The Decimal Deficiency of Pi
Welcome to Aquarius, Volume 16 (January 20, 2008)
The Decimal Deficiency of Infinite Pi
1) I Disclose the Truth Because Secrets are Lies of Omission
2) In the Decimal Place System Many Numerical Values, Such as Pi, Cannot be Written
3) Sample Physical Object
4) Length of side confronts length of smallest possible particle
5) Physical pi varies, up to the limit determined by the smallest possible particle
6) Infinite pi is not the same as the diagonal or square root of three
7) Proposed Revision of Number Theory - Decimal Deficiency Numbers
8) The two forms of numbers that cannot be written in decimal place format are
9) The geometrical number system
10) Pi and science crossing paths with religion
11) The homonym "irrational" meaning "unreasonable" compounds the problem
12) Pi is pi whether curved or straight
13) The best number system for science and mathematics
14) Mathematics Being a Human Invention is Defended Also by Doctorates
1) I Disclose the Truth Because Secrets are Lies of Omission
A) I disclose the truth because it is unforgivable to withhold it. To withhold any important information is a detestable offense, the lie of omission, the lie of a thief who steals the truth.
B) Why hide the history of planet Earth and keep it a secret within stone walls? If you keep any important information secretly to yourself, you must do so because you are afraid of me, the average person.
C) And if you are afraid of me, then I should not trust you and I am inclined to make your fear of me justified, for I do not want you to possess any power if you use your power to conceal the truth which is mine by birth.
D) Therefore I reveal the truth about the history of humankind and by doing so I empower everyone and diminish your power to sustain your lie of omission, which lie is a moral offense equal in its gravity to killing every living child.
E) Those who keep secrets will not be forgiven because they cannot be forgiven. A lie protects only the deceiver and can never protect the deceived. The deceived are led astray and the deceiver is thereby responsible for any and all injuries and damages that follow.
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2) In the Decimal Place System Many Numerical Values, Such as Pi, Cannot be Written:
There can be no infinite pi as a dimension in the real, physical world because we reach the dimension of the smallest possible particle before we reach 10 to the minus 50th decimal place. We can compare physical pi to infinite pi. Physical pi is the outer dimension of a polygon, circle or sphere when we describe or define the object as a regular polygon with the number of sides increasing as powers of 2, beginning with 8 sides, to 16, 32, 64, 128, sides and so on until we get to that number of sides which brings the physical dimension of the side into conflict with the supposed dimension of the smallest possible particle of physical matter.
Using the metric and decimal systems, the decimal fractions are:
1/10 = 0.1 = decimeter, a tenth of a meter, or 10^(-1)
1/100 = 0.01 = centimeter, a hundredth of a meter, or 10^(-2)
0.001 = millimeter, a thousandth of a meter, or 10^(-3)
0.000,001 = micrometer, a millionth of a meter, or 10^(-6)
0.000,000,001 = nanometer, a billionth of a meter, or 10^(-9)
--> Angstrom, 10^(-10) or 0.0000000001
0.000,000,000,001 = picometer, a trillionth of a meter, or 10^(-12)
smaller decimal fractions, each divided by 1,000 (one thousand) would therefore be
0.000,000,000,000,001 or 10^(-15), or fifteen spaces to right of the point
0.000,000,000,000,000,001 or 10^(-18)
0.000,000,000,000,000,000,001 or 10^(-21)
10^(-24), 10^(-27), 10^(-30) and so on.
3) Sample physical object:
We can see, that if we had a sample physical object with a diameter of one meter, and that object was a regular polygon with 10^40 sides, but we designated our object as a Euclidean circle (with infinite points on a circumference), then our simple formula of circumference = pi*diameter gives us infinite pi as the length of the circumference. However, if we continue to designate our object as a regular polygon, then we are confronted with a problem, specifically a conflict between two dogmas of physics. One dogma is that there is a smallest possible particle, and the other dogma is that a physical object can be a sphere with a circumference of infinite pi (infinite pi being a value with infinite decimal digits).
