E-StoreMATHS AGAIN
14 Aug 2008 21:22:24 IST
MATHS AGAIN
This is a fascinating number, studied by the ancients and still a wonder today. It can be found in many sources in nature, most notably in naturally occurring spirals. Mathematically, the number can be found in the ratio between certain segments in a regular pentagon, in ratios between Fibonacci numbers, and can also be generated by some of the simplest sequences in math.
For example, consider the following sequence:
1 + 1 / [1+1 / [1+...] ]
To find the solution of this sequence, we can set it equal to an arbitrary 'L'
L = 1 + 1 / [1+1 / [1+...] ]
Upon looking at this sequence, a pattern stands out
L = 1 + 1 / [L]
What we intend to find is L, so we can form a quadratic equation to solve the sequence
L^2 - L - 1 = 0
Using the quadratic formula, we find that L = (1 + sqrt(5)) / 2
What does this equal? It is phi, the Golden Ratio
Also consider this sequence:
L = sqrt(1 + sqrt(1+ sqrt(1 + ... )))
In simplifying the sequence,
L = sqrt(1 + L)
And we arrive at the same quadratic equation, and the same value for L: phi
I thought it was cool, if you have any more of these sequences please post them!
It's a good question, and I've seen people bang their heads on desks over it. Try to think of it from a purely mathematical standpoint, and not a philosophical one...
Try to solve this before you read the solution!
0.990=1
There are actually many ways to prove that the equality IS true, of course.
1. 9*10-1+9*10-2+...+9*10-inf approaches 1.
2. The density of real numbers: there is no number between .999... and 1, so they
must be the same.
3. This is the most intuitive proof:
Let k = .999...
Then 10k = 9.999...
10k-k = 9.999... - 0.999...
9k=9
k = 1 and k = .999...
So 0.999... = 1
Post any more interesting solutions you know!
For example, consider the following sequence:
1 + 1 / [1+1 / [1+...] ]
To find the solution of this sequence, we can set it equal to an arbitrary 'L'
L = 1 + 1 / [1+1 / [1+...] ]
Upon looking at this sequence, a pattern stands out
L = 1 + 1 / [L]
What we intend to find is L, so we can form a quadratic equation to solve the sequence
L^2 - L - 1 = 0
Using the quadratic formula, we find that L = (1 + sqrt(5)) / 2
What does this equal? It is phi, the Golden Ratio
Also consider this sequence:
L = sqrt(1 + sqrt(1+ sqrt(1 + ... )))
In simplifying the sequence,
L = sqrt(1 + L)
And we arrive at the same quadratic equation, and the same value for L: phi
I thought it was cool, if you have any more of these sequences please post them!
It's a good question, and I've seen people bang their heads on desks over it. Try to think of it from a purely mathematical standpoint, and not a philosophical one...
Try to solve this before you read the solution!
0.990=1
There are actually many ways to prove that the equality IS true, of course.
1. 9*10-1+9*10-2+...+9*10-inf approaches 1.
2. The density of real numbers: there is no number between .999... and 1, so they
must be the same.
3. This is the most intuitive proof:
Let k = .999...
Then 10k = 9.999...
10k-k = 9.999... - 0.999...
9k=9
k = 1 and k = .999...
So 0.999... = 1
Post any more interesting solutions you know!
Comments (1)
VARUN RAJ
Blazing goIITian
Joined: 16 Mar 2008 12:29:41 IST
Posts: 1825
15 Aug 2008 08:36:00 IST
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