The heart of any counting system is ‘Base’ of that number system
A base is the minimum number of symbols used to write a number. The most famous number system which is still used is Hindu – Arabic number system with base 10. The Babylonian civilization had used a sexagesimal number system with the ‘Base’ as far as 60. The Mayan numerals had a base of 20 where all the numbers could be represented by three symbols. Chronologically, civilization has been reducing the ‘Base’ to perform calculations.
We still use Babylonian base 60 in our calculations!
There are various reasons why each civilization used different bases for keeping track of the counts and time. For example; Babylonians had found out that the same day comes again when 60 days/nights happens 6 times – means a complete year have 360 days (to be precise 354 days). They found out that 60 was (is) the smallest number that is divisible by every number from 1 to 6. Also, 6 times 60 equals 360 which was the most convenient way of representing a complete year. These were the reasons for Babylonians to use 60 as the base of their number system. No wonder we still use base 60 in calculating degrees of geometries and minutes and seconds of time.
Mayans calendars, more sophisticated then Gregorian calendar of today!
On the other hand, Mayans used shells, pebbles, and sticks as original counting items (they used them as symbols), and it was easier to remember the count using the fingers and toes of a person. Hence, they used base 20. Their calculations for calendars were so advanced and sophisticated that they were far more accurate than Gregorian calendar used in Europe.
Most significant inventions of modern civilization: Zero & Fraction
Although the Babylonian number system is more convenient and natural in some scenarios (like trigonometric tables), it did not have a symbol for zero. The absence of a symbol for zero makes it difficult for other calculations and representations. The pre-classic Maya and their neighbors had independently developed the concept of zero by at least as early as 36 BCE but, they did not know the concept of fraction. So the Hindu-Arabic number system (Decimal number system) had these advantages over the previous number systems. Hence, it became mainstream. It has ‘0’ (zero) as a symbol to represent nothingness and ‘.’ (point) to represent fractions. We can represent all the base symbols distinctly. Symbols to represent them are not confusing. They are easy to remember.
But the current number system also seems to be incomplete!
The modern civilization has used decimal number system very extensively and has developed algebra, calculus, geometry, logic, and trigonometry around it. However, if you observe carefully, you would realize that the number system still fails to represent some of the problems entirely in one way or other. For example, if you have a pizza and try to divide it into three equal parts as the percentage, then each piece would be equivalent to 33.33 %. The remaining 0.01% will remain unaccounted, how far you try. But, if you consider the pizza as an object of 3600, you can divide it equally into 1200 each! Here the decimal number system (also referred to as natural number system) is failing, and Babylonian number system works. Let’s see some more instances where natural number system fails to represent the naturally occurring events and concepts:
Birthday Paradox is a well-known probability problem. It says that in a room of just 23 people there’s a 50-50 chance of two people having the same birthday. Similarly, in a room of 75, there’s a 99.9% chance of two people having the same birthday. But in a room of 365 persons, the probability is NOT 100%! In theoretical mathematics, it turns out to be equal to (100 − (1.45×10-155)) % – almost 100% but NOT 100%. Doesn’t this mean that the probability of at least two people in a room of 365 persons having same birthday cannot be represented perfectly?
Irrational Number: PI
The number π (Pi) is a mathematical constant. It is the ratio of a circle’s circumference to its diameter. The exciting fact about π is that none of the mathematician or the supercomputer has ever calculated its exact value! Knowing the fact that the circumference of a circle and its diameter are real-world values and their ratios should yield a real value. Isn’t it surprising that their ratio is a non-repeating and non-terminating number?
Mathematical constants which represent other physical entities or the physical phenomenon where the decimal number system does not describe them perfectly. These do not pull one’s imagination, as we have accepted them as fact. Let’s see the values of different mathematical constants. The numeric value of π (Archimedes’ constant) is approximately 3.1415926535. The numeric value of e (Euler’s number) is approximately 2.7182818284. The numeric value of √2 (Pythagoras’ constant) is approximately 1.41429. The numeric value of the Golden ratio, φ (used in geometry), is approximately equal to 1.6180339887498948482. All these mathematical constants are approximations!
The physical constants. We are not able to represent even the physical constants perfectly. The value of Gravitational constant G is 6.67408 × 10-11 Nm2/kg2 which is approaching zero but not the absolute zero. The value of Plank’s constant h is 6.626070040 × 10-34 Js. The value of Magnetic constant μ0 is 4π × 10-7 Tm/A. The value of Electric constant ε0 is 8.854187817 × 10-12 C2/Nm2. Looking at these values you must be intrigued as to why the so-called natural numbers cannot quantify physical constants perfectly?
Isn’t this possible that we have a digit missing from our number system?
So can’t I have an opinion that the number system as we know may be imperfect and we might need to add a new digit or knock off one ? Alternatively, we should try to develop algebra, calculus and other mathematical functions using the Babylonian number system or Mayan number system? There must be a possible way so that the calculations done by human can perfectly represent physical object and natural phenomenon. There must be a way where one can map every aspect of nature to a number or a digit.
One day we might have a number system which is mirror image of nature. Using which it will be easier to find the solution for traveling salesman problem. Supercomputers will be used for predicting weather patterns that never fails. Experts in finances will be able to data mine to gauge and guide stock markets which do not fail to predict next financial crash. We will be able to predict the earthquake, able to predict when an engine or airplane fails and will be able to predict where the next thunder strikes. Until now all the calculations are predictions are an approximation, extrapolation, and probabilities. These are approximations because the decimal number system can not define them perfectly.