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#1 2006-07-14 12:47:38
- John E. Franklin
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- Registered: 2005-08-29
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reciprocals
Reciprocals are fun to memorize.
My favorite one is 8.1
Its reciprocal has no 8 in it, but it has everything else!!
0.123456790123456790123456790123...
The square root of 10 is a good one since we are in base-10 for some reason.
3.16 something becomes .316 something.
And of course the reciprocal of 1.618034... (golden ratio) is 0.618034..., which is nice too.
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#2 2006-07-16 09:31:12
- John E. Franklin
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Re: reciprocals
I think I saw earlier today that 2.4142136... reciprocated is 0.4142136..., which is 1 + square root of 2.
If you ignore the decimal and leading zeros, then you can, if you feel like it,
Create a jumping-pair between 10 and 100 that always straddles 31.622777 or something.
Anything between 50 and 100 jumps to between 20 and 10.
Anything between 20 and 25 jumps to between 50 and 40.
Anything between 25 and 30 jumps to between 40 and 33.33333333
Anything between 30 and 31.622777 jumps to between 31.622777 and 33.33333333
It makes a picture that looks like a rainbow!
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#3 2008-05-03 05:57:38
- John E. Franklin
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Re: reciprocals
I was teaching my
Dad about reciprocals
today with my
fingers in the air and
infinity way up in the sky.
Zero and One were
always fingers, and the
reciprocals were other
fingers.
Here's what I noticed:
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#4 2008-05-03 10:05:20
- JaneFairfax
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- Registered: 2007-02-23
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Re: reciprocals
Here is what Ive noticed.
And the golden ratio φ satisfies the quadratic equation x[sup]2[/sup] − x − 1 = 0.
∴
And last, but not least,
Last edited by JaneFairfax (2008-05-04 02:11:54)
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#5 2008-05-03 10:35:50
- JaneFairfax
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Re: reciprocals
I think I saw earlier today that 2.4142136... reciprocated is 0.4142136...
Have you noticed that
?That is,
?Also
()?And
()?Notice a pattern yet?
()? ()?Offline
#6 2008-05-03 11:01:14
- John E. Franklin
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Re: reciprocals
in post#4, what is the technique used
to deal with repetitive things like .001001001001...
What is the technique applied next?
I think I saw this in college once.
Last edited by John E. Franklin (2008-05-03 11:06:03)
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#7 2008-05-03 11:14:54
- John E. Franklin
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Re: reciprocals
In post#5, I see now that this is true because
Reminds me of x^2-y^2 = (x+y)(x-y), where
x^2 = (n + 1)
and
y^2 = (n)
Last edited by John E. Franklin (2008-05-03 11:15:54)
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#8 2008-05-03 16:20:24
- Identity
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- Registered: 2007-04-18
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Re: reciprocals
Yes, rationalizing the denominator (multiplying the top and bottom by the conjugate of the denominator) also works.
And the method Jane was using to find the fraction is the sum of infinite geometric series:
For first term a and common ratio r (r < 1 or the series will not converge)
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#9 2008-05-04 02:11:33
- JaneFairfax
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- Registered: 2007-02-23
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Re: reciprocals
In post#5, I see now that this is true because
Reminds me of x^2-y^2 = (x+y)(x-y), where
x^2 = (n + 1)
and
y^2 = (n)
Thats right. 
In fact, n does not have to be an integer; it can be any non-negative real number.
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