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Limits

Approximating π

The Formula, Solution

Differentiation

The Derivative at a Point

The Derivative as a Function

Polynomials

Sine and Cosine

Exponential Functions

Logarithmic Function

Demonstrations

Approximating `π`

Archimedes' `π` (≈250 BC)

Perimeter of the Inscribed Polygon < Circumference of the Circle < Perimeter of the Circumscribed Polygon

Archimedes used a 96-sided polygon to get formula

It is easy to use this method by using trigonometry. Archimedes however only used geometry, and Greek numerals!

Liu Hui's `π` (≈250 AD)

Liu Hui used areas and realized that:

He used the fact that if you know the side in a regular n-gon, you can find the side in a regular 2n-gon. (Just as Archimedes he didn’t use trigonometry and he didn’t have decimal numbers)

Liu Hui's method

This isn’t the entire method; it is just a few details. (Google on it for more information)

Start with a circle with radius r and a n-gon with a known side s₀.

Find the side of the 2n-gon, s₁, in terms of r and s₀.

The Recursive Formula

Find the side of the hexagon in terms of the radius r
Pick a radius of your own choice. Let s₀ be the length of a side in a regular hexagon, s₁ be the length of a side in a regular dodecahedron, s₂ the length of a side in a regular 24-gon, and so on. Find a recursive formula for s_n.
Use the recursive formula obtained and a spreadsheet to approximate π. It’s easier to use the circumference than the area.

And then what?

The reason Liu Hui used areas instead of circumferences was that he found a clever way of approximating the area of a polygon with a rational number, thus avoiding having to taking successive square roots.

There is no known record of any approximation of π prior to Archimedes. Those claiming there is, do not refer to any known references.

Archimedes used rational numbers as approximation of square roots. He approximated formula correct to 5 significant figures. There is no record of how he did it.

Archimedes (≈250 BC) used a 96-gon.

Liu Hui (≈250 AD) used a 3072-gon.

Zu Chongzhi (≈500 AD) used a 12 288-gon to obtain π ≈3.1415962, correct to 8 significant figures.

↓ 900 years later

Madhava of Sanganagrama (≈1400 AD) obtained 10 correct significant figures by using a series later called the Leibniz' series.

Leibniz discovered this series ≈1700 AD.

↑ 300 years later