As hard as pi

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Here is a test of your mathematical intuition.  Can you answer the following question in plain English (it is a hard one)?   The natural logarithm, e, is an irrational number that starts with the digits 2.71818…  It is easily derived in a large number of different ways.  But why is it that number and not some other number?

To elaborate, physical constants of interest have a value that is relative to their units.  For instance, the speed of light is 186,000 if the units are miles per second, but 300,000,000 if the units are meters per second.  The boiling point of water is 100 if the units are degrees celcius and 212 if the units are degrees fahrenheit, and so it goes with the gravitational constant, permeability of free space, Planck’s constant, etc.

Then there are pure constants that don’t have any units.  For instance, π (pi), the ratio of the circumference to the diameter of a circle is a pure constant — just another point on the real number line.  Why is it that point and not some other?  One way of explaining why π is that particular point on the real number is that it is relative to Euclidean space.  If we transform to hyperbolic or parabolic space then the ratio of the circumference to diameter is also transformed.  So, the value of π is a property of the space we are operating in.  Euclidean space is just one space, being the boundary between parabolic and hyperbolic space.

Returning to the natural logarithm.  If the value of π is a property of the chosen space, then what is e the property of?  What change in assumption, or reference point or axiom would give a different value of the natural logarithm?  Try to explain it in terms that non mathematicians, like me, can understand.  Obviously π and e are directly related in the equation  exp(iπ) + 1 = 0  and in the central limit theorem and many other fundamental relationships.  But in simple terms what would you have to change to change the natural logarithm?

 

12 Responses to "As hard as pi"
  1. I cheated and looked up Wolfram Mathworld but not sure if this is what you’re thinking of. The area under the graph 1/x and the x-axis between 1 and exp(1) equals 1. Define the ln(x) as the area under 1/t and the t-axis between 1 and x (where t is a dummy variable). Then the properties of ln(x) and exp(x) follow namely ln(e)=1, ln(1)=0, exp'(x)=exp(x), and d[ln(x)]/dx=1/x assuming you accept a number of other theorems are true. So if you want to change the value of e then you have to change something in the original equation and I don’t think you can do that and still come up with the same set of properties which makes e e. That’s my best attempt at it but I’m not the most mathematically inclined so someone with greater maths ability can probably come up with something better.

  2. Off the top of my head I don’t know if there’s an answer, but ‘natural logarithm’ is the name of the function (log to base e), not the constant.

  3. So this is a bit of a thought bubble.  I hope it ends up being clear.

    Let’s use the power series definition of the exponential function:

    exp(x) = 1 + x + x*x/2! + x*x*x/3! + …

    Then e is just the exponential function evaluated at x=1.  

    There are two things you could change here: the sort of thing that x is, and the multiplication operation.  You could, if you wanted to, change the *’s into +’s and get an “additive exponential function”.  The leading term becomes a zero because zero is the additive identity:

    addexp(x) = 0 + x + (x+x)/2! + (x+x+x)/3! + … = e*x.

    Then you could plug in the additive identity (zero) and get addexp(0) = 0.  This is pretty useless and you couldn’t use 0 as a base for an “additive logarithm”, but at least it’s an idea of what you could change to get a different “e”.

    The other way you could change things is to change x from being a real number to something different, like a matrix.  If you did that you’d end up with the regular e times the identity matrix.

  4. I’m not a mathematician, but I think you may find the hyperreals interesting (numbers with infinitesimals). exp(dx) might be something like an “infinitesimal version” (in the halo of 1?) of the usual e=exp(1).
    I mean, usually I think of e as the # s.t. e^x has derivative e^x. However I’m not sure that’s still true with the usual e when x is hyperreal.
    Another thought: in the case of 2×2 matrices,
    exp(I) = eI = [e 0;0 e] = “E”?
    So that might be the “E” for matrices. Something similar holds for infinite dimensional linear operators, too.

  5. For logarithms, e is the logarithmic base that gives you the smallest value of logs.  This allows you to have a smaller slide rule, and simpler addition as there are less likely to be as many significant digits.

    For interest, e is the maximum value of compound interest if you compound continuously, so if you compound $1 for N years, the value of the compound interest will be e to the N.  Which again is easily worked out using e based logarithms.

    So e becomes the simplest number avaialble for calculating exponential functions, i.e. non-linear problems in a linear (euclidian) space.  Moving out of the space might affect the value of e, but I can’t escape the feeling that e would still be a constant in a non-euclidian space.

  6. Adding to my original comment, you could change the original equation (say to the area under the graph 2/x and the x-axis between 2 and exp(2)) and redefine the number 2 (to be the ’new’ 1) to have same the same properties of 1, that is given a constant k, k*2=k and k*(a/2)=a*k (where a is some constant) so that ln(e)=2, and ln(2)=b (where b is a constant which is a ‘new’ 0 i.e. k*b=b and k+b=k) then the area is equal to 2 and now you have e which has a new value but with the same properties of the original e but working under a different set of axioms.

  7. Just to clarify, no matter what the value of e is, even if it the same visual representation ie. 2.71828…, the number 2, 1, 0, and b (whatever you decide b to be) will have different ‘meanings’.

  8. A mathematician has set me straight. Turns out e is still e in the hyperreals. The derivative of real exp is exp, and the derivative of hyperreal exp is defined as the hyperreal function induced by the derivative of real exp (exp). Real exp is e^x, and the hyperreal function it induces, hyperreal exp, is also e^x.
     
    Hyperreals are identified as sequences of real numbers. The hyperreal function induced by a real function is just the function applied to each term in the sequence. So, for example, sequence (1/n) is an infinitesimal, and e^(1/n) is the hyperreal number (e^(1/n)) (a sequence), which is in the halo of 1.

  9. There are many constant that are just what they are, with no underlying “reason” for it.  For example, Brun’s constant is the limit of the sum of the reciprocals of the twin primes, and is estimated to 1.9021605.
    e may be in this class.

  10. kme, Sam’s question as applied to Brun’s constant has a fairly straightforward answer – instead of working with prime numbers in the positive integers, you work with prime elements of some other ring.

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