# arrow_back The golden ration known as the most irrational number. Is there proof?

31 views
I do work on continuous fractures, I came across that the golden ratio is the most irrational, is there any proof of that?

We can prove the irrationality of the golden ratio by contradiction.

To do this, let the golden ratio, $\phi$, be rational.

We know $\phi>1$ so if it is rational, we could write
$\phi=\frac{a}{b}$
where $a>b>0$ are integers and $\operatorname{gcd}(a, b)=1$. Then using the relation $\frac{1}{\phi}=\phi-1$ gives
$\frac{b}{a}=\frac{a-b}{b}$
which is a contradiction since $\operatorname{gcd}(a, b)=1$ by construction and $a>b$
by Gold Status
(30,445 points)

## Related questions

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Why is the golden ration known as the most irrational number?
Why is the golden ration known as the most irrational number?Why is the golden ration known as the most irrational number? ...
close
Classify each number as being a natural number $(\mathrm{N})$, whole number $(\mathrm{W})$, integer $(\mathrm{I})$, rational number $(\mathrm{Q})$, and/or irrational number $\left(Q^{\prime}\right)$.Classify each number as being a natural number $(\mathrm{N})$, whole number $(\mathrm{W})$, integer $(\mathrm{I})$, rational number $(\mathrm{Q})$, an ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
If $\frac{1}{3}$ can be written as $0,\dot{3}$ or as $0,\bar{3}$ then is it irrational?
If $\frac{1}{3}$ can be written as $0,\dot{3}$ or as $0,\bar{3}$ then is it irrational?If &nbsp;$\frac{1}{3}$ can be written as &nbsp;$0,\dot{3}$ or as $0,\bar{3}$ then is it irrational? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove that the following number is irrational $(2+\sqrt{5})$
Prove that the following number is irrational $(2+\sqrt{5})$Prove that the following number is irrational $(2+\sqrt{5})$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Prove that the following number is irrational $(2-3 \sqrt{5})$
Prove that the following number is irrational $(2-3 \sqrt{5})$Prove that the following number is irrational $(2-3 \sqrt{5})$ ...
close
Let $p$ be the statement " $x$ is an irrational number", $q$ be the statement "y is a transcendental number", and $r$ be the statement ' $x$ is a rational number iff $y$ is a transcendental number".Let $p$ be the statement &quot; $x$ is an irrational number&quot;, $q$ be the statement &quot;y is a transcendental number&quot;, and $r$ be the state ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What is the definition of an irrational number?
What is the definition of an irrational number?What is the definition of an irrational number? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Irrational numbers ($\mathbb{Q'}$) are numbers that cannot be written as...
Irrational numbers ($\mathbb{Q'}$) are numbers that cannot be written as...Irrational numbers ($\mathbb{Q'}$) are numbers that cannot be written as... ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What are the most commonly used number systems?
What are the most commonly used number systems?What are the most commonly used number systems? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Determine whether the simplified expression is rational or irrational: $\sqrt{-16+4(5)+5}$.
Determine whether the simplified expression is rational or irrational: $\sqrt{-16+4(5)+5}$.Determine whether the simplified expression is rational or irrational: $\sqrt{-16+4(5)+5}$. ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Which ONE of the following numbers is irrational?
Which ONE of the following numbers is irrational? Which ONE of the following numbers is irrational? ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational?
For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational? For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational? $1$ $-10$ &nbsp; &nbsp; $\sqrt{2}$ &nbsp; &nbsp; ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Is $\frac{2}{11}$ irrational?
Is $\frac{2}{11}$ irrational?Is $\frac{2}{11}$ irrational? ...
For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational?
For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational? &nbsp;For which of the following values of $a$ is $\frac{a}{14}$ rational or irrational? &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; $1$ &nbsp; &nb ...