Amazing Number 108

## Some Interesting Observations

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Introduction

The number 108 is considered auspicious in Indian culture and has its significance in many rituals followed in India.
Vedic sages of later era devised a Vyjanti mala (rosary) with 108 beads/stones to keep count of mantras (hymns) in their daily ritual of meditation.

The sacred rosaries have 108 beads/Stones. Mantras like Mahamritunyaj mantra and Gayatri mantra are chanted 108 times. Most popular of all mala i.e. Rudraksha has 108 sacred beads.

The number 108 is considered holy and auspicious in Hinduism. Not only Hinduism, number 108 has its profound significance in Sikhism, Buddhism, and Jainism. In the Jain religion, 108 are the combined virtues of five categories of holy ones, including 12, 8, 36, 25, and 27 virtues respectively. The Sikh tradition has a mala of 108 knots tied in a string of wool, rather than beads.

The Chinese Buddhists and Taoists use a 108 bead mala, which is called su-chu, and has three dividing beads, so the mala is divided into three parts of 36 each. Chinese astrology says that there are 108 sacred stars. Some Buddhists carve 108 small Buddhas on a walnut for good luck. Some ring a bell 108 times to celebrate a new year.

What is so interesting about 108

Let us see how fascinating the number 108 from mathematical point of view is.

• 108 is the product of the first three numbers raised to themselves:

108 = 11 . 22 . 33

So it is the hyperfactorial of 3 since it is of the form 11 . 22 . 33.

• 108 is the smallest number whose divisors contain every digit at least once.

The divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.

• 108 is the largest known number n such that the decimal expansion of 2^n does not contain the digit 9 i.e.

2108 = 324518553658426726783156020576256

It can be noted that 2108 contain all digit from 0 to 8 but not 9 while each number 2n for n = 109, 110, 111 etc contain it.

• 108 is the smallest positive integer which can be written as a sum of a cubic number and a perfect square in two distinct ways i.e.

108 = 23 + 102 = 33 + 92

• 108 is sandwiched between 107 and 109 which are twin primes.

The concatenation of 108 with 107 and 109 is prime, i.e., 108107 and 108109 are primes.

• 1082 = 11664, where 1, 16 and 64 are squares..

1082 = 11664,   and    1 + 16 + 64 = 92,

1082 = 11664,   and    11 + 6 + 64 = 92,

1082 = 11664,   and    11 + 66 + 4 = 92,

1082 = 11664, 1292 = 16641 and 2042 = 41616. It can be noted that all these three squares contain same digits.

• 108 is the smallest number which consists of digits which are all distinct cube digits.

The cube digits are 0, 1 and 8.

• The number of palindromic numbers less than 1000 is 108.
• 108 is the smallest number which can be partitioned into six distinct primes such that the sum of any five is prime:

108 = 5 + 7 + 11 + 19 + 29 + 37.

It can be noted that sum of any five out of 5, 7, 11, 19, 29 and 37 is always prime.

5 + 7 + 11 + 19 + 29 = 71

5 + 7 + 11 + 19 + 37 = 79

5 + 7 + 11 + 29 + 37 = 89

5 + 7 + 19 + 29 + 37 = 97

5 + 11 + 19 + 29 + 37 = 101

7 + 11 + 19 + 29 + 37 = 103

It can be seen that 71, 79, 89, 97, 101 and 103 are all primes.

• 108 is an abundant number, tetranacci number and a semiperfect number. It is evil. powerful and practical.
• 108 equals the sum of the first 9 multiples of 3, viz. 0, 3, 6, 9, 12, 15, 18, 21 and 24.
• 108 is also divisible by the total number of its divisors (12), hence it is a refactorable number.
• 108 is sum of the squares of the three sixes making up the Biblical-Apocalyptic number of the Beast, 666.

108 = 62 + 62 + 62

• 108 is a number that is divisible by the value of its Eulers phi function, which is 36.
• The sum of the prime factors of 108 is 2 + 2 + 3 + 3 + 3 = 13.
Of all the integer partitions of 13, the partition {2, 2, 3, 3, 3} yields the maximum product when you multiply the individual parts.
The partition {2, 2, 2, 2, 2, 3} has a product of 96, the partition {1, 3, 3, 3, 3} has a product of 81, and the partition {4, 4, 5} has a product of 80.
• In base 10, it is a Harshad number (number divisible by the sum of its digits) and a self number.

Geometric connections of 108

• The interior angles of a regular pentagon measure 108 degrees each.
• A chord of a circle, with an angle of 108 degrees, is in golden proportion to the radius of the circle. If the radius i.e. side AC or BC has unit length, then the chord AB has a length equal to Golden Ratio i.e. 1.61803...

• There are 108 different types of free heptomino, a shape made by fitting together seven squares.

A heptomino (or 7-omino) is a polyomino of order 7, that is, a polygon in the plane made of 7 equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there are 108 different free heptominoes i.e. 108 free polyominoes of order 7.

This is the smallest polyomino to contain a hole as shown below. It is in the form of a 3 by 3 square with one corner and the central square missing.

