Yes, it is correct mathematicians think differently and extraordinarily. It is expected of them as traditional approach will lead to traditional results and halt the process of evolution of new results and formulae. It reminds me of Johann Carl Friedrich Gauss, sometimes referred to as Princeps mathematicorum. He was a German mathematician and also contributed to many fields including number theory, algebra, statistics, analysis, differential geometry, geodesy, geophysics, mechanics, electrostatics, magnetic fields, astronomy, matrix theory and optics.When Gauss was studying in school, to inflict punishment upon him for misbehaving, his school teacher gave him a cumbersome and time consuming problem of addition of all integers from 1 to 100. But to the astonishment of the teacher, he added the quantities with in few seconds and showed the result. The result was correct and it earned appreciation for young Gauss in the heart of the teacher.

What Gauss did even baffle now to most of us. He first wrote the sum as

S = 1 + 2 + 3 + ………………………………………up to 100 and again wrote it but in a different way

S = 100 + 99 + 98 +………………………………………………up to 1 and added both as

2S = 101 + 101 + 101 + ……………………up to 100 times and then

2S = 101 x 100 = 10100 and

S = 5050.

That produced the result wonderfully in a neat and clean manner. Had there been some other student, he would have adopted the normal procedure of adding 1 with two then with 3 and then with four until he would have reached the integer 100. Even the teacher expected it so from Gauss but Gauss was not a simple student, he was extraordinary.

In the same way, another student at the university of Copenhagen was asked in examination how the height of a skyscraper can be measured with barometer. Naively, he replied by tying the barometer with a rope and then hanging it from top of the tower touching the ground and measuring the length of rope plus the height of barometer that will give the height of the skyscraper. Not impressed with the answer, the examiner awarded zero marks to this answer.

On appeal by the student, it was held that answer is correct but there is no physics involved and the student was called for oral examination. When the student was asked to give answer to the same problem with in five minutes, he spent most of the time thinking for answer. Ultimately, he was warned the time was running out and he should put forth his answer immediately. However, the student replied, he has many answers to the problem and was thinking which answer was the best. When he was asked to give all the answers, he politely replied,” Height of skyscraper can be known by throwing the barometer from top and noting the time taken by it to touch the ground. Since it may cost barometer, the height can alternatively be known by measuring the length of barometer and also its shadow and then measuring the shadow of the skyscraper, ratio of barometer to its shadow will equal ratio of height of building to its shadow and simple arithmetics will give height of skyscraper. He continued, there is yet another way, difference of barometer reading at top and bottom of the sky scrapper will equal to atmospheric pressure equivalent to height of skyscrapers. He still added, If one is too scientific, one can note down the time period of a pendulum at the base of the skyscraper and and also at the top of skyscraper and difference in time period will be a measure of height.”

That student was genius mathematician and physicist Neil Bohr who propounded quantum theory later in his life.

It is also said of Bohr that during a foot ball match where Bohr was a goalkeeper, the ball remained in the opponent half but when the attacker brought the ball to Bohr’s half, a spectator shouted, ” Be aware Bohr, the ball is in your half,” and at that time, Bohr was found solving mathematical problem on the ground.

Mathematicians think differently from an ordinary person. I am certain, I will be correct if I say, most of inventions have been made when unconventional route was adopted.

Let us take the case of determination of value of pi written as π which is a mathematical constant and is defined as a quantity when multiplied with diameter of a round object gives its circumstance. Normal thinking leads us to determine its value by measuring circumference of a round figure and then dividing it by diameter.

Method is correct but error in measurement of circumference and diameter will result in error in value of pi. And pi so calculated will culminate in large error in the area or circumference of a round object if that happens to be of large diameter. That led mathematicians to determine pi independent of measurement of its circumference or its area. This is, thinking differently.

Without going into the history of pi, I will attempt to explain how earlier mathematician and physicist thought of pi. Pi, according to mathematician and physicist Archimedes, can be calculated by averaging the area of a polygon circumscribed and inscribed in a circle. His thought points to the fact that circumference of a circle is equal to half the circumference of polygon just outside plus half the the circumference just inside the same circle. That was an apt thinking and gave great approximation of value of pi provided sides of regular polygon are quite large. Probably, such thought would generally not come to ordinary person.

Idea that occupied to most of the mathematicians in the subject of determination of pi was approximating regular polygon of extremely large number of sides with circle. Extremely large number here is that number which is greater than the greatest.

Let us attempt finding value of pi by above said approximation.

I submit, a point forms an angle of 2. Pi around it and let us take polygon or Triangle of three equal sides AB, BC, CA. That will make, its three corners or points as equidistant from each other. Let there be a point O inside the triangle such that these points are also equidistant from point O ie OA = OB = OC = radius r. That means a circle with centre O , circumscribes triangle ABC. Let us find out perimeter of triangle ABC which is equal to sum of AB, BC and CA.

Side BC = twice length BD and BD = OB x sine of angle BOD.

Since ABC is a polygon of three sides, therefore angle BOC = 2.pi/3.

If regular polygon has n sides then angle BOC = 2.pi/n.

For n =3, angle BOD = ½. 2.pi/3 = pi/3.

