Editing Golden ratio (section)

===Golden ratio conjugate<!--'Golden ratio conjugate' redirects here--> and powers===
The [[Conjugate (square roots)|conjugate root]] to the minimal polynomial {{tmath|\textstyle x^2-x-1}} is

<math display=block>-\frac{1}{\varphi}=1-\varphi = \frac{1 - \sqrt5}{2} = -0.618033\dots.</math>

The absolute value of this quantity ({{tmath|0.618\ldots}}) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, {{tmath|b/a}}).

This illustrates the unique property of the golden ratio among positive numbers, that
<math display=block>\frac1\varphi = \varphi - 1,</math>

or its inverse,
<math display=block>\frac1{1/\varphi} = \frac1\varphi + 1.</math>

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with {{tmath|\varphi}}:

<math display=block>\begin{align}
\varphi^2 &= \varphi + 1 = 2.618033\dots, \\[5mu]
\frac1\varphi &= \varphi - 1 = 0.618033\dots.
\end{align}</math>

The sequence of powers of {{tmath|\varphi}} contains these values {{tmath|0.618033\ldots}}, {{tmath|1.0}}, {{tmath|1.618033\ldots}}, {{tmath|2.618033\ldots}}; more generally,
any power of {{tmath|\varphi}} is equal to the sum of the two immediately preceding powers:
<math display=block>
\varphi^n = \varphi^{n-1} + \varphi^{n-2}
= \varphi \cdot \operatorname{F}_n + \operatorname{F}_{n-1}.
</math>

As a result, one can easily decompose any power of {{tmath|\varphi}} into a multiple of {{tmath|\varphi}} and a constant. The multiple and the constant are always adjacent Fibonacci numbers. This leads to another property of the positive powers of {{tmath|\varphi}}:

If {{tmath|1= \bigl\lfloor \tfrac12n - 1 \bigr\rfloor = m}}, then:
<math display=block>\begin{align}
\varphi^n &= \varphi^{n-1} + \varphi^{n-3} + \cdots + \varphi^{n-1-2m} + \varphi^{n-2-2m} \\[5mu]
\varphi^n - \varphi^{n-1} &= \varphi^{n-2}.
\end{align}</math>