I have a completely different point of view. Since there is no direct text in any scripture or common consensus explaining why 108 is important, so instead of making speculations we will have to make the most logical guess. Why? Well Hinduism is supposed to be the most scientific religion so if there is a genuine reason why 108 is important then the reason cannot be illogical.
First let me address my concerns with other others answers.
1. There are 108 X in Y so 108 is important
Some of the other answers here have tried to justify the importance of 108 by arguments which can be summed up as - 108 is important There are 108 X in Y e.g there are 108 Pradakshinas. These arguments are incorrect they have reversed the cause and the effect of things. The fact that many things in Hinduism are related is the result of the fact that 108 was already considered auspicious. Therefore it is wrong to say that 108 is important because many things in Hinduism are related to 108.
2. 108 is a multiple of 9 or 18 or 27 etc so it is important.
Firstly, yes the number 9 appears in things like the navgrahas and the navratri etc. But we also have other numbers more popular than 9 e.g. like 10 avatars of Vishnu or 10 hands of Durga or 10 heads of Ravan, like wise 3 eyes of Shiva and Durga, 4 heads of Bhramha, 4 arms of most of the prominent gods and goddesses, 7 horses of Surya etc. So there is no particular reason why 9 should be the preferred multiple.
Secondly, once again we are mixing cause and effect here. Fact that 108 is a multiple of 9 or 18 or 27 etc is a consequence of the fact that 108 was chosen as the auspicious number because once you have chosen 108, automatically it will be the multiple of all its divisors.
Thirdly, numbers like 1008, 10008, 100008 etc have no special significance in Hinduism so it would not be logical to make any argument using them.
3. 108 is special in astronomy
I think again that this is the effect of having selected 108 as an important number first, and not the cause. Secondly, some say that 108 is important because the ratio of the distance between earth and the sun to the diameter is 108. Well this is close the ratio varies between 107 to 109. But most people don't know earth is very slightly drifting away form the sun. So billion of years back during Staya Yug, the ratio would have been much less than 108.
Moreover we found the ratio of suns distance to earth's diameter to be close to 108 only in past few centuries. But if you look at all the ancient Hindu texts, no where is the number 108 mentioned as the ratio of these distances.
How would god choose an auspicious number?
If god has to make one particular number as an auspicious number what criteria would god put in order to select that number. I am sure god would not have randomly selected any number that came to the mind.
The number must be neither too small nor too large. Why? Think practically. You can't make a garland with only 10 flowers and you would have a tough time if you make a tried to make a garland with 1000 flowers. So a number in the range of 100 should be ideal and manageable
The number must have a lot of divisors. Why? The more the number of divisors, the more number of ways it can be divided into groups of equal sizes which will help in stacking, arranging or dividing among people etc. For example if you have 97 or 107 gold coins you cannot divide it equally among 2, 3, 4 or any number of people because these are both indivisible numbers
Based on these criteria, god must have immediately thought of the number 100. It look perfect size and you can divide it into six equal groups of size 2, 4, 5, 10, 20 and 50.
But after sometime god must have checked a few other number near 100 and realized the number 108 is even better it can be divided into ten equal groups of size 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 so you have much more options.
How would have god chosen between 100 and 108?
In terms of equal division facility 108 has a clear advantage over 100 i.e. 10 possible groups with 108 to only 6 possible groups with 100
But the main significant advantage of 108 over 100 is that 3 is the smallest number missing from the list of divisors of 100 while 5 is the smallest number missing from the list of divisors of 108. Which one would people miss more form a utility point of view? Clearly more people are likely to have 3 children or 3 close friends as compared to 5 children or 5 close friends. So people are more likely divide things into three groups more often then divide things into five groups. So missing 3 is worse than missing 5. So it is advantage 108.
Based on these considerations, god would have selected 108 to be the auspicious number of Hindusim and once the selection was done, many things as describe in other other was connected to 108 and its popularity grew.
108 number has a very special significance in our religion. We have Ashtottara Sata Namavali, 108 pradakshinas etc.
In our religion we have 27 nakshatras and each nakshatram has 4
paadas. The veda states that there are 84 lakhs of types of jivas. Any
jiva among all these must be born among these 27 nakshatras only. 27
stars each containing 4 paadas= 27x4 = 108. Anyone who recites
Ashtottara Satanamavali prays knowingly or unknowingly the welfare of
all the jivas in the entire universe as Sanatana Dharma always says
lokāḥ samastāḥ sukhino bhavantu.
The number 108 is important because it is a multiple of the number 9.
Numbers 18, 108, 1008, 10008 are all multiples of 9 which is a mystic
number. All multiples of 9 added together ultimately become number 9.
This can be verified (16x9=144; 1+4+4=9).
The mystic number 9 is arrived at thiswise: The universe is constituted
of the three factors - time, space and causation. The universe is
constituted of the three Gunas (ingredients) - sattva, rajas and
tamas. The universe is constituted of the three functions - creation,
preservation and destruction. This three times three making nine has
made nine a mystic number. It exhausts the definition of the
phenomenal universe. Twice nine or eighteen makes the Mahabharata
scheme complete. The eighteen Parvas define in detail the career of
man on earth. The eighteen chapters in the Gita make Yoga philosophy
complete. The eighteen day warfare makes the warrior's exploits
Eighteen are the divisions of the armies of the contending parties
-Pandavas and Kauravas. The one having seven and the other eleven. Thus all the available human forces mobilized were eighteen in number.
The Mahabharata is thus an exposition of the human possibilities and
achievements graded into eighteen, the first multiple of nine, a
mystic number. The higher multiples of it [like 108] signify further
ranging into divine regions.
REF:Swami Chidbhavananda in his commentary on the Gita
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