subtract
1/10^9,

(10/81) - 1/1000000000 = 9999999919/ 81000000000 =

0.123456789 123456790 123456790 123456790 and so on forever.

But this too is less than perfectly cool since the first 9 is not followed by a 0
or a 10. Show that one can turn 10/81 into a very cool number by subtracting
(m/n)/10^9 from it so that the result looks like

0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . . .

for a LONG way. The idea is to continue the integer sequence for as long
as you can using no more than 4-digit numbers for m and n.

Source: Cool Irrational Numbers and Their Rather Cool Rational
Approximations, by Francesco Calogero, The Mathematical Intelligencer,
Fall 2003, p. 72-76.

=======================================================

One of my favorite recent realizations is that any number divided by 9 (repeated for the length of that number) is that number as a repeating decimal. For example, 1/9 = .111... 12/99 = .12121212... 1234567890/9999999999 = .12345678901234567890...

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soham chokshi

Cool goIITian

Joined: 2 Jul 2007 16:18:39 IST

Posts: 30

5 Jul 2007 19:02:58 IST

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comment guys and girls !!!

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