The Golden Ratio
Since the days of old, artists, as well as mathematicians, have known that there is a special, aesthetically pleasing rectangle with width $1$, length $x$, and the following property:
When a square of side $1$ is removed, the remaining rectangle has the same proportions as the original rectangle. Since the new rectangle has a width $x-1$ and a length of $1$, the equivalent of the proportions means that
$$\dfrac{x-1}{1} = \dfrac{1}{x}$$
The number $x$, which is called the golden ratio, must therefore satisfy the equation
$$x^2-x-1 = 0 \cdots (1)$$
This equation can be solved by the quadratic formula $x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$
$$ \begin{align}
x &=\dfrac{-(-1) \pm \sqrt{(-1)^2-4 \times 1 \times (-1)}}{2 \times 1} \\
&= \dfrac{1 \pm \sqrt{5}}{2}
\end{align} $$
We take the positive root of this equation.
$$ x = \dfrac{1 + \sqrt{5}}{2} \approx 1.168 \cdots$$
Now, we try to find a rational number that approximates the irrational number $x$. To do this, we note that the equation $(1)$ allows us to write that
$$ \begin{align} \displaystyle
x^2-x-1 &= 0 \cdots (1)\\
x-1-\dfrac{1}{x} &= 0 &\textit{divide by }x \\
x &= 1 + \dfrac{1}{x} &\textit{substitute itself} \\
x &= 1 + \dfrac{1}{1 + \dfrac{1}{x}} \\
x &= 1 + \dfrac{1}{1 + \dfrac{1}{1 + \dfrac{1}{x}}} \\
x &= 1 + \dfrac{1}{1 + \dfrac{1}{1 + \dfrac{1}{1 + \cdots }}} \\
\end{align} $$
By carrying out further substitutions of the same kind and by omitting the last $\dfrac{1}{x}$, we arrive at the following serious numbers;
$$1, 1+\dfrac{1}{1}, 1+\dfrac{1}{1+\dfrac{1}{1}}, 1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1}}}, 1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{1}}}},\cdots$$
To present this convergence phenomenon, we can write a series of the Golden ratio.
In practice, it is tedious to continue to calculate these numbers indefinitely, but the Golden Ratio is nevertheless possible to see that they get closer and closer to the exact value of
$$ \large \dfrac{1+\sqrt{5}}{2}$$
If the Golden Ratio were rational, it would be the ratio of sides of a rectangle with integer sides. But it would also be a ratio of integer sides of the smaller rectangle obtained by deleting a square. The sequence of decreasing integer side lengths formed by deleting squares cannot be continued indefinitely because the integers have a lower bound, so the Golden Ratio cannot be rational.
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Chain Rule Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses