Happy π-Day: What is Pi?

What is π, or “pi” as we spell it out. The mathematical definition is the ratio of a circle's circumference to its diameter:

\[ π = { C \over d } \]

It is one of many mathematical constant. Commonly reffered to the value 3.14, which is inaccurate to say it mildly. Contrary to popular belief, π wasn’t discovered by a greek philosopher. Some of its earliest references are from Babylon and ancient Egypts, accurately enough calculated at the time. A babylonian clay tablet dated 1900-1600 BC used the following approximation:

\[ π = { 25 \over 8 } = 3.125 \]

In the Egyptian Rhind Papyrus dated around 1650 BC, which is a copy from another papyrus dated to 1850 BC, the formula is treated as:

\[ π = ({ 16 \over 9 })^2 ≈ 3.16 \]

These approximations was close enough at the time.

The knowledge about this had, probably through trade, reached China by mid-first millennium, where its accuracy had been calculated to seven decimal places.

Now let us head over to Grece and fast forward a little, approximately to the year 250 BC. The greek mathematician Archimedes created an algorithm to calculate π through the use of polygons. Basically draw a circle with diameter 1, draw a square around it where the edges touches but not intersects circumference, draw another square inside the circle where the corners touches the circumference. Ok, this didn’t give a good answer, but it give you an idea of how the game goes. By comparing the perimeters of the polygons he could find that π was in between these numbers. Archimedes added sides to this polygon until he reached a 96-sided regular polygon, and calculated the perimeters of these polygons:

\[ { 223 \over 71 } < π < { 22 \over 7 } \]

In decimal numbers:

\[ 3.1408 < π < 3.1429 \]

This stood the test as the true value of the constant for a very long time. π was hard to calculate any further before mechanical and later electronic calculators came in as aid. Since any other approach to solve π except the polygonic algorithm approach required advanced calculations to the nth decimal.

In decimal number system, this is the best representation I have been able to find: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 ...

Because of the size of the number, let us present it in hexadecimal (base 16) 3.243F 6A88 85A3 08D3 1319 ...

Some ancient cultures used a sexogesimal number system (base 60), the number π would then be represented as 3;8,29,44,0,47. Mark that each of the decimal numbers would have its own symbol.

Happy π-day.

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