4) Length of side confronts length of smallest possible particle:
A fraction of a meter that is 1/10 raised to the 30th power, or 40th power, is obviously a very small dimension. It is so small it raises the question as to whether the smallest possible particle is the same length, or even longer than the supposed length of the side of our sample physical object. For purposes of the argument, let's say that the smallest possible particle has a length of 10^(-38) meter and the length of the side of our sample physical object is 10^(-40) meter. This presents us with a physical impossibility. How could the side of our sample polygonal object be shorter than the length of our smallest possible particle? This cannot be.
5) Physical pi varies, up to the limit determined by the smallest possible particle:
The length of the side of a regular polygon object cannot be shorter than the length of the smallest possible particle. Then, we see that if we contemplate our regular polygon object and designate it to be a circular object, it still has the same number of smallest possible particles that comprise the dimension of the perimeter, or circumference. This means that regarding our sample object as a regular polygon or as a circle does not actually change the outer dimension, perimeter or circumference. And, because that outer dimension is not changed from the outer dimension of a regular polygon, it is a finite dimension with a finite number of decimal places. It is physical pi. The perimeter of the physical regular polygon equals the circumference of the physical circle because the circle circumference cannot have more sides or more length. If we add a smallest possible particle to it, it is then slightly larger, but larger only to the extent that we have added one particle length to the perimeter/circumference. The perimeter of the regular polygon and the circumference of the circle are still exactly the same and that dimension is still physical pi and not pi to an infinite number of decimal places.
6) Infinite pi is not the same as the diagonal or square root of three:
If a physicist or mathematician argues that the circumference of a circle is infinite pi, then they are discrediting the concept of a smallest possible particle. They are taking a position that can be sustained only if there is no smallest possible particle and physical matter is comprised of particles that are smaller and smaller down to nothing. This concept is difficult to defend, because it implies that everything physical is comprised of particles of nothing, or null=1. We accept the definition of the diagonal and the square root of three as being endless decimal digits, but this is a convention of our decimal place number system. Number systems are inventions, artifacts. The decimal digit system has a limitation, a deficiency. It cannot be used to write the value of the length of the diagonal or certain square roots or the trigonometric ratios. Physicists and mathematicians who defend infinite pi as a real physical dimension are treating the decimal digit number sytem as though it is reality. It is not. It is a method for writing numerical values. There is no scientific basis to assert that reality conforms to a specified method of writing numbers. The written number for the square root of two, using the decimal place system, may have an infinite number of digits, but the diagonal line does not have infinite length. It has finite length, and therefore the non-compliance between the written number and the length is obvious, and the limitation of the number system should be acknowledged. Otherwise, the belief in infinite pi is a religious belief, not a description of physical reality.
7) Proposed Revision of Number Theory - Decimal Deficiency Numbers:
I seriously propose that the institution of mathematics, and the general public, acknowledge that the decimal place number system is deficient. There are numerical values that simply cannot be written - expressed - using the common decimal place system. Currently, our Number Theory designates some fractions as "rational" fractions, meaning they can be written as a "ratio" of one whole integer divided by another whole integer, such as (3/5) = 0.6. Certain other fractions are designated as "rational" if they can be written as an endless repeating decimal place number. Other fractional values are designated as being "irrational" fractions, meaning the many numerical values that are written as endless non-repeating decimal places and these non-repeating decimals cannot be written as the ratio of one whole integer divided by another whole integer. Therefore, endless decimal place numbers occur in two forms. An example of a "finished fractional number" written in decimal place is (5/16) = 0.3125. There are many such numbers.
8) The two forms of numbers that cannot be written in decimal place format are:
First, those rational fractions that cannot be written as a finished number but are infinitely repeating decimal digit values (1/7) = 0.142857142857142857142857142857143..., the "set" of decimal digits that is endlessly repeated can be detected here as (...142857...). The practice accepted in mathematics is to write a truncated (cut-off) version, such as 0.142857... And that format is deemed to give expression to (1/7), but it does not, only the infinite version is truly equal to (1/7), and therefore, in the decimal place system, the truth is that (1/7) cannot be written, or given expression.
