International π Day

March 14th (3.14)

Pi is the ratio of a circle’s circumference to its diameter i.e.   

    \[ \pi=\frac{C}{d}\]

The value of π (pi) has been known for almost 4000 years. [If we calculated the number of seconds in 4000 years and worked out the value of π to that number of decimal places, it would still only be an approximation for the actual value of π.]

One Babylonian tablet (ca 1900-1680 BCE) has a value π = 3.125.

The “Rhind Papyrus” (ca 1650 BCE) gives an insight to Egyptian Mathematics and they came up with a value of π = 3.1605.

The first calculation of π was completed by Archimedes of Syracuse (287-212 BCE). His method came up with a value that π was somewhere between 3\frac{1}{7}  and 3\frac{10}{71}. The value of 3\frac{1}{7} =\frac{22}{7} was commonly used in schools before calculators were in widespread use. (It explains why the radius and diameter of circles were always given as multiples of 7.)

Zu Chongzhi (429-501) from China, calculated the value as \pi=\frac{355}{113} (not very nice to work with so it didn’t catch on).

The symbol that we use to represent π was introduced by the Welshman William Jones (Ynys Mon, 1675-1749) in 1706. Initially, it was not popular.

The symbol was popularized by Leonhard Euler (Swiss, 1707-1783) who adopted it in 1737.

In the eighteenth-century, a French mathematician named Georges Buffon (1707-1788) devised a way to calculate π based on probability (see Buffon’s needle).

Π is defined as an irrational number which means that you cannot write it as an exact fraction (top heavy in this case). This means that it is an infinite non-recurring decimal number.

How many digits do we need for calculations?

In school we tend to work with 3 digits most of the time (3.14).

NASA only uses 15 digits for π in its calculations for sending rockets into space. (See the film “Hidden Figures” about the life of Katherine Johnson.)

To measure the universe to an accuracy that is less than the size of an atom you would only need to use the first 40 digits of π.

These last three points does make you ask why we bother to investigate the accuracy of π any further.

Teacher Resources on Line (cleavebooks.co.uk)

Some notable dates in the development of π are in the table below.

DateDigitsComment
≈2000 BCE4Babylonian
≈800 BCE7 
480 CE8 
140010 
159620 
1706100Symbol introduced
1853261 
1853440 
1946620Most by hand
Jan 1947710First by desk calculator
19491120Last by desk calculator
19492037First by computer – 70 hours
Jan 195810,0001.7 hours
1961100,2658.7 hours
1967500,00028 hours
19731,001,25023.3 hours
19812,000,036 
Jan 1987134,214,700 
Aug 19891,011,196,691First to a billion
29th Apr 20092,576,980,377,52429.09 hours
28th Dec 201312,100,000,000,05094 days
14th Aug 202162,831,853,071,796108 days