BC = 2.BD = 2r.sin pi/3. For n sided regular polygon, it would equal 2r.sin pi/n.

AB + BC + CA = 3. BC = 6.r. Sin pi/3. For n sided regular polygon, perimeter would be 2.n.r sin pi/n.

First approximation is made by equating perimeter of three sided regular polygon with circumference of circle (2.pi.r). That is

6.r. Sin pi/3= 2.pi.r.

Or pi = 3. under root 3/2 = 3. (.866)= 2.698.

Next approximation would be by taking n = 4,

pi = 4.(sin pi/4)= 4./2^1/2 = 2.828.

Next approximation when n= 6,

pi = 6. Sin pi/6 = 3.

It is clear from above as n increases, value of pi approximates better to actual value of pi.

For n sided regular polygon,

pi = n.sin pi/n……………………………………………………………(1)

If we put x = pi/n, pi equals pi.sin x/x when x is very small.

It has been the endeavour of mathematician to find value of sin x/x as it contains the value of pi in it.

In Europe, first attempt to determine value of pi was made by François Viète*, a French mathematician. His work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry 3 and Henry 4 of France. He attempted equating polygon with circle by Geometry and finally gave a formula for pi in nested radicals. What he practically solved was determination of value sin x/x. Sin x is a trigonometric ratio of perpendicular to hypotenuse in a right angled triangle with angle x facing perpendicular.

He geometrically reached the formula for sin x as

sin x = x.cos x/2.cos x/4. Cos x/8. Cos x/4…………………………………up to infinity with last term as 1.

When x equals pi/4, equation takes the form,

(½)^1/2 = pi. cos pi/8. cos pi/16. cos pi/32………………………

Where cos pi/8. cos pi/16. cos pi/32 are known from double angle formula,

cos pi/8 = ½.(2+ 2^1/2)^1/2),

cos pi/16 = ½{2+(2+ 2^1/2)^1/2}^1/2,

cos pi/32 = ½[2+ {2+(2+ 2^1/2)^1/2}^1/2]^1/2,

………………………………….

On putting these values,

1/pi = 2^1/2. ½.(2+ 2^1/2)^1/2). ½{2+(2+ 2^1/2)^1/2}^1/2. ½[2+ {2+(2+ 2^1/2)^1/2}^1/2]^1/2…….(2)

There came then most eminent mathematician of 18th century Leonhard Euler. He thought differently to determine the value of pi from sin x/x. He factorised sin x in infinite product.

sin x/x = {1- (x/pi)^2}.{1- (x/2pi)^2}.{1- (x/3pi)^2}…………. up to infinity………………..(3)

Sin x can also be written in series expansion as x – x^3! + x^5!-…………. up to infinity.

Or sin x/x = 1- x^2/3! + x^3/5!-…………………………………………………….(4)

Comparing coefficient of x^2 of equations (3) and (4),

1/pi^2.[1/1 + ½^2 + 1/3^2+………]= 1/3!

Or pi^2= 6.[1/1 + ½^2 + 1/3^2+………]

Another English mathematician John Wallis thought differently from Euler and Viete, he, in stead, straight way considered equation (3) of sin x/x and put x = pi/2. That gave the value of pi as

2/pi = (1-1/4).(1- 1/16).(1-1/36).(1-1/64)……………

Or pi = 2/1.4/3. 16/15. 36/35. 64/63……………..

Thus it can be summarised as equation of a circle in trigonometric ratios is pi = n.sin pi/n where n is number of sides of regular polygon. Also sin x/x where x = pi/n, when equals 1 then n tends to infinity. If it does not equal 1, it means, n has finite value and sin x/x has also value other than 1. Thus (sin x)/x is an equation that contains value of pi. Or in other words pi is determinable from (sin x)/x.

Different ways have been adopted by mathematician to calculate pi from (sin x)/x and that had given different formulae of pi. If an attempt is made to find (sin x)/x, new formula will originate. Is it not then worth trying to find some other way to express (sin x)/x?

I submit, mathematicians are not restricted to sin x divided by x to determine pi, statistical methods, probability theory and many more have already been introduced and pi freshly calculated. Still more methods will be put forth because mathematicians think differently.

References:

1. Viète* For further reading, ‘When Law Met Mathematics’ at https://narinderkw.wordpress.com/2017/08/07/when-law-met-mathematics/

2. https://en.m.wikipedia.org/wiki/Carl_Friedrich_Gauss

3. http://felix.physics.sunysb.edu/~allen/Jokes/bohr.html

4. https://en.m.wikipedia.org/wiki/François_Viète

5. http://www2.mae.ufl.edu/~uhk/EULER-SINE.pdf

6. Figure courtesy https://www.quora.com/Find-the-perimeter-of-an-equilateral-triangle-of-side-l-cm-is-inscribed-in-a-circle-with-radius-r-cm

7 Cover photo courtesy artist Domenico Fetti, Italian Painter, current location Gemaldegalerie Alte Meister, Source/Photographer http://archimedes2.mpiwg-berlin.mpg.de/archimedes_templates/popup.htm

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About author:

Writer is an 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.