Begin2.DVI

Special Numbers

Rational Numbers

All those numbers having the form p/q, where pand q are integers and q is

understood to be different from zero, are called rational numbers.

Irrational Numbers

Those numbers that cannot be written as the ratio of two numbers are called

irrational numbers.

The Numberπ

The Greek letterπ(pronounced pi) is an irrational number and can be defined

as the limiting sum^1 of the infinite series

π= 4

(

1 −^13 +^15 −^17 +^19 − 111 +···+(−1)

n

2 n+ 1+···

)

Using a computer one can verify that the numerical value ofπto 50 decimal places

is given by

π= 3. 1415926535897932384626433832795028841971693993751 ...

The numberπ has the physical significance of representing the circumferenceC of

a circle divided by its diameterD. The symbolπfor the ratioC/Dwas introduced

by William Jones (1675-1749), a Welsh mathematician. It became a standard nota-

tion for representingC/Dafter Euler also started using the symbolπfor this ratio

sometime around 1737.

The Numbere

The limiting sum

1 +^1

2!

+^1

3!

+···+^1

n!

+···

is an irrational number which by agreement called the numbere. Using a computer

this number, to 50 decimal places, has the numerical value

e= 2. 71828182845904523536028747135266249775724709369996 ...

The number e is referred to as the base of the natural logarithm and the function

f(x) =exis called the exponential function.

(^1) Limits are very important in the study of calculus.