Second, those irrational fractions that cannot be written as a finished number but are infinitely non-repeating decimal digit values (an example is the diagonal of a square, which numerical value is sort(2): 1.4142135623730950488016887242097... . There is no repeated set of decimal digits in the fractional part of this numerical value. The practice accepted in mathematics is to write a truncated (cut-off) version, such as 1.414213562... and that format is deemed to give expression to sqrt(2), but it does not, only the infinite version is truly equal to sqrt(2), and therefore, in the decimal place system, the truth is that sqrt(2) cannot be written, or given expression.
9) The geometrical number system:
Geometry can give expression to the square root of two. Simply by drawing, or "constructing" a square and then the diagonal, the diagonal is in fact not only a line but also a "symbol" that writes, or gives expression, to sqrt(2) or 1.4142135623730950488016887242097... going on to "complete" an infinite number of non- repeating decimal digits. This means that if the institution of mathematics accepted geometry itself as a number system, as the ancient Pythagoreans did, we could "write" both "endlessly" repeating and non-repeating fractional numbers, which tells us that our decimal place system is actually "deficient" because it cannot "give expression" to numbers that can be given expression in the geometrical number system. Based on this limitation of the decimal place system, I declare that our decimal place system is the sole source of the arbitrary designation of endless decimal fractions as "rational" and "irrational" required to be truncated in order to be written and entered into calculations. Mathematics and the general public should acknowledge that this is an example of "progress egotism" where we assume that our technology must be superior to any technology used in the past. Therefore, the essence of my proposal is that: Any numerical value that requires an infinite train of decimal digits be designated as a decimal deficiency number because the true value cannot be written or given expression using the decimal place system.
10) Pi and science crossing paths with religion:
Face the realtiy that mathematics is the religion of western civilization and infinite pi is worshipped as though it gives expression to the "mystery" of the unknown in nature. Mathematicians and physicists adhere to religious beliefs and give their doctrines and dogmas scientific names. Our society worships mathematics. Whenever there is a political or economic or technological controversy, the contenders love to make the claim that they "did the numbers," which implies, of course, that if any argument is in some way confirmed by mathematics it cannot be an error.
To offer another form of argument in support of this proposition, here is a test question that can be made part of any test for mathematicians or physicists:
Part 1: Is A or B below true?
A) The decimal place system equals reality
B) The decimal place system is not equal to reality
Part 2: Write up to 1,000 words to defend your choice of A or B.
Part 3: If you selected (A), using the decimal place system, write the square root of two. Ask if you need more time.
11) The homonym "irrational" meaning "unreasonable" compounds the problem:
If the goal of the institution of mathematics is to disseminate mathematical knowledge, then the language of mathematics needs to be developed so that it promotes clairty and not confusion. Use language that clarifies the meaning of mathematical concepts rather than obscuring mathematical concepts. The linguistic issue that compounds the problem in Number Theory addressed here is that in common usage "irrational" means "not reasonable" or in some way emotional or "outside of the world of reason." But in the language of mathematics this same word (the homonym) means "not able to be expressed as one whole integer divided by another whole integer." And such a number is not by any means "not reasonable" or "outside of the world of reason." This practice causes the general public and students of mathematics to believe that there is something wrong with such a number, or that such numbers are "mystical." Although pi is defined by mathematicians as "transcendental," it still falls within the scope of "irrational" because infinite pi is defined as an endless non-repeating decimal number. Therefore, pi belongs to that set of numbers that cannot be written, not given expression, using the decimal place system. Further, there is no scientific basis to assume that because our decimal place system possesses this limitation and deficiency that no other number system is able to express such so-called "irrational" numbers. All irrational numbers can be given expression in the geometrical number system, as lines. The homonym problem is most likely the same in the Italian, French, Spanish, German and Russian, which are the languages of origin in which mathematical advances have been developed since the calculus. Of course, mathematical advances have also originated in Arabic and other languages of Southern Asia and the Orient, as well as Africa.
