Golden ratio

''This article uses &#966; (U+03C6) for the golden ratio and its capital form (U+03A6) for $1⁄&#966;$. &#x03D5; is another form for the golden ratio.''

The golden ratio is an algebraic irrational number equal to $${1+\sqrt{5}\over2}={1\over2}+\sqrt{1{1\over4}}$$.

The golden ratio is known for use in nature, and the ratio of two consecutive numbers in the Fibonacci sequence is the approximation of &#966;.

Mathematical properties
&#966; is a constructible number and is used in pentagons.

Due to its continued fraction expansion, the golden ratio is equal to √1 + &#966;. Therefore, the golden ratio is also equal to $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+...}}}}}}$$.

'Golden ratio is the only number to be the square root of itself added to 1' Proof
There is a proof that there are only two solutions for this: &#966; and &#966; - 2.

Assume the golden ratio is unknown.

Then, $$\varphi = \sqrt{\varphi + 1}$$.

Due to this, we can square both sides to get $$\varphi^2 = \varphi + 1$$, therefore rearranging it to the equation where the quadratic formula can be used:

$$\varphi^2 - \varphi - 1 = 0$$

Using the quadratic formula, which is $$\varphi = {{-b \pm \sqrt{b^2 - 4ac}}\over{2a}}$$ for $$a\varphi^2 + b\varphi + c = 0$$. a is equal to 1, b = -1 and c = -1 in this equation.

From here, we get two solutions:

$$\varphi = {{-(-1) + \sqrt{(-1)^2 - 4(1)(-1)}}\over{2}}= {1+\sqrt{1 - (-4)}\over2} = {1+\sqrt{5}\over2} = 1.61803398874...$$ and $$\varphi = {{-(-1) - \sqrt{(-1)^2 - 4(1)(-1)}}\over{2}}= {1-\sqrt{1 - (-4)}\over2} = {1-\sqrt{5}\over2} = -0.61803398874...$$

However, we generally use the positive solution for measurement purposes.