Wednesday, December 6, 2023



Philosophy behind origin of some Mathematical Numbers – by Dr. SS Verma

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We generally make use of the ancient knowledge system in our daily life without bothering to into deep about its origin and importance as we always make use of 60 seconds in a minute, 60 minutes in an hour, 24 hours is a day & night (1 day), 7days in a week, 12 months in a year, 360 degrees in a circle etc., just few to mention here. Under Indian knowledge system (IKS), Indian mythological literature (Vedas) is always assigned as a source to the origin of present day maximum scientific and mathematical knowledge.

However, philosophy and importance behind some common scientific facts and mathematical numbers used and respected by human all over the world in general and in India in particular, is always under debate. There are many mathematical numbers like 7, 108, 12, 24, 60 and 360 etc. which is an integral part of our daily life and here we will discuss to the origin of some these numbers according to the information available on google.

There is even evidence that a circle in the Rigveda from India was divided into 360 parts as “Twelve spokes, one wheel, navels three. Who can comprehend this? On it are placed together three hundred and sixty like pegs. They shake not in the least.”- Dirghatamas, Rigveda 1.164.48.

However, their origin as per Indian Knowledge system can be ascertained by reading the Vedas which is well beyond the scope of a common man as well as not well accepted by the presently scientifically developed world. Therefore, here the origin of these numbers is briefly mentioned based on the present knowledge system.

The origin of the division of time

To find the origin of the division of time, and thus understand why there are 60 seconds in a minute, we must go back to the Egyptian and Babylonian antiquity, to about 3,000 years BC. Based on the observation of the phases of the moon, which last about 29 and a half days, our distant ancestors established the months. Egyptian astronomers also calculated, by observing the sky, that a year lasts 365 days.

Around 2800 BC, the year was divided into three seasons of four months, and each month was composed of three weeks of ten days, which were later transformed into weeks of seven days, probably to allow workers to rest more often on the last day of the week, and also for religious reasons. In connection with this calendar of 12 months of 30 days, Babylonians and Greeks divided the circle into 360 degrees (12 times 30).

At the same time, around 200 BC, the Egyptians divided the day into 12 hours of day and 12 hours of night, probably because the year was already divided into 12 months. This means that one hour of summer daylight lasted longer than one hour of winter daylight, and vice versa for the night. So minutes and seconds also had a random duration. So, we were not yet on our traditional division of 60 seconds which give one minute, and 60 minutes one hour!

The ancient Egyptians used stars to calculate the time at night. They did this by paying close attention to a group of 36 special stars called “decans.” These stars rose in the sky in a pattern and were used to keep track of the hours. However, only about 12 of these stars out of 36 were visible at a given time, and which stars appeared changed with the seasons and the calendar year. Now, the Egyptians made special charts called star calendars to keep track of the time using stars correctly.

These charts had 36 columns for each decan and 12 rows for each hour that could be counted during the night. Eventually, this timekeeping process was standardized with days and nights, each having 12 hours. However, this system only worked well during equinoxes, when the Sun was directly above the equator, dividing days and nights into equal lengths. At other times, the length of an hour would vary. For example, hours during the day in summer would be longer, while shorter in winter.

However, at that time, the hours did not have a fixed length. The Greek astronomers who were then trying to find answers to the queries of the universe, existence, and stars and galaxies found it difficult to manage calculations with the prevailing method. Then, Greek mathematician Hipparchus gave us the “Equinoctial hours” by proposing the division of a day into 24 equal hours. Even then, for a long time, ordinary people kept using the seasonally varying hours. It wasn’t until the 14th century in Europe, when mechanical clocks came into use that commoners began using the system that we practice today.

Sumerians to Babylonians: The division of the hour into 60 minutes and of the minute into 60 seconds comes from the Babylonians who used a sexagesimal (counting in 60s) system for mathematics and astronomy. They derived their number system from the Sumerians who were using it as early as 3500 BC. The use of 12 subdivisions for day and night, with 60 for hours and minutes, turns out to be much more useful than (say) 10 and 100 if we want to avoid having to use complicated notations for parts of a day.

