Incidentally, we are apt to carelessly think things in 'addition';
however, in fact, most of events in the natural world vary in 'multiplication
= with multipliers'. In other words phenomena in the natural world
vary not in linear lines or with velocity but in curved lines or
with acceleration such as parabolas and hyperbolas.
Now the 'e', 'I' and '', which appear in the 'Euler's Equation',
are keys to the 'wonder in the natural world'.
The 'e' is called the 'Napier's Constant' after its finder, however
it was Euler who made the secret of the 'e' clear, and it is mathematically
called the 'base of natural logarithm'. Apart from such mathematical
details, the 'e' plays an important role when we describe the changes
of various phenomena in the natural world.
Unless we can describe the movement in the natural world, I don't
say it is entirely impossible, but various types of inconvenience
arise when we forecast the weather or fly an airplane well.
The '' is the 'circular constant' as you know it very well. While
the discovery or determination of the 'circular constant' has a
long past from the Pharaonic era, I have to raise my hat to the
efforts of geniuses who have tried to convert a 'circle', one of
the 'perfect curved lines = nature, into a 'straight line = human
knowledge' at all costs.
Everybody can draw a 'circle' on the ground, the word 'Geometry'
was created combining 'ground (Geo)' and 'measuring (Metric)'; however
the '' becomes indispensable when we want to match a circle to
a building composed of a straight line. After all, the sun and the
earth as well is round.
The 'i' is an imaginary figure which is a fictitious one existing
in a brain only, while a 'real figure' which can practically be
used for counting an apple, two apples and so on by touching or
seeing them.
As it is a fictitious figure, it does not seem to be directly related
to us. On the contrary the 'i' exists close by our life. For instance,
the 'i' always appears in the structural calculation of a reinforced
concrete building and makes it possible to construct a safe building
on the earth.
The greatness of the 'Euler's Equation' is that it has admirably
combined entities which seem to be a fanciful 'sign' only with the
most fundamental figures which we can realise immediately, i.e.
'1 = the minimum unit of existence' and '0 = inexistence'. Saying
'1 and 0', they also compose a 'digital world' which is indispensable
to a computer.
Ms. Ogawa Yoko beautifully depicts such great but charming 'figures'
as being lurking in the natural world in her 'The equation the doctor
loved'. I strongly recommend you to read it.
The attached illustration is superfamous 'The back of waves off
the coast of Kanagawa' by Hokusai. His greatness is that he intuitively
detected the 'e', '' and 'i', and made them up into a skilful ukiyoe.
Was Hokusai a 'transcendental mathematician' or was Euler a 'supereminent
artist'? Probably they were both.
