"Simple" math question: How can (x+1)=x^2?

Question

This is true when dealing with the golden section (1.618 approx.). This constant (phi) is arrived at through the fibbonacci sequence, i.e 0,1,1,2,3,5,8,13,21,34,55....
Then you make fractions: 1+1/2+2/3+3/5+5/8+8/13+....
and you get 1.618 (I think I did this right.)

so how can this infinite sequence^2 (1.618)(1.618)=2.618
and 1.618+1=2.618.

Also, how can (Phi^2 * 6)/5=pi? 2.618*6/5=3.1416?

weird, because there is the other equation e^(pi*i)=1?

Can you substitiute e^([Phi^2*6/5]*i)=1? i=sq rt of (-1).

Answer 1

Not quite right. You want to take the limit of succeeding terms, not the sum:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ...
Notice the terms alternate too high and too low around the actual value of phi:
1, 2, 1.5, 1.666, 1.6, 1.625, ...

You can see this in the following way: Call the limit of the ratio described above R, and let f(n) be the nth fibonocci number. For any n, f(n - 1) + f(n) = f(n + 1). Divide by f(n): f(n - 1)/f(n) + 1 = f(n + 1)/f(n). Now let n go to infinity. In this case, f(n + 1)/f(n) goes to its limit, R, and f(n - 1)/f(n) goes to its limit, 1/R (you see why this is 1/R, right?) So the equation becomes 1/R + 1 = R. Since R != 0 multiply through by R to get 1 + R = R^2.

Also, I don't think I believe phi^2*6/5 = pi. I have never seen this, and since phi is an algebraic number but pi is trancendental, I think this is just a good approximation, not an identity.

(edit) There is a great fibonocci page at http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html

Answer 2

Yes, phi is a fascinating number. There is a lot of crap that is exagerated about it (e.g., Dan Brown et al) but even so there is still plenty of amazing things left over about it.

Mario Livio wrote a great book, The Golden Ratio. Check it out.

Answer 3

x + 1 = x^2
0 = x^2 - x - 1
x = [1 +- sqrt(5)]/2

[1 + sqrt(5)]/2 seems to be the answer you are looking for, your phi above. The other answer is negative.

Answer 4

I don't get the question.. You asked how (x+1) can = x^2 yet you showed an example explaining it.. eh I'm confused


YORDANKA- no not really...