OulunLyseonlukio / Galois club 2010-2011 / On the extraordinary constant e / TL
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1. Where does the constant e come from?
Euler’s number, denoted by e,also known as Napier’s number, is one of the most important constants in all mathematics. Below we sketch a few natural contexts in which the constant e emerges.
Example 1.1. (Continuous compound interest)
Assume a hypothetical bank account with an annual interest rate 100 %. Assume thatat the beginning of the year the account holds exactly 1 euro (or any monetary unit). Assume that the interest is credited to the account n times a year at regular intervals. We calculate the bank account’s final saldoafter one year forn = 1, 2, 3.
n = 1:
n = 2:
n = 3:
You notice the pattern that arises. We have in general
[equation 1.1]
It seems that the final amount might actually increase towards infinity as we reduce the crediting period towards zero (in which case we may speak of continuous compounding of interest ), or equivalently let ? However, armed with the formula [1] and calculator we may convince ourselves that this is not the case. We have, for example:
monthly crediting>
weekly crediting>
daily crediting>
hourly crediting>
minutely crediting>
END OF EXAMPLE 1.1
Motivated by the above examplewe define
or equivalently
It will be shown later that this is a valid definition i.e. that the limit really exists.
The number is irrational (to be proved later) and its approximation to two decimal places is 2.72 as indicated already in the example above.
The symbol refers to Euler, a Swiss mathematician (1707-1783), who was the first to see its enormous significance in many fields of mathematics although the Scottish mathematician John Napier, aka Neper (1550-1617), had introduced earlier as a natural basis for logarithms (which he actually invented).
Example 1.2. (Continuous compound interest in general)
Assume a bank account with an annual interest rate %. Let be the corresponding growth factor (e.g. if then). Assume again that the account holds 1 euro at the beginning of the year and the interest is credited to the account n times a year at regular intervals. Below we calculate the final amount f after one year for values of n = 1, 2, 3.
n = 1:
n = 2:
n = 3:
In general we have
Whenthe interest is compounded continuously and the final amount will be
.
For example, if and , we have
while
and .
END OF EXAMPLE 1.2
Example 1.3. (Bernoulli trials)
Jacob Bernoulli (1654-1705), another Swiss mathematician and Euler’s predecessor, considered the following problem:
In a bowlthere are n balls of which all except one are white (the exceptional one being black). A gambler picks one ball at random, checks its colour and puts it back into the bowl. This is repeated n times. What is the probability that the gambler never gets the black ball? What limit does this probability approach when n increases towards infinity?
At each draw the probability of not getting the black ball is . Because successive draws are independent of each other we may apply the product rule to obtain the required probability (that the same results is repeated at each of the n draws)
.
For example
.
It may be proved that
.
2. Infinite series representation for e
The number e has many remarkable properties. Besides the limits of sequences (or functions) considered above, the number e can be expressed as infinite sums, of which the most useful is
[equation 2.1]
This follows from a more general result that the exponential function can be represented as a power series
[equation2.2]
It can be proved that this representationholds true for all real values of x.
Note that from [equation 2.2] we obtain, by setting ,
.
The equation 2.1 can also be justified (if not rigorously proved) directly by applying the binomial theorem to the definition .
By the binomial theorem we have
and consequently
Analogous justification works for the more general equation 2.2 when we notice (as we already did in the example 1.2 where we had for ) that
3. Irrationality of e
Theorem 3.1. The Euler-Napier number e is irrational
Proof: Assume to the contrary that e is rational, that is for some positive integers . Using the power series representation of e we have then
and consequently
Now multiply both sides of this equation by to get
where the right hand side is positive and can also be evaluated from the above as follows
So we have
But then the equation
isimpossible because the left hand side is an integer while the right hand side lies between 0 and 1.
But this equation followed from our assumption that e is rational. Hence the assumption must be false.
QED
Note: Charles Hermite, a French mathematician proved in 1873 that the number e is not only irrational but even transcendental (non-algebraic). Nine years later, in 1882, Ferdinand von Lindemann, a German mathematician, proved that another important mathematical constant is also transcendental. Georg Cantor, creator of the modern set theory, had proved in 1874 that actually almost all real numbers are transcendental.
4. Number erepresented as a simple continued fraction
Remember that simple[1] continued fractions are of the form
, , ,
which are usually denoted by a special square-bracketed coordinate notation as follows
[2; 2] , [2; 2, 3] , [2; 2, 3, 7] , [2; 2, 3, 7, 2] , [2; 2, 3, 7, 2, 4]
All positive rational numbers can be expressed as finite (or terminating) continued fractions (like those above).
Irrational numbers, in contrast, can be expressed as non-terminating (infinite) continued fractions like
[1; 2, 2, 2, . . . ]
It can be shown that the Euler number e also has a quite regular representation
[2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, . . . ]
1
[1]A continued fraction is simple if all of its numerators are equal to 1.