The golden ratio is an irrational number. Below are two short proofs of irrationality:
This is a proof by infinite descent. Recall that:

If we call the whole ⁠⁠ and the longer part ⁠⁠, then the second statement above becomes

To say that the golden ratio ⁠⁠ is rational means that ⁠⁠ is a fraction ⁠⁠ where ⁠⁠ and ⁠⁠ are integers. We may take ⁠⁠ to be in lowest terms and ⁠⁠ and ⁠⁠ to be positive. But if ⁠⁠ is in lowest terms, then the equally valued ⁠⁠ is in still lower terms. That is a contradiction that follows from the assumption that ⁠ ⁠ is rational.
Another short proof – perhaps more commonly known – of the irrationality of the golden ratio makes use of the closure of rational numbers under addition and multiplication. If ⁠⁠ is assumed to be rational, then ⁠⁠, the square root of ⁠⁠, must also be rational. This is a contradiction as the square roots of all non-square natural numbers are irrational.
The golden ratio is also an algebraic number and even an algebraic integer. It has minimal polynomial

This quadratic polynomial has two roots, ⁠⁠ and ⁠⁠.

The golden ratio is also closely related to the polynomial ⁠⁠, which has roots ⁠⁠ and ⁠⁠. As the root of a quadratic polynomial, the golden ratio is a constructible number.
The conjugate root to the minimal polynomial ⁠ ⁠ is

The absolute value of this quantity (⁠⁠) corresponds to the length ratio taken in reverse order (shorter segment length over longer segment length, ⁠⁠).

This illustrates the unique property of the golden ratio among positive numbers, that

or its inverse,

The conjugate and the defining quadratic polynomial relationship lead to decimal values that have their fractional part in common with ⁠⁠:

The sequence of powers of ⁠⁠ contains these values ⁠⁠, ⁠⁠, ⁠⁠, ⁠⁠; more generally, any power of ⁠⁠ is equal to the sum of the two immediately preceding powers:

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

If ⁠⁠, …

Read more on Wikipedia

The Swiss architect Le Corbusier, famous for his contributions to the modern international style, centered his design philosophy on systems of harmony and proportion. Le Corbusier's faith in the mathematical order of the universe was closely bound to the golden ratio and the Fibonacci series, which he described as "rhythms apparent to the eye and clear in their relations with one another. And these rhythms are at the very root of human activities. They resound in man by an organic inevitability, the same fine inevitability which causes the tracing out of the Golden Section by children, old men, savages and the learned."

Le Corbusier explicitly used the golden ratio in his Modulor system for the scale of architectural proportion. He saw this system as a continuation of the long tradition of Vitruvius, Leonardo da Vinci's "Vitruvian Man", the work of Leon Battista Alberti, and others who used the proportions of the human body to improve the appearance and function of architecture.

In addition to the golden ratio, Le Corbusier based the system on human measurements, Fibonacci numbers, and the double unit. He took suggestion of the golden ratio in human proportions to an extreme: he sectioned his model human body's height at the navel with the two sections in golden ratio, then subdivided those sections in golden ratio at the knees and throat; he used these golden ratio proportions in the Modulor system. Le Corbusier's 1927 Villa Stein in Garches exemplified the Modulor system's application. The villa's rectangular ground plan, elevation, and inner structure closely approximate golden rectangles.

Another Swiss architect, Mario Botta, bases many of his designs on geometric figures. Several private houses he designed in Switzerland are composed of squares and circles, cubes and cylinders. In a house he designed in Origlio, the golden ratio is the proportion between the central section and the side sections of the house.
Leonardo da Vinci's illustrations of polyhedra in Pacioli's Divina proportione have led some to speculate that he incorporated the golden ratio in his paintings. But the suggestion that his Mona Lisa, for example, employs golden ratio proportions, is not supported by Leonardo's own writings. Similarly, although Leonardo's Vitruvian Man is often shown in connection with the golden ratio, the proportions of the figure do not actually match it, and the text only mentions whole number ratios.

Salvador Dalí, influenced by the works of Matila Ghyka, explicitly used the golden ratio in his masterpiece, The Sacrament of the Last Supper. The dimensions of the canvas are a golden rectangle. A huge dodecahedron, in perspective so that edges appear in golden ratio to one another, is suspe…

Read more on Wikipedia

Examples of disputed observations of the golden ratio include the following:
• Specific proportions in the bodies of vertebrates (including humans) are often claimed to be in the golden ratio; for example the ratio of successive phalangeal and metacarpal bones (finger bones) has been said to approximate the golden ratio. There is a large variation in the real measures of these elements in specific individuals, however, and the proportion in question is often significantly different from the golden ratio.
• The shells of mollusks such as the nautilus are often claimed to be in the golden ratio. The growth of nautilus shells follows a logarithmic spiral, and it is sometimes erroneously claimed that any logarithmic spiral is related to the golden ratio, or sometimes claimed that each new chamber is golden-proportioned relative to the previous one. However, measurements of nautilus shells do not support this claim.
• Historian John Man states that both the pages and text area of the Gutenberg Bible were "based on the golden section shape". However, according to his own measurements, the ratio of height to width of the pages is ⁠⁠.
• Studies by psychologists, starting with Gustav Fechner c. 1876, have been devised to test the idea that the golden ratio plays a role in human perception of beauty. While Fechner found a preference for rectangle ratios centered on the golden ratio, later attempts to carefully test such a hypothesis have been, at best, inconclusive.
• In investing, some practitioners of technical analysis use the golden ratio to indicate support of a price level, or resistance to price increases, of a stock or commodity; after significant price changes up or down, new support and resistance levels are supposedly found at or near prices related to the starting price via the golden ratio. The use of the golden ratio in investing is also related to more complicated patterns described by Fibonacci numbers (e.g. Elliott wave principle and Fibonacci retracement). However, other market analysts have published analyses suggesting that these percentages and patterns are not supported by the data.
The Great Pyramid of Giza (also known as the Pyramid of Cheops or Khufu) has been analyzed by pyramidologists as having a doubled Kepler triangle as its cross-section. If this theory were true, the golden ratio would describe the ratio of distances from the midpoint of one of the sides of the pyramid to its apex, and from the same midpoint to the center of the pyramid's base. However, imprecision in measurement caused in part by the removal of the outer surface of the pyramid makes it impossible to distinguish this theory from other numerical theories of the proportions of the pyramid, based on pi or …

Read more on Wikipedia