Question

3.
Search the internet regarding "Golden Ratio". Explain its history, significance
and find its value in fraction/ decimal. Give at least five examples from naturel
surroundings/ human body/ architecture etc manifesting the concept of Golden ratio.
(06 Marks)
4.
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Answer 1

Answer:

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. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

Line segments in the golden ratio

A golden rectangle with long side a and short side b adjacent to a square with sides of length a produces a similar golden rectangle with long side a + b and short side a. This illustrates the relationship {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi }{\frac {a+b}{a}}={\frac {a}{b}}\equiv \varphi .

{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}{\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}

where the Greek letter phi ({\displaystyle \varphi }\varphi or {\displaystyle \phi }\phi ) represents the golden ratio.[1][a] It is an irrational number that is a solution to the quadratic equation {\displaystyle x^{2}-x-1=0}x^{2}-x-1=0, with a value of:

{\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .}\varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .[2][3]

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[4][5] Other names include extreme and mean ratio,[6] medial section, divine proportion (Latin: proportio divina),[7] divine section (Latin: sectio divina), golden proportion, golden cut,[8] and golden number.[9][10][11]

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[12] The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing this to be aesthetically pleasing. These often appear in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio.

Answer 2

Answer:

If the ratio of two numbers is the same as the ratio of their sum to the bigger of the two numbers, then two numbers are said to be in the golden ratio.

Step-by-step explanation:

1)   If the ratio of two numbers is the same as the ratio of their sum to the bigger of the two numbers, then two numbers are said to be in the golden ratio.

2)   The golden mean or golden section are other names for the golden ratio. Other names for this ratio include the extreme and mean ratio, medial section, divine ratio, golden ratio, and golden number.

3)   Due to the golden ratio's frequent presence in geometry and the significance of the "extreme and mean ratio" (the golden section) in the geometry of regular pentagrams and pentagons, ancient Greek mathematicians were the first to study it.

4)  The golden ratio can be seen in phyllotaxis, and psychologist Adolf Zeising claimed that this indicates that the golden ratio is a universal law. A global orthogenetic law of "striving for beauty and wholeness in the domains of both nature and art" was proposed by Zeising in 1854.

5)  Many alleged examples of the golden ratio in nature, particularly in relation to animal dimensions, have been refuted, according to some