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Pi has fascinated scholars for ever

The value for the ratio of the circumference of a circle to its diameter has fascinated scholars for several thousand years.  This ratio has been termed pi (symbolized by the Greek letter π.  Over 5000 thousand years ago, Babylonians gave it a value of 25/8 or 3 and 1/8 while Bible lists its value as 3 (http://www.csun.edu/~hbund408/math%20history/math.htm), schools today commonly use the ratio 22/7 which was given as an estimate in 1246, Qin Jiushao (another Chinese mathematician). Today we all know that π is an irrational number which means it cannot be written as an exact ratio of two integers.  All these ratios are only estimates.

Liu Hui’s Pi

In the year 263 AD a Chinese mathematician named Liu Hui derived the value of this constant. He determined the value of π based on the thinking that a circle is an infinite sided regular polygon.  He did his calculations up to 96 sided regular polygons.  Based on this work he calculated:

π = 3 + 1/10 +4/100 + 1/1000 + 6/10000 or

π = 3 + 1/101 +4/102 + 1/103 +6/104

John Napier and decimals

As you can see, this was a complicated way of writing such a number. John Napier, a Scottish mathematician standardized a different way of writing such numbers in 1619. This became the basis of the modern decimal point system.  In this system, a decimal point was used after the whole number. The place value of the number after the decimal point determined its value.  Here are some examples.

3.1 = 3 + 1/10

3.14 = 3 + 1/10 + 4/100

3.141 = 3 + 1/10 + 4/100 + 1/1000

Now, the equation π = 3 + 1/10 +4/100 + 1/1000 +6/10000 could be written as π = 3.1416

The exact value of the constant π cannot be determined.  One can get only estimates.  Today with the help of modern computers this estimate is known to 100,000 decimal places.  Can you imagine writing this number as sum of fractions of multiples 10? Here is this value with only 10 digits in it.  How would you prefer to write it as

π = 3 + 1/10 +4/100 + 1/1000 +5/10000 + 9/100000 + 2/1000000 + 6/10000000 5/10000000 + 3/100000000 + 5/1000000000,

or π = 3.1415926535?  You decide.

By the way, the numbers  with the decimal points can be added, subtracted, multiplied or divided with the same principles as the integers you have learned.  Most fractions can also be converted to decimal point numbers just as most decimal point numbers can be converted to ordinary fractions.

Challenge

Another irrational number that you may learn about in high school or in college is called the Euler’s constant (e).  It can only be estimated. Here is an estimate of e to 10 decimal places: 2.7182818284.

Use this value of e to determine the values for 10 x e, 100 x e, 1000 x e , 0.1 x e,      and 0.01 x e.

Solution:

e = 2.7182818284, therefore 10 x e = 27.182818284

e = 2.7182818284, therefore 100 x e = 271.82818284

e = 2.7182818284, therefore 1000 x e = 2718.2818284

e = 2.7182818284, therefore 0.1 x e = 0.27182818284

e = 2.7182818284, therefore 0.01 x e = 0.027182818284