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What is e? 

The number e is the base of the natural logarithms, and is one of the most important numbers in mathematics and physics.  It is an irrational number for which the first few terms in the decimal expansion are: 

            e = 2.71828 1828 … 

In physics, the number e is important for many reasons, for example because solutions of differential equations of the form 

take the following form: 

It turns out that many physical processes can be modelled through differential equations of this type.  One such process is radioactive decay, though in this case the constant a < 0. 

In mathematics, e has numerous important properties.  One such property follows from the differential equation above, with a = 1.  This shows that the derivative of the function 

with respect to x is simply the function itself. 

Another interesting property is that the following limit is equal to e: 

Try it with your calculator – (1+0.00001)¹⁰⁰⁰⁰⁰ = 2.71827.  Continue increasing n and you will get ever closer to the value of e given above.

Proof that e is Irrational

Our aim is to prove that e is irrational.  The statement we need to prove is therefore: 

S = The number e is irrational. 

The converse of the statement S is 

T = The number e is rational. 

If statement T is true, then our original assertion in statement S is false.  Similarly, if T is false, then S must be true.  So, let us assume that statement T is true, and see if we can find a contradiction.  If e is rational, then we can define integers a and b such that 

where a and b are integers with no common factors.  To proceed, we need another interesting result of the number e, which is that it can be represented by the following infinite series: 

where n! means the “factorial of n”, which is the product of all the integers less than or equal to n.  Thus, for example, 4! = 24.  By convention 0! = 1.  Noting that b is an integer, we can split this expression into two terms as follows: 

Now let’s multiply both sides of this equation by b!: 

The second step follows from our assumed definition of e as a rational number.  Rearranging slightly gives us 

The first term on the right-hand side is an integer.  The second term is also an integer, since n is less than or equal to b in all terms in the series.  Thus, we must have that 

provided that we assume e is a rational number.  Writing this sum out explicitly, we find that 

Now here comes the clever bit.  We can see that 

But we can re-write the right-hand side of the above expression as follows: 

where the penultimate step follows from the well-known result that 

Thus, we have deduced that 

This provides us with the contradiction that we seek, since we have already seen above that the sum on the left-hand side of this expression is an integer.  But, since b is an integer greater than 1, the condition above shows that the sum on the left-hand side is less than unity.  Also, the sum is clearly greater than zero.

This is an impossibility - there are no integers that lie between 0 and 1.  Therefore the statement T must be false. 

Hence, statement S is true, and e is an irrational number.

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