Fibonacci Numbers

1. Sep 20, 2006

AAZZ

Legend: ~ = root sign
e.g. ~10 = Root 10
^ = power
e.g. ^2 = power of 2

The equation is:

Show that (1 + ~5/2) ^ 2 = (1 - ~5/2) +1

Last edited: Sep 20, 2006
2. Sep 20, 2006

AAZZ

Legend: ^ = power
e.g. ^2 = power of 2

I did the following:

(1 + ~5/2)^2
(1 + ~5/2) (1 + ~5/2)

1^2 + (~5)^2 + 2 ~5
___________________
2^2

1 + 5 + 2 ~5
____________
4

6 + 2 ~5
__________
4

Am I going the right way? At the end, I am still unable to prove that both are equal to each other.

3. Sep 20, 2006

BSMSMSTMSPHD

The original statement:

$$( 1 + \frac{ \sqrt{5}}{2} ) ^2 = ( 1 - \frac{ \sqrt{5}}{2} ) + 1$$

is not true. So, unfortunately, showing that it is true will be impossible.

However, if you change the + sign on the left side to a - sign, and put the 1's in the numerators, you'll have more luck.

$$( \frac{ 1 - \sqrt{5}}{2} ) ^2 = ( \frac{ 1 - \sqrt{5}}{2} ) + 1$$

Last edited: Sep 20, 2006
4. Sep 20, 2006

AAZZ

How so? What is the reason? I don't only want the answer, I want to understand the statement. Please explain if you don't mind. :)

5. Sep 20, 2006

AAZZ

But my teacher said it is. These are Fibonacci Numbers. Numbers that repeat themselves. I think i'm going nuts.

"The Fibonacci Sequence is 1,1,2,3,5,8,13,21,..... It is based on the idealized reproduction rate of little rabits. It is named after the first known to have devised it.

For centures it had recursive difinition only. The definition is:
a1=1; a2=1; an+1=an + an-1"

6. Sep 20, 2006

BSMSMSTMSPHD

I heard a teacher tell his kids the other day that 0 is not a real number since you can't divide by it, and that vairables that are represented by different letters (like x, y and z) cannot be the same numbers.

Teachers are not always right.

The number you are working with is called the golden mean. It's numerical value is:

$$\frac{1 + \sqrt{5}}{2}$$

It is one of two values (the other has a - sign) that have the following property: they satisfy the equation:

$$x^2 = x + 1$$

That is, when you square the number, you get the same thing as when you just add 1. Notice the structure of this equation compared to that of your numerical statement. It's not the same.

Last edited: Sep 20, 2006
7. Sep 20, 2006

vsage

plug in the left and right side of the equation. They equal different things. I encourage you to convert your equation to LaTeX so as to not confuse, also.

Last edited by a moderator: Sep 20, 2006
8. Sep 20, 2006

BSMSMSTMSPHD

Do you mean evaluate the left and right side of the equation in a calculator? What should he "plug in"? He sounds like a young kid, so I don't think he'll be learning LaTex overnight.