Have you ever thought (2/3) would be the approximation of logarithm of 2, (2/3 + 2/5) would be that of 3 and (2/3 + 2/5 + 2/7) would be that of 4? It is too simple to believe but simplicity leads to ultimate sophistication. This is not what I am saying but has been said, more than five hundred years back, by Leonardo Da Vinci (Leonardo of town Vinci) Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor and architect.
In order to reply opening line query, one wants to know in the first instance what this fuss logarithm is. Before explaining, I will narrate a short fable out of many, popularly known ‘Akbar Birbal Ke Kisse’ in common parlance or ‘Fables of Akbar and Birbal’. Akbar was a Mughal Emperor who once ruled India. In a chess game, when Birbal defeated Emperor Akbar, he, enthralled by the exceptional skill, asked Birbal to state whatever he liked as a reward and his wish would be fulfilled. Hesitatingly, Birbal expressed that he might be given a few grains of rice on chessboard itself in a such a way that one grain be put in first square, two grains in second square, four grains in third square so on and all the sixty four squares be filled up in this fashion. To the surprise of the emperor, the entire rice stock of the empire fell short, even when half of the squares could not be filled up. Witty Birbal knew the hugeness of exponentiation, the inverse of which is logarithm.
In mathematics, serial number of the square is analogous to logarithm of the grains kept in that square and multiplying factor 2 that makes 2, 4, 8 grains in second, third, fourth squares, is analogous to the base of the logarithm. If serial number of the square is 11, then logarithm of the grains kept in this eleventh square, is 10. Similarly, we can state that logarithm of grain, kept in first square is 0. Had Birbal wished for putting 1 grain in first square, 10 in second, 100 in third so on, then the base of the logarithm would have been 10. In scientific use, base of logarithm is taken as ‘Euler’s Number’ abbreviated ‘e’ which equals 2.71828. Such logarithm to the base ‘e’ is called natural logarithm and is denoted by ln. For example, logarithm of 100 to the base ‘e’ is written as ln 100.
In what may lead to an important research work in Number Theory, two amateur mathematicians from India, have innovated the simplest ever method of determining natural logarithm of a real number. Briefly stating, for determining natural logarithm of number 2, one will have to find odd numbers in 4 i.e double of 2. These are obviously 1 and 3. Ignoring odd number 1 in all cases, natural logarithm of 2 will be 2/3. In this way, natural logarithm of 5, will be (2/3 + 2/5 +2/7 +2/9). To determine logarithm of a number ‘n,’ odd numbers (ignoring 1), i.e, 3, 5, …, 2n-1 are first written and, then double the sum of reciprocal of these odd numbers except one, called odd harmonic series, yields value of logarithm of n written as ln (n). Mathematically,
ln (n) = 2/3 + 2/5 + 2/7 + …+ 2/(2n-1).
Proceeding in the same manner, natural logarithm of 5/2 is 2/5 + 2/7 +2/9 and in general
ln (n/m) = 2/(2m+1) + 2/(2m+3) + 2/(2m+5) + …+ 2/(2n-1).
These determinations are approximates and to achieve precise value of natural logarithm, the paper “APPROXIMATION OF LOGARITHM, FACTORIAL AND EULER-MASCHERONI CONSTANT USING ODD HARMONIC SERIES, ” prescribes multiplication and division of the real number by a large number. For example, number, ‘n’ should be written as (100n)/100 and the formula modifies to
ln (n) = 2/201 + 2/203 + 2/205 + …+ 2/(200n-1). (1)
More the value of multiplier and divider (it is 100 in the example), the better would be the result. “It arises the curiosity to know the theory that culminates into determination of ln (n) by Equation (1). The authors explain in the paper,
“It will not be out of context to state that it bears a strong analogy with human cells. Human cells trillions in number compose the body and NBB’s, in the same way, though limited to (x− 1) in number, compose a number x. To illustrate how NBB’s compose a number, we present the relation between x and NBB’s.
x= (2/1) · (3/2) · (4/3) … {x/(x− 1)}.
It is obvious, NBB’s (2/1), (3/2), (4/3), … , {x/(x− 1)} are (x− 1) in number and when multiplied generate a number x. In normal practice, a number x is envisaged as that which has magnitude equal to what we get when 1 is added x times. This concept of numbers by addition, arises on account of the fact, we are taught mathematics starting with ‘counting of the numbers,’ in kindergarten and basing thereupon, we distinguish one number from the other on account of weight acquired on accumulation of unities in it. We do addition and subtraction, corresponding integers to our fingers that is why fingers are called digits. With these strong prejudices, we are unable to envisage an integer as product of numbers. Thinking out of box, we have considered, in this paper, an integer to be product of NBB’s. Based upon that we give some examples. (101/3) is product of NBB’s (4/3), (5/4), (7/ 8) … , (101/100) and 20 is product of NBB’s (4/3), (4/3) , (5/4), (5/4), (7/ 6),…,(36/35). Using NBB’s, a number say x can be represented in infinite ways. NBB is the abbreviation of number building block. This is the crux of the research highlighted in the paper.”
Euler Mascheroni constant is equal to (1/1+1/2 +1/3+… 1/p)- ln (p) when p extends to infinity. Using number building blocks its, approximate value is calculated as 0.5736309333.
Before parting I would say, when a complicated problem is analysed in basic and simple ways, the success would never be far away. Steve Jobs was not a qualified person, he was not even Graduate and had overwhelming interest in calligraphy of computer writing. But what he had was, probably does not exist in most. He believed in creativity, thinking differently from common lot with unbiased and unprejudiced mind. He never believed, university education makes one creative but was advocate of the fact, it otherwise biases the thinking in particular ways the others have thought and scribed the books. When the mind is unbiased, it has innumerable options to explore but when it is biased, it has only one option that has been studied.
“They teach you how other people think, during your most productive years,” he said. “It kills creativity. Makes people into bozos.” Lisa Brennan-Jobs daughter of Steve jobs recalls her father in her book and its abstract was published in Times Of India.
Steve Jobs was a believer not only in one religion but also in many. He had kept “Bhagavad Gita” in his home daughter Lisa writes in the book. Inspired by the ideals of Steve Jobs, this article as well as the research paper now published in the journal Mathematical Forum abstracted in Mathematical Reviews (USA), were written.
References:
1) Narinder Kumar Wadhawan, Priyanka Wadhawan APPROXIMATION OF LOGARITHM, FACTORIAL AND EULER-MASCHERONI CONSTANT USING ODD HARMONIC SERIES, Mathematical Forum, July, 2021, Issue 28(2) at http://mathematical-forum.org/wp-content/uploads/2021/07/10.-MF592020.pdf. 2) Leonardo Da Vinci from Wikipedia at https://en.m.wikipedia.org/wiki/Leonardo_da_Vinci. 3) Title picture courtesy Update Punjab from article Calculating Logarithm On Fingertips : A study by Narinder Kumar Wadhawan IAS (Retd.) at https://updatepunjab.com/punjab/calculating-logarithm-on-fingertips-a-study-by-narinder-kumar-wadhawan-ias-retd/. 4) Second picture courtesy WordPress article “Was Steve Jobs Right” by Narinder Kumar Wadhawan at https://narinderkw.wordpress.com/2019/01/18/was-steve-jobs-right/
About Author:
I am Electronics and Electrical Communication Engineering graduate and was earlier Scientist, then Instrument Maintenance Engineer, then Civil Servant in Indian Administrative Service (IAS). After retirement, he writes on subjects, Astronomy, Mathematics, Yoga, Humanity etc.
Narinder 