Golden ratio

Definition

The golden ratio, often denoted by $\phi$ (phi), is an irrational number approximately equal to 1.618033988749895. It is defined algebraically as $\phi = \frac{1 + \sqrt{5}}{2}$.

5 Must Know Facts For Your Next Test

1. 1. The golden ratio can be derived from the equation $x^2 - x - 1 = 0$, which has solutions $x = \frac{1 + \sqrt{5}}{2}$ and $x = \frac{1 - \sqrt{5}}{2}$. Only the positive solution is considered the golden ratio.
2. 2. In sequences, particularly in the Fibonacci sequence, the ratio of consecutive terms approaches the golden ratio as the terms increase.
3. 3. The continued fraction representation of the golden ratio is unique: $\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}$.
4. 4. The decimal expansion of the golden ratio is non-repeating and non-terminating, making it an irrational number.
5. 5. In geometry, a rectangle with side lengths in the proportion of $\phi$ is called a golden rectangle; dividing a square from it leaves another smaller golden rectangle.

Review Questions

• What is the algebraic definition of the golden ratio?
• How does the Fibonacci sequence relate to the golden ratio?
• What kind of number (rational or irrational) is represented by the golden ratio?

Related terms

Irrational Number: A number that cannot be expressed as a fraction of two integers and has a non-repeating, non-terminating decimal expansion.

Fibonacci Sequence: A sequence where each term is the sum of the two preceding ones, typically starting with 0 and 1.

Continued Fraction: An expression obtained through an iterative process of representing numbers as sums of their integer parts and reciprocals.