Expert: Paul Soderman - 2/3/2010

Question
QUESTION: what is the meaning of Euler number and what is its application and why Euler get this number and how //did it have a relation with the pi number // i asked an expert in the math section but i want the answer in the point of view of physics rather than mathematics i.e the sens

ANSWER: Euler was a Swiss mathematician who developed the Euler equation that relates forces in a fluid (shear stress, pressures, weight) to the velocity gradients (time and position).  This was done using Newton's law that equates force to mass times acceleration.  Along the way, Euler and others developed non-dimensional analysis, which is a mathematical technique for formulating various physical equations of state using non-dimensional quantities that are deemed to be important.  For example, the momentum equation in fluid mechanics can be written in terms of a differential equation with terms multiplied by non-dimensional parameters such as m/rVL (Reynolds number) and p/(rho V^2).  The last term is the Euler number. I'm not sure if he derived it or if it is named after him because of his pioneering work in fluid mechanics.  The Euler number is the non-dimensional ratio of pressure to inertia forces in a flowing fluid.  Pi does not enter.
Paul

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QUESTION: sorry i was talking about the number "e"=2.71828.........

ANSWER: You mean you want to know the origin or e, or do you want to know if e is in Euler's number (it's not) ?

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QUESTION: yes i want to know the definition of the mathematical constant "e"

Answer
e is an irrational number approximately equal to 2.71828.  It was discovered during the early days of logarithm development, but it really came to be known as a special number when the mathematics of finance was developed in the 17th century by Jacob Bernoulli.  He found that in one year and 100% interest rate, 1 dollar would grow to 2 dollars.  If the interest was compounded every 6 months, the sum grew to $2.25.  If compounded quarterly, the sum grew to $2.44.  And if the sum were compounded every second or less, the sum grew to $2.71828 - the limiting case is the magic number e.  From there the number found its way into many exponential laws of financial or physical growth and decay as well probability distributions and many other areas unrelated to finance but very important to science.

Along the way, Euler devised what is known as the most remarkable formula of all mathematics:

e^(i*pi) + 1 = 0

Why it is called that is beyond me.  Suffice it to say that mathematicians have found magical ways to use the number e.
Paul