In mathmatics, how is the number of pi derived?

Question

Providing a mathmatic proof is not necessary, but the answer should still be justified and mathmatically valid.

Answer 1

Simply put, a string can be used to go around the circle. Mark where the first end meets the other then measure on a ruler. Record that. Then u measure the diameter ( the line that parts in two equal parts). Divide the diameter into the circumference and u r supposed to get 3.14 or close to that which is pi. So pi is the ratio of the diameter to circumference. Simply put!!!

Answer 2

After a lot of research, mathematicians discovered that the ratio between the diameter and the circumference of any circle is constant i.e. 22/7 & it is called as "pi".

Answer 3

The value of π has been known in some form since antiquity. As early as the 19th century BC, Babylonian mathematicians were using π=25/8, which is within 0.5% of the true value.

The Egyptian scribe Ahmes wrote the oldest known text to give an approximate value for π, citing a Middle Kingdom papyrus, corresponding to a value of 256 divided by 81 or 3.160.

It is sometimes claimed that the Bible states that π=3, based on a passage in 1 Kings 7:23 giving measurements for a round basin as having a 10 cubit diameter and a 30 cubit circumference. Rabbi Nehemiah explained this by the diameter being from outside to outside while the circumference was the inner brim; but it may suffice that the measurements are given in round numbers. Also, the basin may not have been exactly circular.
Principle of Archimedes' method to approximate π.
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Principle of Archimedes' method to approximate π.

Archimedes of Syracuse discovered, by considering the perimeters of 96-sided polygons inscribing a circle and inscribed by it, that π is between 223/71 and 22/7. The average of these two values is roughly 3.1419.

The Chinese mathematician Liu Hui computed π to 3.141014 (good to three decimal places) in AD 263 and suggested that 3.14 was a good approximation.

The Indian mathematician and astronomer Aryabhata in the 5th century gave the approximation π = 62832/20000 = 3.1416, correct when rounded off to four decimal places.

The Chinese mathematician and astronomer Zu Chongzhi computed π to be between 3.1415926 and 3.1415927 and gave two approximations of π, 355/113 and 22/7, in the 5th century.

The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of π/4 into the form

π = √12 (1 - 1/(33) + 1/(532) - 1/(733) + ...

and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3.14159265359. By adding a remainder term to the original power series of π/4, he was able to compute π to an accuracy of 13 decimal places.

The astronomer Ghyath ad-din Jamshid Kashani (1350-1439) correctly computed π to 9 digits in the base of 60, which is equivalent to 16 decimal digits as:

2π = 6.2831853071795865

The German mathematician Ludolph van Ceulen in 1615 computed the first 32 decimal places of π. He was so proud of this accomplishment that he had them inscribed on his tombstone.

In 1789, the Slovene mathematician Jurij Vega improved John Machin's formula from 1706 and calculated the first 140 decimal places for π of which the first 126 were correct [1] and held the world record for 52 years until 1841, when William Rutherford calculated 208 decimal places of which the first 152 were correct.

The English amateur mathematician William Shanks, a man of independent means, spent over 20 years calculating π to 707 decimal places (accomplished in 1873). In 1944, D. F. Ferguson found (with the aid of a mechanical desk calculator) that Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were fallacious.

Answer 4

I have measured pi with a group of 12-13 year olds to great accuracy

I asked them to measure out circle diameters and their circumference of all different sizes

then i asked them to repeat for 5 or 6 different shapes each.

then find the mean

then I collated all there means and came up with a class mean

i did this with two classes of pupils (60 pupils in all)

the mean of both classes value of pi came out to precisely 3.14 to 2DP

so taking the mean of both classes means actually honed down the answer to the precise value of pi as the Greeks told us

amazing eh?

some stories just cant be made up

Answer 5

by definition, it is the ratio of the circumference to the diameter of a circle.

due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions - therefore numerical calculations must use approximations of π.

Answer 6

Seies enlargement is the proper way. you're able to desire to get a chain that converges rapidly. for example: pi/4 = [one million/2-one million/3(one million/2)^3+one million/5 (one million/2)^5 -...] +[one million/3-one million/3)^3 +one million/5(one million/3)^5 -..]. Logarithms , sin, cos, tan and e are all computed by utilising sequence enlargement.

Answer 7

It is the ratio of twice the circumference of a circle to it's radius

Answer 8

well Circumfrence=pi*D so to get pi you need the circumfrence and divide by the diameter, i'm not sure how they get the circumfrence in the first place, but you're at least one step closer to solving the problem ;)

Answer 9

it's the ratio to of the diameter of a circle to it's circumfrenece. It is a constant non variable to all circles.

Answer 10

it is done by computers these days. They have calculated pi to millions of decimal places.