The actual value of $\phi$

Question

I was reading into the Golden Ratio, and saw that Wikipedia, along with many other more reliable sites, said that the Golden Ratio was equal to $\frac {\sqrt{5}+1}{2}=1.618033\ldots$.

But when I asked my friends that, they said the Golden Ratio was equal to $0.618033\ldots$ or $\phi-1$.

I'm confused at the moment, because the Internet says $\phi=1.618\ldots$ but my friends said $0.618\ldots$. So which value is the correct value?

Note: My friends also claimed that $\frac {1-0.618\ldots}{0.618\ldots}=0.618\ldots$. This seems very interesting.

Doing some basic math, I find that $0.618\ldots=\frac {2}{\sqrt{5}+1}$. While $\phi=\frac {\sqrt{5}+1}{2}$, which is the reciprocal of the other value.

Answer 1

According to Wikipedia:

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

Hence we need:

$$\frac{a+b}{a}=\frac{a}{b}=\phi>1$$

$$1+\frac{b}{a}=\frac{a}{b}=\phi>1$$

Notice

$$\frac{a}{b}=\phi$$

Substitute this in and solve to get:

$$\phi=\frac{1 \pm \sqrt{5}}{2}$$

But one of this is negative and as a result of the definition of the golden ration can't be the actual golden ratio.

Thus if we take this definition you are correct.

Furthermore the golden ratio is one solution to:

$$x^2-x-1=0$$

Dividing both sides by $x$ we have:

$$x-1-\frac{1}{x}=0$$

So it shouldn't really be that much of surprise that:

$$\phi-1=\frac{1}{\phi}$$

From where:

$$\frac{1-(\phi-1)}{(\phi-1)}=\phi-1$$

Follows.

Answer 2

Both are right depending on how the source defines it.   I usually encounter it as the larger number though.

And yes, $(1.618033\ldots)^{-1} = (0.618033\ldots)$ is the interesting property of the golden ration.

$$\begin{align} \phi =&~ \dfrac{\surd 5~+1}{2} \\[2ex] \phi^{-1} =&~\dfrac{2}{(\surd 5+1)}\cdot\dfrac{(\surd 5 -1)}{(\surd 5-1)} \\[1ex] =&~ \dfrac{\surd5~-1}{2} \\[1ex] =&~ \phi-1 \end{align}$$

Answer 3

the formal definition of the golden ratio is :

the golden ration $\varphi$ is the positive solution of $$ x^2=x+1$$ and is $$ \varphi=\frac{1+\sqrt5}{2}$$

and so the other solution (the negative one) is : $$ -\varphi^{-1}=\frac{-1+\sqrt5}{2}=1-\varphi $$ which explain the confusion

Answer 4

The golden ratio is any two nonzero numbers $x$ and $y$ which satisfy the equation $\frac xy=\frac {x+y}{x}$ Defining the ratio $\phi=\frac xy>0$ the equation gives: $\phi=1+\frac 1\phi$. You can then multiply by $\phi$ to obtain $\phi^2=\phi +1$ or $\phi^2-\phi-1=0$. Using the quadratic formula gives a positive solution $\frac {1+\sqrt5}{2}\approx1.618$

It is however a ratio so inverting it ($\frac 1\phi=\frac yx\approx 0.618$) just depends on which two quantities you're comparing. In other words, its a ratio of two quantities so they're both correct. It just depends on the order taken. Think of me asking you to give me the ratio of boys to girls at a school. You could give me $\frac{\text{num.boys}}{\text{num.girls}}$ or $\frac{\text{num.girls}}{\text{num.boys}}$