INVESTING BY THE NUMBERS

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HaloulakosDr. V. E. “Bill” Haloulakos
Aerospace Science Consultant/Professor
AIAA National Distinguished Lecturer and Legacy Member
Distinguished Engineering Educator Award Winner
E-mail: imveh@sbcglobal.net

 When we invest money we are always playing a numbers game.

  • The amount invested is a number
  • The interest rate is a number
  • The period of investment is a number
  • The final value of the total amount accrued is a number
  • The number of compounding periods in a year is also a number

We then feel justified to raise the question as to exactly WHAT IS A NUMBER?

We shall discuss the concept of numbers and also lay out the peculiar behavior of two special numbers, namely ZERO and INFINITY, and also show how INFINITY plays a special role in an interesting investment problem.

When we need to talk about the numbers one, two, etc, we usually show the corresponding number of fingers or we point to a corresponding number of material items. But in the number system there are two very special numbers with rather elusive behavior. They are the numbers of ZERO and INFINITY. When we need to talk about zero all we can do is make a circle with our thumb and the index finger. There’s no other way to indicate the idea or concept of nothing, which the number zero indicates. On the other end of the numerical spectrum lies infinity which is impossible to describe in a simple understandable manner. All we can say is that it is BIG, very BIG. Actually exactly as say that infinity is something very BIG we can also say that zero is something very small. How BIG and how SMALL is only understood by crossing over into the fields of Physics and finance, which we shall do below.

Mathematically both zero and infinity have precise mathematical definitions but they are not all that clear to the ordinary person.

The definition of zero is:

  1. When zero is added to any number there is no change. i.e.
    a + 0 = a
  2. When a number is multiplied by zero the result is zero. i.e.
    a x 0 = 0
  3. Division of any number by zero is undefined. One may say that it is something very, very large, i.e. infinity, but that’s not quite true.

The definition of infinity is:

  1. When infinity is added to any number the answer is infinity.
  2. When any number is multiplied by infinity the answer is infinity.
  3. When any number is divided by infinity the answer is zero.

All this is well and good but is too abstract for the common man in the street. Where and when mathematics displays its glory is its applications into explaining the world around us, i.e. its use in the field of Physics and that’s where we will go in order to explain the intricate subtleties of both zero and infinity in a meaningful manner to the layman.

We will also tackle a special investment problem that will surprise the reader!

In the world of physics we deal with physical objects and processes. We conduct controlled experiments, collect and interpret data and draw conclusions, which apply everywhere. So, let us now consider the process of addition involving a physical process. In particular, let us examine the meaning of the first definition of zero where when we add zero to any number there is no change, i.e. a + 0 = a.

In a physical process we say that when we add something to an object and there is no measurable change then that “something” added is ZERO. So, let us now consider the process of a bulldozer driver moving a big pile of dirt and we go and throw a couple of handfuls of electrons on the dirt. The driver will absolutely not notice any change in the mass, or weight, of the dirt. Therefore, the mass of the added electrons is ZERO, i.e. the mass of the individual electrons is ZERO. Now let us move, with this concept of the electron mass being ZERO, to an experiment involving one hydrogen atom and add one electron there. The result will be very noticeable, so, the electron mass is NOT ZERO. So, in the world of physics the concept of ZERO depends on the process involved.

In a more “down-to-earth” application in money investment, let us consider the case of adding one thousand dollars ($1000) to the assets of billionaires Bill Gates or Warren Buffet. Are we to say that either one of them will notice a change on the total value of their holdings? More than likely they will not. So, the conclusion is that one thousand dollars for them is like ZERO, but for the rest of us this is not the case. So, the conclusion is that the concept of ZERO is relative to the situation under consideration.

Similar arguments can be made for the number infinity. For example the distance to the sun for ordinary human activities is like infinity but for the astronomer it’s around the corner. The astronomers talk of thousands or millions of light years as being finite distances. Back to earth now, for an ant the distance of a couple of miles is like infinity. So, the conclusion here again that in the world of physics infinity depends on the problem under consideration.

INTERACTIONS OF ZERO AND INFINITY BETWEEN THEMSELVES AND WITH EACH OTHER

There are situations that arise where these two numbers interact among themselves and with each other that require special consideration. In particular there is specific set of such operations that are termed “Indeterminate” not because their value cannot be determined but because they have different values depending on their origin, i.e. from what functional operations were obtained and require special handling.

These sets of operations are:

Zero divided by Zero, Infinity divided by Infinity, Zero multiplied by Infinity, Infinity subtracted from Infinity, Zero raised to the Zero power, Infinity raised to the Zero power and One raised to the Infinity power.

It’s not the purpose here to fully discuss all these operations but we shall limit our discussion to the last case for it has a unique application to a special money investment problem.

In high school Algebra we learn that one raised to any power is one. But this does not include the case of raising one to the power of infinity. One might ask, why would we want to raise one to the infinity power? The answer is that we do not it but it manifests itself in normal mathematical operations, most notably in a special investment problem involving compound interest, which goes like this.

Invest one dollar at 100% per annum interest compounded an infinite number of times for one year. HOW MUCH MONEY WILL YOU HAVE AT THE END OF THE YEAR?

The formula for compound interest calculations is P = P0[1 + (i/n)]En, where i is the interest rate and n is the number of compounding periods in one year. For the case of one dollar invested at 100% per annum this becomes P = [1 + (1/n)]En and as we let n become infinite this becomes P = one raised to the power of infinity. Answer: $2.72.

Actually this particular function approaches, what we call in mathematics, a limiting value of e = 2.7182818284…, which is the basis of the natural logarithms and there is hardly a physics problem that does not involve e in its answer. The detail calculations for this investment problem are shown in the attached Appendix.

Are you surprised? Investing at 100% per annum interest and compounding it an infinite number of times for the year, were you not expecting to accumulate more than $2.72 on the dollar?

NOTE: “En” denotes raising to the exponent n.

APPENDIX

Invest one dollar at 100% per annum interest compounded an infinite number of times for one year. HOW MUCH MONEY WILL YOU HAVE AT THE END OF THE YEAR?

SOLUTION

The formula for this problem is: P/P0 = [1+ (1/n)]En, where n is the number of compounding times during the year. So, the computations are:

n = 1          P/P0 = (1 + 1)E01 = 2.00, i.e. in one year the money doubles

n = 2          P/P0 = (1+ 0.5)E02 = 2.25, i.e. it increases substantially if   compounded twice/yr

We continue to more and more compoundings per year.

n = 4                   P/P0 = (1 + 0.25)E04 = 2.44

n = 5                   P/P0 = (1 + 0.20)E05 = 2.49

n = 10                  P/P0 = (1 + 0.1)E10 = 2.59

n = 100                P/P0 = (1+ 0.01)E100 = 2.70

n = 500                P/P0= (I + 0.002)E500 = 2.71557

n = 1,000             P/P0 = (1 + 0.001)E1,000 = 2.71692

n = 10,000           P/P0 = (1 + 0.0001)E10,000 = 2.71815

n = 100,000          P/P0 = (1 + 0.00001)E100,000 = 2.71827

n = Infinity                  P/P0 = e , The base of natural logarithms!

i.e. we approach the very interesting mathematical limit

Untitled1
= 2.71828 18284 59045 etc.

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