e = 2. · PDF filee = 2.718281828459 . . . 1­4 The number e, like the number π, appears in many different contexts in mathematics. Especially in the context of the very basic

e = 2.718 . . .

1­E Precalculus

IntroductionIntroduction

http://www.flickr.com/photos/andreapreda/2571391498/

Imagine  the  following  situation:   We  lend  some  money  to  somebody,say 1 Euro,  and  we  manage  that  he  agrees  to  100 %  interest  per  year.After  a  year  we  will have  1 + 1 = 2  Euro.

If  we  are  more  clever,  we  can  try  to  get  agreement  on  50%  interest,but  now  for  each  half  year.  That  sounds  as  being  the same,  half  theinterest,  but  to  be  payed  twice  a  year.  But  it  is  not  the  same.

1­1 Precalculus

http://www.n24.de/media/_fotos/8verbraucher/2008_1/juli_3/Zinsen-dpa-gr.jpg

After  six  months  we  have                Euro , and  again  six  months  later   1 12

our  possession  increased  again  by  a  factor                ,  i.e.  after  the year  we  have                               Euro. 

1 12

We  get  even  more,  if  the  interest                contributes  three  times  per  year:    3313

%

1 13

3

= 2.37 Euro.

1­2

1 12

2

= 2.25

Precalculus

IntroductionIntroduction

Is it possible to make this way in one year a fortune?

No, it is not.

n = 1, 1 1n

n

= 2

n = 10, 1 1n

n

= 2.59374

n = 100, 1 1n

n

= 2.70481

n = 1000, 1 1n

n

= 2.71692

n = 10000, 1 1n

n

= 2.71815

The reason is, that the expression

1 1n

n

has a finite limit for n versus infinity.

1­3 Precalculus

IntroductionIntroduction

The  number  The  number  ee

This limit is

e = 2.718281828459 . . .

1­4

The number e, like the number ,π appears in many differentcontexts in mathematics. Especially in the context of the verybasic question, how fast something is changing.

Precalculus

FindingsFindings

http://www.beastcoins.com/RomanImperial/II/Hadrian/Hadrian-NIR-COSIII-Crescent7Stars-Eastern.jpg

For centuries, scholars assigned to prehistorical objects a rel-ative time scale only. It was based on the simple observation,that archaeologist go back in time when digging deeper anddeeper. But what about the absolute age of some old bonesfor example?

2­1 Precalculus

Traces  of  dinosaursTraces  of  dinosaurs

http://www.monstersandcritics.de/downloads/downloads/articles3/152817/article_images/dino1.jpg

Since the sixties, age determinations are possible by means of radiocarbondating. The relevant formula is

A t = A0 e− t ,

where A(t)  is the radioactivity of the sample, t is the time in years sincethe death of the animal, and λ is the decay constant of C14.

2­2 Precalculus

Height  of  frothHeight  of  froth

3­1

http://www.elektronik-kompendium.de/public/schaerer/bilder/salgreib.jpg

One can check experimentally that the hight of froth in a glassdecreases exponentially with time. A measurement resulted in

h = h0 e−k t , h0 = 6 cm , k = 0.022 s−1

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

4­1

The number e plays an important role in context of instabilities

http://img.fotocommunity.com/photos/9011788.jpg

Precalculus

Imagine for example a small drop falling into a dish of milk withsputtering milk. First the drop is more or less ball-shaped. Due tothis initial symmetry one may expect that the surface of the milkreacts symmetrically.

4­2

http://gallery.photo.net/photo/10040340-lg.jpg

Fig.  1­1:  A  drop  dropping  into  a  dish  of  milk

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

4­3

http://gallery.photo.net/photo/6312824-lg.jpg

http:///blog.photosandponderings.com/wp-content/uploads/2009/04/dsc_6757.jpg

Indeed, this is what one sees first: the surface rises at the point of the collision forming a smooth circular ring. Its thin wall is slightlybent outwards.

Fig.  1­2:  Reaction  of  the  milky  surface  to  the  clash

Shortly afterward, without obvious reason, valleys and tops show up atthe rim of the thin wall. It looks then like a little crown. The prongsrapidly get longer, and finally they eject a tiny droplet of milk.

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

4­4

http://www.ljclark.com/seeing/Images/MilkDropW2.jpg

Fig.  1­3:  Tiny  droplets  of  milk  are  ejected

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

4­5a

We show once more how an instability develops in a glass of milk.

Fig.  2­1:  The  drop  falls  into  a  glass  of  milk

http://www.nies.ch/misc/index.de.php/image200509-d70-5633.php

Instabilities  in  a  glass  of  milkInstabilities  in  a  glass  of  milk

Precalculus

4­5bFig.  2­2:  Surface  of  milk  at  the  clash 

Http://www.nies.ch/misc/index.de.php/image200509-d70-5628.php, http://www.nies.ch/misc/image200509-d70-5673.jpg

Precalculus

Instabilities  in  a  glass  of  milkInstabilities  in  a  glass  of  milk

4­6

Why such a pattern of prongs?

Why do the thin walls of milk not keep their circular symmetry?

http://www.scantips.com/speed.html

In principle it is possible that the wall grows thereby preservingits symmetry, but such development is very unlikely. There are al-ways small disturbances which rapidly increase and lead in the end to very different shapes.

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

4­7

The  original  symmetric  movement  is  unstable,   like  balancing  a  pencilon  its  tip.  The  important  point  is,  that  when  the  instability  sets  in, theincrease  of the  disturbance  is  proportional  to  the  already  existing  distur­bance.  It  is  here,  where  the  number  e   is  back  in  the  game.

Precalculus

Number  e  and  InstabilitiesNumber  e  and  Instabilities

e = 2.718 . . .

Precalculus