Twelve is divisible by 2, 3, 4, 6 and 12 itself – whereas 10 has only three devisers – whole numbers that divide it a whole number of times. Sixty has 12 devisers and because 60 = 5 x 12 it combines the advantages of both 10 and 12. In fact both 12 and 60 share the property that they have more devisers than any number smaller than themselves.

Babylonians to Greek: In today’s world, the most widely used numeral system is decimal (base 10), a system that probably originated because it made it easy for humans to count using their fingers. The civilizations that first divided the day into smaller parts, however, used different numeral systems, specifically duodecimal (base 12) and sexagesimal (base 60). Thanks to documented evidence of the Egyptians’ use of sundials, most historians credit them with being the first civilization to divide the day into smaller parts.

The first sundials were simply stakes placed in the ground that indicated time by the length and direction of the resulting shadow. As early as 1500 B.C., the Egyptians had developed a more advanced sundial. A T-shaped bar placed in the ground, this instrument was calibrated to divide the interval between sunrise and sunset into 12 parts.

This division reflected Egypt’s use of the duodecimal system–the importance of the number 12 is typically attributed either to the fact that it equals the number of lunar cycles in a year or the number of finger joints on each hand (three in each of the four fingers, excluding the thumb), making it possible to count to 12 with the thumb.

The next-generation sundial likely formed the first representation of what we now call the hour. Although the hours within a given day were approximately equal, their lengths varied during the year, with summer hours being much longer than winter hours.

Why is a full Circle of 360 Degrees?

Hypothesis 1- Mathematical Reasons: The number 360 is divisible by any number from 1 to 10, except the 7th. It is actually divided into 24 different numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360 itself. These 24 numbers are called the divisors of the number 360. This is the highest number of divisors for each positive integer up to its own value of 360.

In contrast, the number we often wish would be the value for the full circle-100-has only 9 divisors. This property of the number 360 makes it a strongly composite number. Numbers are said to be highly composite if they are positive integers with more divisors than any smaller positive integer has. Highly composite numbers are considered good base numbers with which to perform common calculations.

For example, 360 can be divided into two, three and four parts and the resulting number is a whole number. The resulting numbers are 180, 120 and 90. However, dividing 100 by three doesn’t end in a whole number. Instead, it provides a decimal value of 33.3 recurring, which makes performing calculations difficult. Calculations using 360 actually become pretty simple once you’re smart enough to do them in your head and can put down the calculator.

Hypothesis 2- The Length of a Year: There are exactly 365 days a year i.e. earth complete its one rotation around Sun in 365 days and there is no mathematical reason for this, but it is just an observation made by our ancestors, and these observations also contributed to a circle being closed by 360 degrees. The ancient astronomers, especially the Persians and Cappadocians, noticed that it took the sun 365 days to return to exactly the same position.

In other words, the sun advances one degree every day on its elliptical orbit. Persians had a leap month every 6 years to compensate for the 5 extra days. Moreover, the lunar calendar has a total of 355 days, while the solar calendar has 365 days. And, the number sits perfectly between these two and is a composite number is 360.

Hypothesis 3- Historical Reasons: Another theory that suggests why a full circle is 360 degrees comes from the Babylonians. The Sumerians and Babylonians famously used the Sexagesimal number system. The sexagesimal system is one with a base value of 60, while the current system we use is known as the decimal system and has a base value of 10. So, once we reach the 10th number, we start repeating the symbols of earlier numbers from 0 to 9 to form new numbers.

If we were to draw an equilateral triangle with the length of sides equal to the radii of the circle and place one of its vertexes at the centre of the circle, then we could fit a total of 6 such equilateral triangles inside a circle. Since the Babylonians used the sexagesimal numeral system, they considered each triangle to have a base value of 60. Thus, 6 triangles x 60 base value again gives us a value of 360.

The mathematical numbers discussed here play an important role in our education, knowledge and applications in daily our daily life and these answers may or may not satisfy the quest of any individual about the origin of time division or angles of a circle, however, every one is free to learn about the origin of these important numbers and corelate it to any knowledge system.

Dr. SS Verma, Department of Physics, S.L.I.E.T., Longowal, Distt.-Sangrur (Punjab)-148106

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