• In four-dimensional space, there are exactly 108 four-dimensional cuboids with integer side lengths such that the hyper-volume equals the hyper-surface area.

Fibonacci sequence and 108

• The digital root of an integer is obtained by summing the digits repeatedly till a single digit remains. For example, the digital root of 473 is 5 (4+7+3=14, 1+4=5).

The first 24 numbers of the Fibonacci Sequence are:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657.

The digital roots of the sequence of Fibonacci numbers yields a pattern that repeats as follows:
(0), 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1.

The first term in the sequence is a zero in the first cycle but becomes a 9 every cycle thereafter. This difference is shown as highlighted below:

First sequence
(0), 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1

Every sequence thereafter
(9), 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1

On adding the repeating 23 digits , surprisingly the number 108 is obtained.

1 + 1 + 2 + 3 + 5 + 8 + 4 + 3 + 7 + 1 + 8 + 9 + 8 + 8 + 7 + 6 + 4 + 1 + 5 + 6 + 2 + 8 + 1 = 108

• The value of 2 sin(108/2) results in the Golden ratio.

Astronomical Coincidences of the Number 108

• Do you know why sun and moon look equally big in the sky?

The distance between the earth and the sun equals about 108 (actually 107-odd) times the sun's diameter. Likewise, it so happens that the distance between the earth and the moon equals about 108 (actually 110-odd) times the moon's diameter. So the sun and moon look equally big in the sky because of their having almost the same ratio between distance and diameter.
It is just a coincidence that the solar and lunar distances happen to match the number 108. Many ratios in Moon-Earth-Sun astronomy coincidentally converge near the number 108. Some of these are:

• The ratio of the Earth-Sun distance to the Sun's diameter is about 107.5.

So the distance between earth and sun is about 108 times sun-diameter

• The ratio of the Sun's diameter to Earth's diameter is about 109.1.

So the diameter of the sun is about 108 times the earth diameter

The ratio of the solar diameter (km 1,392,000) to Earth diameter (km 12,756) is 109.1, close to 108 with an error less than 1%.

Increase of solar diameter is suggested by measurements taken during over a century at Greenwich Observatory; hence in earlier times the above ratios could have been much closer to 108.

• The ratio of the Earth-Moon distance to the Moon's diameter is about 110.5.

So the distance between earth and moon is about 108 times moon-diameter

It has been discovered that the moon's radius is literally shrinking over the years. Hence in earlier times the ratio could have been much closer to 108.

The 108 Emergency Ambulance Service in India

• The 108 Emergency Ambulance Service is a 24x7 service, which anyone can avail by dialing the number 108 on their phone during the case of any emergency.

Why Ambulance Service Named as 108 ?

The number 108 was chosen for the toll-free service because it is considered an auspicious number in multiple religions prevalent in India.

• In the Holy Bible Matthews Chapter 10 Verse 8 (108) reads::: Heal the sick, raise the dead, cleanse the lepers, cast out demons; freely ye received, freely give. some believe this as the reason for choosing the number 108.

• 1-0-8 is a technologically acceptable number enabling the processing of calls to occur faster.

Some Interesting Observations

• The official baseballs used in Major League Baseball have 108 double stitches.

• 108 is the name of a community seeking collaboration from open-source developers, created by Red Hat.
• Stonehenge is a stone monument situated in the south of England. The pre-historic monument Stonehenges Sarsen Circle is about 108 feet in diameter.
• In astrology, the metal silver is said to represent the moon. The atomic weight of silver ( i.e. 107.8682) = 108
• Interestingly enough, a leap year consists of 366 days and 3 x 6 x 6 gives 108.
• FM Radio frequency upper limit is 108 MHz. The frequency of radio waves used for broadcasting on the FM band is set between 88 and 108 MHz.
• The temple bells ring 108 times at year end in Japan

On New Years Eve one of the most important ceremonies in Japan is Joya no Kane, the ringing of the temple bells 108 times at midnight. It is believed that each ring represents one of 108 earthly temptations a person must overcome to achieve nirvana and ringing the bell purifies us and sets things right for the new year.

• The first manned space flight on April 12, 1961 by Yuri Gagarin, a Soviet cosmonaut lasted for 108 minutes in his Vostok 1 spacecraft.
• 108 degrees Fahrenheit is the internal temperature at which the vital organs of human body begin to fail from overheating.
• 108 sacred water taps in Muktinath Temple (Nepal)

Muktinath, a temple town in Nepal is a sacred place both for Hindus and Buddhists. The main shrine of the temple is Pagoda-shaped and is dedicated to Lord Vishnu. Near the Vishnu temple the water of a holy source is distributed by a system of 108 outlets. The outer courtyard has 108 water spouts shaped as cows heads , where the devotees take their sacred bath even in the freezing temperatures.

