History of Ram Gautam Loras College

Professor: Dr. Angela K.

History and Progress of

History and Progress of continue….

Calculating the value of before and after calculus era:

Calculate of π before calculus era:

• When Newton and Leibnitz discovered calculus in 17th century, this helped to find new formula to find the value if π as:

…….

Substituting x=1 from Gregory-Leibnitz formula (1671-74)

• …….Using Gregory Series the value of pi is calculated as,

…….). Hence the value of pi is calculated as 3.141592453589793238464643383279502784197305820…….

Calculate of π during calculus era:• The value of π in modern area is given from the Euler’s trigonometric identity,

• which produces a geometrically convergent rational series,

•

• The above equation can be written as follows which is given by John Machin. )

The value of π can easily be calculated using machine.

Formulas used to compute the value of :

Conclusion

The goal of computing in different time periods is to check the consistency in characteristics and value of pi . Hence different mathematicians have given their own idea to compute the value and characteristics of. Today all the mathematicians have reached to a common point that is an irrational and transcendence number as well as algebraic- the root of a polynomial with integer coefficients. Since is applied in various field of study from mathematics and engineering to farming it is necessary to calculate the value of and assure that calculated value is consistent with the change of time.

Works Cited

• Peter, Beckman, A History of PI, Barnes & Noble, Inc., New York, 1971• Borwein M. Jonathan, The Life of Pi: From Archimedes to Eniac and Beyond, Australian

Research Council., 2011.• Allen G., Doland, Pi A Brief History, Texas A&M University College Station, Texas.• Dunham, William, Journey Through Genius, Penguin Books, 1991• http://egyptonline.tripod.com/history.htm• http://mercury.educ.kent.edu/database/eureka/documents/DiscoveringPiCOPs.pdf• Wilson, David, The History of PI, History of Mathematics, Rutgers, 2000• http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80

Introduction

The Greek letter is the symbol used by mathematician to represent the ratio of a circle’s circumference to its diameter which can’t be expressed as a fraction. is a

number, starting with 3.1415926535… ad infinitum; a very common

approximation is 3.14. is an irrational number. Approximate value of in

fractions, decimal, binary, hexadecimal, and sexagesimal includes:

• Fraction: , , , , …….

• Decimal: The first 100 decimal digits are 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 ....

• Binary: 11.001001000011111101101010100010001000010110100011 ....

• Hexadecimal: The base 16 approximation to 20 digits is 3.243F6A8885A308D31319 ....

• Sexagesimal: A base 60 approximation is 3;8,29,44,1

Uses of in physics:

• cosmological constant:

• Heisenberg's uncertainty principle:

• Einstein's field equation of general relativity:

• Coulomb's law for the electric force:

• Magnetic permeability of free space:

• Period of a simple pendulum with small amplitude:

Where letters have their usual meaning.

Uses of in Mathematics:

• Where C is the circumference of a circle, r is the radius and d is the diameter.

• Where A is the area of a circle and r is the radius.

• Where V is the volume of a sphere and r is the radius.

• Where SA is the surface area of a sphere and r is the radius.

• The initial idea of was obtained from the bible which says, “Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about” (Old Testament, 1 Kings 7:23).

• Mathematical Derivation of π given by Babylonians:Babylonians used hexagon to find the value of.

They approximated that the perimeter of a hexagon

is equal to the six times the radius of the circumscribe

d circle and hence they chose to divide the circle into

360 degrees. Therefore, the ratio can be used

where C is circumference of circle and r is the radius. The

circumference of a circle is given as. Using, we have,

Egyptians gave the value of π as. Ahmes assumed that the area of a circular field with a diameter of 9 units is the same as the area of a square with a side of units. But the area of circle is which yields

And hence the Egyptian value of was derived.

• Wallis :

• Machin :

• Ferguson :

• Euler :

• Euler :

• Euler :

• Borwein and Borwein :

• Chinese gave that. In 264 A.D. Liu Hui used a variation of Archimedean

inscribed polygon to estimate the value of. He used the polygon of 192 sides and found

that and with a polygon of 3,072 sides he found that.

Archimedes’ method of calculation • Archimedes was the first one who gave a method of calculating to any desired degree of accuracy. “It

was based on the fact that the perimeter of a regular polygon of sides inscribed in a circle is smaller than circumference of the circle, whereas the perimeter of a similar polygon circumscribed about the circle is greater than its circumference” ( Beckman, History of Pi, 64).

He gave that Let be the angle subtended

by one side of a regular polygon at the center of the circle, then if the

length of the inscribed side is

and Therefore it can be said that

Now dividing by, Suppose the original number of sides is double times and we will get, Suppose is

arbitrarily large then the lower and upper bounds of will be arbitrarily close. Archimedes used Archimedean spiral to get the upper and lower bounds. From the measurement of circle following result is obtained:

Newton's approximation of

Newton computed up to 15 digits using the formula Newton began with a semicircle having its center C at and radius , as shown in the figure below: Using the circle’s equation:Solving for y gives the equation of the

upper semicircle as, . Using the binomial expression gives the

following equation of the semi-circle as:

He assumed B at the point (1/4, 0) as indicated

in Fig. and drew BD perpendicular to the semicircle’s diameter AE. He then worked with the shaded area ABD by fluxions using the rules of De Analysis to find the values of. According to Newton, the area ABD was . He also told that, "If the value of y be made up of several terms, the area likewise shall be made up of the areas which result from every one of the terms."

Using the same rules of the De Analysis the shaded area can be written as:

Simplifying above equation at gives the following result;

and and so on. Thus the shaded area (ABD) gives the first nine terms as an approximation of 0.07677310678. Thus solving gives the value of up to 15 digits.

AcknowledgmentsDr. Angela L. Kohlhaas

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