The scientists, mathematicians, and engineers come across zero, circles & infinity in a way or two in their respective fields. Mathematics is regarded as the language of the Universe. We can decode some of the most interesting mysteries of the universe using mathematics. Mathematics is the root of all the advancements in the field of science, so this mathematical field can also be utilized to unlock the meaning of this entire cosmos.
“Learn how to see. Realize that everything connects to everything else”Leonardo Da Vinci
Exceptions, contradictions or unexpected results exist in life whether it is Chemistry or Mathematics. The detailed research and interpretations of some of these phenomena have always lead to discoveries/inventions and results.
The Number Line
We have always assumed that the number line is a straight line running from minus infinity to plus infinity. There are infinite numbers between two given real numbers. The sum of two opposite numbers for example +1 and -1 or +8 and -8 is always zero, so we can say that the zero separates two opposite worlds in the mathematical plane and all these numbers unite at a point called zero.
Nature of Infinity
For a given system the value or perception of infinity changes, for example- for a semiconductor system the normal value of voltage level is very less as compared to an electrical power system, so for a semiconductor system Kilo Volts is like infinity and for an electrical power system, Kilo Volts is a normal operating range. In the same way, our perception of infinity can be different from other’s perceptive. The nature of infinity in form of odd or even is not known, but if we can explore mathematical equations utilized or acceptable for explaining the universe then we can come close to defining the nature of infinity.
May read: COVID-19 to Strengthen Scientific Temper
Take the sum of natural numbers up to infinity i.e. 1+2+3+……+∞ this sum is equal to -1/12, a quite unexpected result since the sum of positive numbers can never be a negative number. Interestingly, this equation is used in Quantum Physics and also in the derivation of the famous String Equation.
The proof of this sum (Ramanujan’s Summation) is as follows-
Let S = 1+2+3+4+5+6+7.
Consider S1 = 1-1+1-1+1-1+1-1…..
Now, this sum should be 0 or 1 based on the number of natural numbers taken. If infinite numbers are even, S1=0, if odd S1=1. But, the Riemann zeta function gives it a value of ½. Mathematical community too agrees that the sum is ½. You can also see a depth proof of S1 if not satisfied.
Now, S1 = 1-1+1-1+1-1+1…..
1-S1 = 1-(1-1+1-1+1-1+1…)
1-S1 = 1-1+1-1+1-1+1…..
1-S1 = S1
So, S1 = 1/2
Now, Let S2 = 1-2+3-4+5-6+7…..
S2 = 1-2+3-4+5-6+7-8+9…..
S2 = 1-2+3-4+5-6+7-8……. (Shifted the RHS by a unit position)
So, 2S2 = 1-1+1-1+1-1+1…..
Hence, 2S2 = S1
Therefore, S2 = 1/4
Let’s come back to our sum of infinite numbers.
S = 1+2+3+4+5+6+7+8+9…..
S2 = 1-2+3-4+5-6+7-8+9….
So, S-S2 = 4+8+12+16+20…..
Hence, S-S2 = 4(1+2+3+4+5+6+7+8….)
S-S2 = 4S
So, -S2 = 3S
And, S = -S2/3 = -1/12
Hence, 1+2+3+…………+∞ = -1/12
We can interpret from this result that an infinite number of the opposite sign must be connected to this sum i.e. why we getting a negative number.
Let us take zero on the number line. We can write 0 as the sum of -1 and +1 or -2 and +2 or -∞ and +∞. Now, connect two zeros through a line having the distance ’d’ between them as shown in Figure 1.
Now, reaching the two zeros through two different paths i.e. from 0 to +1 to +2 to +3….up to +∞ and from 0 to -1 to -2 to -3 up to -∞ as shown in Figure 2.
Thus, the shape so formed is in the form of a circle and the most interesting thing derived from this circle is pi which is equal to circumference/diameter. So, we are reaching the same point through two different paths and that gives us the ultimate number π. Pi is the most mysterious number of all as we can see from the facts associated with it.
Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever meet, the date, time and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those two points. All information that has ever existed or will ever exist, the DNA of every being in nature.
All contained in the ratio of a circumference and a diameter!
As we know that zero was given by India and the very shape of it has something to say about the numbers that we are familiar with. If we observe then we can say that ranging from the smallest atom to the huge universe, all are in the shape of a circle. All the modern technological advancement in the field of science has been done because of the study of an atom which is a circle and scientists are studying the universe for knowing about our origin. So, from the model of the circle so formed (fig 2) out of nothingness has given us the information of the entire universe in the form of pi. So, nothingness and everything are connected.
So, we can say that +∞ and -∞ are connected and form a circle that contains all the information (in the form of π = circumference/diameter) of this cosmos.
About the authors :
Above is a piece of work by Chandan Bhai Rawal and Prof. (Dr.) Yaduvir Singh.
Prof. Yaduvir Singh is Professor and Head of Department of Electrical Engineering at School of Engineering, HBTU, Kanpur.
Disclaimer: The views expressed in this article are of the authors solely. The Rise.co.in neither endorses nor is responsible for them.