• The modern day home of the popes, the Vatican city is the Smallest independent sovereign State in the World ( declared in 1929). The area of the Vatican city is approx 108 acres .
• The main Buddhist temple in Urga, Mongolia, has 108 columns.
• The Upanishads are the last part of the Vedas (the Vedas being the scriptures that form the foundation of the early religious beliefs of Hindus) and also the ultimate teachings. Traditionally it is considered that there are 108 major Upanishads.
• The theatre of Sabrata, Libya, beautifully located in face of the sea, is adorned with 108 fluted Corinthian columns.
• According to Ayurveda, there are 108 pressure points in the body, where consciousness and flesh intersect to give life to the living being. The Chinese school of martial arts agrees with the South Indian school of martial arts on the principle of 108 pressure points.
• The sacred River Ganga spans a longitude of 12 degrees (79 to 91), and latitude of 9 degrees (22 to 31). 12 times 9 equal 108.
• 108 sacred books constitute the holy writings for Tibetans.
• International Day of Yoga is celebrated annually on 21 June i.e. 21/6 and 216 is twice the auspicious number 108.
• Rishi Panchami is a day of fasting observed traditionally by (married) women to pay their homage to the Sapta Rishis. According to Hindu calendar, Rishi Panchami is celebrated on the fifth day of the Shukla Paksha in the Hindu month of Bhadrapada (August- September). On the day of Rishi Panchami, Devotees perform ritual bath in holy rivers or other water bodies. They clean their hands and mouth 108 times.
• There are 108 cards in a deck of UNO cards.
• In yoga, it is traditional to perform 108 Sun Salutations (Surya Namaskar) specially at the change of seasons.
• In the book Natya Sastra (a Sanskrit Hindu text on performing arts attributed to sage Bharata Muni), 108 postures are described that can be combined to give the various movements of dance.
• Bodhnath Stupa near Kathmandu is the biggest stupa in Nepal and is the holiest Tibetan Buddhist site outside of Tibet. Around the base of the stupa at ground level there is a brick wall with 108 images of the meditational Buddha inset behind prayer wheels. Worshippers come here to circumambulate the stupa while spinning prayer wheels and praying to the 108 small images of the Buddha surrounding the structure.
• Lord Vishnu is one of the most important Hindu Gods to many devotees across the world. The 108 most sacred temples of Lord Vishnu are known as the Divya Desams, according to Hindu mythology. Of the 108 temples, 105 are in India, one is in Nepal, and last two i.e. Thirupalkadal and Paramapadam are believed to be outside this world where lord Vishnu presides.
• 108 auspicious symbols of Buddha

Wat Pho is world famous temple in Bangkok for the huge image of the reclining Buddha (made of gold leaf). The soles of the feet of the Buddha are each divided into 108 unique sections, displaying 108 auspicious symbols by which Buddha can be identified, such as flowers, dancers, white elephants, tigers etc

Next to the statue lie 108 bronze bowls in the corridor representing the 108 auspicious symbols on the Buddhas foot. Visitors may drop coins into these bowls as it is believed to bring good fortune. The money collected is used to help maintain the temple.

• Vat Savitri Vrat or Vat Purnima Vrat is the Indian Hindu fasting day for the married women who pray to banyan tree for the well being of their husbands. As per rituals, ladies tie white/red cotton threads around the holy banyan tree while walking around the tree for 108 times.
• Astrological connections of number 108

There are 12 zodiac signs based on the movement of sun and moon . There are 9 planets whose positions represents the events that will occur in our life. Multiplying 12*9, we get 108. There are thus 108 possible ways the planets can be placed. The 27 nakshatras or lunar constellations spread over the 4 directions - north, south, east, and west. This also constitutes the whole of existence. 27 x 4 = 108.

• All of us are referred to as Shri by our rishis. Vedic sages used the term Shri Shri before the name of yogis and highly learned gurus. It is interesting to note that numerical equivalent of Shri Shri is 108.

Consider A=1,B=2...H=8,I=9,...R=18,S=19,..Z=26 so

SHRI SHRI = 19+8+18+9+19+8+18+9 =108

• 108 comprised of numbers 1, 0 and 8.

1- Stands for supreme truth or god consciousness.

0- Stands for nothing or emptiness.

8- Stand for eternity or infinity. Note that the symbol of infinity is obtained by rotating 8 by 90 degrees.

Thus the sacred number 108 tells us that we are at once something, nothing and everything.

• 108 in π and e.

In the decimal expansion of π (3.14159265358979323846264338327950288419….), the sum of first 22 digits after decimal equals 108.

1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 + 3 + 5 + 8 + 9 + 7 + 9 + 3 + 2 + 3 + 8 + 4 + 6 + 2 + 6 = 108

In the decimal expansion of e (2.7182818284590452353602874713526624977572….), the sum of first 24 digits after decimal equals 108.

7 + 1 + 8 + 2 + 8 + 1 + 8 + 2 + 8 + 4 + 5 + 9 + 0 + 4 + 5 + 2 + 3 + 5 + 3 + 6 + 0 + 2 + 8 + 7 = 108

(submitted by K D Bajpai from India vide his email dated 31 January 2018)

• Interesting Expressions for 108.

-1 + 2 + 3*4 + 5*6 + 7*8 + 9 = 108

1 - 2 - 3 + 4*5*6 - 7 + 8 - 9 = 108

-2! * 3! + 5! = 108

(submitted by K D Bajpai from India vide his email dated 25 December 2017)

If you find any new and interesting observation about number 108, please email me.

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