3.8 Important constants

1 Introduction 2 Mathematics is not only about computing 3 Basic concepts 3.1 Note on mathematical notation 3.2 Sets and set operations 3.3 Number sets 3.4 Significant subsets of real numbers 3.5 Propositions and logical connectives 3.6 Abbreviated writing of sums and products 3.7 Factorial and binomial coefficient 3.8 Important constants 4 Elementary functions 5 Analytical geometry 6 Warning 7 List of the used symbols Index Bibliography

In applications we often encounter the need to use Euler's number $e$ and Ludolph's number $π$ . Approximate values of these constants with precision of one thousand decimal places are given below.

\begin{aligned} π \approx 3. & 14159265358979323846264338327950288419716939937510582097494459230781 \\ 64062862089986280348253421170679821480865132823066470938446095505822 \\ 31725359408128481117450284102701938521105559644622948954930381964428 \\ 81097566593344612847564823378678316527120190914564856692346034861045 \\ 43266482133936072602491412737245870066063155881748815209209628292540 \\ 91715364367892590360011330530548820466521384146951941511609433057270 \\ 36575959195309218611738193261179310511854807446237996274956735188575 \\ 27248912279381830119491298336733624406566430860213949463952247371907 \\ 02179860943702770539217176293176752384674818467669405132000568127145 \\ 26356082778577134275778960917363717872146844090122495343014654958537 \\ 10507922796892589235420199561121290219608640344181598136297747713099 \\ 60518707211349999998372978049951059731732816096318595024459455346908 \\ 30264252230825334468503526193118817101000313783875288658753320838142 \\ 06171776691473035982534904287554687311595628638823537875937519577818 \\ 57780532171226806613001927876611195909216420199 \dots \end{aligned}

\begin{aligned} e \approx 2. & 71828182845904523536028747135266249775724709369995957496696762772407 \\ 66303535475945713821785251664274274663919320030599218174135966290435 \\ 72900334295260595630738132328627943490763233829880753195251019011573 \\ 83418793070215408914993488416750924476146066808226480016847741185374 \\ 23454424371075390777449920695517027618386062613313845830007520449338 \\ 26560297606737113200709328709127443747047230696977209310141692836819 \\ 02551510865746377211125238978442505695369677078544996996794686445490 \\ 59879316368892300987931277361782154249992295763514822082698951936680 \\ 33182528869398496465105820939239829488793320362509443117301238197068 \\ 41614039701983767932068328237646480429531180232878250981945581530175 \\ 67173613320698112509961818815930416903515988885193458072738667385894 \\ 22879228499892086805825749279610484198444363463244968487560233624827 \\ 04197862320900216099023530436994184914631409343173814364054625315209 \\ 61836908887070167683964243781405927145635490613031072085103837505101 \\ 15747704171898610687396965521267154688957035035 \dots \end{aligned}

The definition of Euler's number will be discussed in detail in BIE-ZMA. It is not necessary to stress out the importance of $π$ . One application of $e$ is related to its being used as the base of the natural logarithm that we will be discussing in section 4.10.

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