In mathematical philosophy, formalism is the position that regards mathematics as the study of formal deductive systems. Mathematical truth is simply provability in the system, and there is and can be no ultimate meaning to mathematics other than the operation of naked symbols according to fixed rules.Philip J. Davis & Reuben Hersh
Welcome to the blog Math1089 – Mathematics for All.
Numbers are an integral part of mathematics, as well as various numbers in Indian culture. One such number is 108, which is considered auspicious in Indian culture and holds significance in many rituals practiced in India.
The number 108 is considered sacred and auspicious in Hinduism, Sikhism, Buddhism, and Jainism. It is used in various religious practices such as chanting mantras and counting beads. In Hinduism, a mala with 108 beads is used for meditation. Chinese Buddhists and Taoists also use a 108 bead mala. Similar traditions exist in Sikhism, Buddhism, and Jainism. The number 108 holds symbolic importance in these religions and is associated with virtues, stars, and rituals for good luck.
It’s now time for mathematics and we will see how fascinating the number 108 from mathematical point of view is.
• The number 108 is consisting of the digits 0, 1 and 8. It is made up of three distinct cube digits.
• The divisors of 108 are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108; and 108 is the smallest number whose divisors contain every digit at least once.
• 108 is divisible by the total number of its divisors (12 in number, which are 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108).
• 108 equals the sum of the first 9 multiples of 3. In fact,
108 = 0 + 3 + 6 + 9 + 12 + 15 + 18 + 21 + 24.
• Since 108 = 4 × 27 = 22 × 33, it is the product of a square and a cube.
• 108 is the product of the first three numbers raised to themselves (in fact, it is the hyperfactorial of 3):
108 = 11 × 22 × 33
• 108 can be written as a sum of consecutive naturals: 35 + 36 + 37.
• The number 108 is divisible by the sum of its digits (= 1 + 0 + 8 = 9):
• 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 = 8 + 100 = 23 + 102
108 = 27 + 81 = 33 + 92
• 108 is sandwiched between the twin primes 107 and 109. More interestingly, the concatenation of 108 with 107 and 109 are also primes (i.e., 108107 and 108109).
• The number of palindromic numbers less than 1000 is 108.
• Square of the number 108 has many interesting properties as listed below:
1082 = 11664, where 1 = 12, 16 = 42 and 64 = 82 are squares.
1082 = 11664, and 11 + 66 + 4 = 81 = 92, a perfect square
1082 = 11664, and 11 + 6 + 64 = 81 = 92, a perfect square
1082 = 11664, and 1 + 16 + 64 = 81 = 92, a perfect square
• Note that 1082 = 11664, 1292 = 16641 and 2042 = 41616. All these three squares contain same digits in different orders.
Various Mathematical Representations of 108
108 = 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9
108 = ‒1 + 2 + 3 × 4 + 5 × 6 + 7 × 8 + 9
108 = 1 ‒ 2 ‒ 3 + 4 × 5 × 6 ‒ 7 + 8 ‒ 9
108 = 9 + 8 + 76 + 5 + 4 + 3 + 2 + 1
108 = 9 + 8 × 7 + 6 × 5 + 4 × 3 + 2 ‒ 1
108 = ‒1 × 7 + 2 + 91 ‒ 7 + 29
108 = 5! ‒2! × 3!
• 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.
The possible sums are as follows:
5 + 7 + 11 + 19 + 29 = 71, a prime.
5 + 7 + 11 + 19 + 37 = 79, a prime.
5 + 7 + 11 + 29 + 37 = 89, a prime.
5 + 7 + 19 + 29 + 37 = 97, a prime.
5 + 11 + 19 + 29 + 37 = 101, a prime.
7 + 11 + 19 + 29 + 37 = 103, a prime.
• 108 is divisible by the value of its Euler’s phi function [ϕ(108) = 36].
• 108° is the measure of each interior angles of a regular pentagon.
• There are 108 different types of free heptomino (is a polygon in the plane made of seven equal-sized squares connected edge-to-edge).
• The value of 2 sin (108°/2) results in the Golden ratio (Ф).
• In case of a leap year, there are 366 days in a year and 3 × 6 × 6 gives 108.
Single digit representations of 108
This blog is as much yours as it is mine. Would you like to contribute an exceptional, non-routine article and have it published on Math1089? Perhaps you have a preliminary idea that you wish to see in its published form—please share your ideas by dropping us a line.
We wholeheartedly welcome your contributions and eagerly anticipate featuring your ideas on “Math1089 – Mathematics for All” in our next captivating mathematics blog post. Thank you for being a part of our journey, and we look forward to your involvement in shaping the future content of Math1089. See you soon for another intriguing exploration into the world of mathematics!
Like this:Like Loading…
Related
