Pi is said to be discovered by ancients

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In summary, Pi is an irrational number that is infinite in length and its decimal expansion does not have a repeating pattern. The ancients' determination of its value was just an approximation, as is ours. It is difficult to prove precisely using geometry that all circles are similar, but one way is to imagine a circle as a regular polygon with an infinite number of sides. Another way is to use the definition of similarity in terms of scaling.
  • #1
JamesU
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Pi is said to be discovered by ancients, but if even we don't know the EXACT number of pi, how could they, or was theirs just a close estimate of pi?
 
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  • #2
Pi is just the constant defined by the circumference over the diameter. Recognizing that the ratio doesn't change between circles qualifies as the discovery of pi. It's an interesting exercise. Also try to show the same constant relates the radius to the area. And be careful that you're arguments aren't circular (poor pun intended). You can't use any theorems that depend on what you're trying to show.

Looking at the discoveries of the ancients often makes you feel like the lady who walks out of one of Shakespeare's plays and say "I don't get what the big deal is. All he did was string a bunch of famous cliches together." Some of what they did seems simple but try to be the first guy to do it without modern machinery.
 
  • #3
Good point, I never thought opf it that way
 
  • #4
Pi is an irrational number. This means that it is non-repeating, non-terminating. In other words, it is infinite in length and, unlike 1/3 = 0.33333..., it does not have some infinitely repeating pattern of digits in its decimal expansion. So, in short, the ancients determination of the value of pi was, just as ours is, just an approximation. No one may know exactly the value of pi. But, for all practical purposes, an approximation is sufficient.
 
  • #5
How would one prove precisely (using geometry) that all circles are similar? The only way that I can think of is to imagine a circle as a regular polygon with an infinite number of sides. Since it's easy to prove that any two n-sided regular polygons are similar, it should therefore extend to circles. However, this isn't very rigorous. Anyone know of a way to prove it rigorously using classical geometry?
 
  • #6
Manchot said:
How would one prove precisely (using geometry) that all circles are similar? The only way that I can think of is to imagine a circle as a regular polygon with an infinite number of sides. Since it's easy to prove that any two n-sided regular polygons are similar, it should therefore extend to circles. However, this isn't very rigorous. Anyone know of a way to prove it rigorously using classical geometry?
If you accept a definition of similarity in terms of scaling :

1. translate the center, c1 to new center c2
2. scale by a factor r2/r1 about center (since every point is an equal distance from the center, scaling the radius by a factor x scales the distance of every point from the center by this same factor)

As for the OP : http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html
 

Related to Pi is said to be discovered by ancients

1. What is Pi and why is it important?

Pi is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is approximately equal to 3.14159 and is used in many mathematical and scientific calculations involving circles and spheres.

2. Who discovered Pi?

The concept of Pi has been studied and used by many ancient civilizations, including the Egyptians, Babylonians, and Chinese. However, the first known calculation of Pi as 3.1416 was done by the ancient Greek mathematician Archimedes around 250 BC.

3. How did the ancients discover Pi?

The ancients discovered Pi through various methods, including measuring the circumference and diameter of circles, inscribing polygons inside circles, and using geometric constructions. They also used approximations and ratios to calculate Pi.

4. How accurate were the ancients' calculations of Pi?

The accuracy of the ancients' calculations of Pi varied depending on the method and tools they used. The ancient Egyptians were able to calculate Pi with an accuracy of up to 3.16, while the ancient Chinese had an accuracy of up to 3.125. The most accurate calculation of Pi by the ancients was done by Archimedes, who was able to approximate it to 3.1416, which is accurate to four decimal places.

5. How is Pi used in modern science and technology?

Pi is used in many modern scientific and technological applications, including engineering, physics, and astronomy. It is also used in computer science for calculations involving circles and spheres. Pi has also been used to test the accuracy of supercomputers and to break records for calculating the most digits of Pi.

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