Mammoth Memory

Irrational Number

Irrational numbers when written as decimals, do not terminate (nor do they repeat). The number goes on, indefinitely.

An irrational number is a number that never terminates such as a decimal number

I have got a rash on (irrational) my body. I’ve even got ear rash (irrational) and the spots go on and on (the number goes on and on).


Example 1

1.41421356………. Is an irrational number, and is also known as the surd `sqrt2`

2.7370508075…… Is an irrational number, and is also known as the surd `sqrt3`

3.333333333…….. Is rational as it can be turned into a fraction `1/3` and is not irrational because the same number repeats, i.e. 3.


Example 2

The most famous irrational number is  `pi = 3.142`… on and on and on.


Example 3

Another irrational number is the golden ratio `phi=(1+sqrt5)/2 =1.6180`... 

The golden ratio can be linked to the Fibonacci spiral, which is used by artists, and is often found in nature and physics. The spiral patterns seen on shells, or the pattern produced by water as you flick your hair are both examples of the Fibonacci spiral.

The golden ratio is an irrational number and is linked to the fibonacci spiral

The golden ratio of 1:1.618 is used to create rectangles which divide the composition of an image into ever decreasing/increasing rectangles of these proportions. A spiral can then be drawn from one corner of each rectangle to the opposite corner, (known as the golden spiral). The line this spiral forms is used to draw the viewer’s eye to elements the artist wants to emphasise.

A wave is calculated by the golden ratio and Fibonacci spiral

The golden ratio and Fibonacci spiral are used to create the ideal proportions, and to position pieces within a composition to give the most pleasing image. The Great Wave of Kanagawa by Katsushika Hokusai fits the spiral created by the golden ratio, giving the wave a natural shape. It is also said to produce aesthetically pleasing proportions, which may explain why many artists have works that conform to the golden ratio.

The human body is calculated by the golden ratio and fibonacci spiral

The golden ratio is found between many human body parts and is used by plastic surgeons and artists to ensure things are proportioned correctly. The most aesthetically pleasing ratio of total height to naval height is 1.618:1 as is the ratio of the wrist to tip of your fingers to the forearm length.

The Mona Lisa was painted with the golden ratio 

The proportions of the Mona Lisa’s head to body conforms to the golden ratio of 1:1.618.

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