12) Pi is pi whether curved or straight:
If we take a modern rubber and fabric fan belt, or power transmission belt, which can be shaped as a circle, we can see that it must possess a circular dimension that is pi times the diameter of the circle. If we cut that belt at one location, using a sharp edge, not a saw, so that we do not remove any material but only separate it, then the resulting straight line length of the belt has to still be pi times the diameter. Therefore, if we designate the diameter to be one unit, then the straight line length of the belt is pi. Thus, here again we have a line length of pi that is similar to the square root of two in that it is a finite length but the written number to express that numerical value does not possess finite length. The decimal system of writing numbers requires that the expression of pi (as well as many other numbers) be an infinite train of decimal digits, which of course cannot be produced. No matter how brief the time required to write one of the digits, the time required to write the infinite number of digits is forever. This is not a trivial or strictly semantic issue. The decimal place number system is not equal to the physical reality of the universe and in order to move forward in our understanding of the real physical universe we must acknowledge that our decimal place number system is deficient. There should be no number values designated as being "irrational" but rather as being "decimal deficiency" numbers, meaning with rather simple clarity that such numbers are as real and as reasonable as any other, but our decimal place system is limited and deficient in that it does not enable writing this class of number. The real pi in the real world is variable, and when written with the decimal place system the written number pi has a finite number of decimal digits that end prior to 10 raised to the 50th power.
13) The best number system for science and mathematics:
The ancient Pythagoreans were right, the best number system for science is the geometrical number system which enables written expression, with line lengths, of any and all proportional values that occur in Nature meaning in the real physical universe. The decimal place system is not the best system for science but it is a very good system for counting money and measuring weights and volumes in the marketplace, which are the purposes for which it was originally adopted.
14) Mathematics Being a Human Invention is Defended Also by Doctorates:
I have conducted by research of the past thirty years based in part on my conviction that our mathematics is a human invention, believing that I was alone. But I am not alone. I am far from being the only person arguing the evidence that mathematics is "our mathematics," meaning a uniquely human invention. Some neuroscientists have identified evidence that our way of counting and measuring, the origin of our arithmetic and mathematics, grows out of the sensory and motor experience of the human body as well as the neurons of the brain. I would say that there is a kind of "mathematics reform movement" in process. This reform movement is or will be similar to the Protestant movement to reform the Christian (Catholic) Church in Europe. The church reform movement is deemed to have begun with the 95 Theses of Martin Luther in 1517. In our time, the "Church of Mathematics" exercises rigid authority over our thinking in a manner that is the same as the Catholic Church in Medieval Europe. That is why I call mathematics "the last anthropomorphic religion." It is the last, unless we invent another, which would not be healthy for human civilization. Psychologists and neuroscientists understand how we learn, how our brains function, and they can see from their studies that the human brain invents mathematics as the primary technology. Our ability to add and subtract, multiply and divide, to count and measure, is necessary before we can exercise any other technology that is more advanced. Several research projects supporting the conclusion that our mathematics is a product of our species and of our evolutionary process are mentioned in a research survey by Drake Bennett, in:
"Don't Just Stand There, Think," Ideas, The Boston Sunday Globe, January 13, 2008.
Mr. Bennett identifies ten scientists at eight different universities, that support the proposition that the universe is not mathematical, but we are. Those scientists are: Shaun Gallagher, Director of Cognitive Science Program, University of Central Florida; Rolf Pfeifer, Director of the Artificial Intelligence Lab, University of Zurich; scientists at the Laboratory of Embodied Cognition, University of Wisconsin; Sian Bielok, Assistant Professor of Psychology, University of Chicago and Lauren Holt; Alexander Lleras and Laura Thomas, psychologists at the University of Illinois; Linguist George Lakoff, University of California Berkeley, and Rafael Nunez, a cognitive scientist at the University of California San Diego, who "have for several years advanced the argument that much of mathematics, from set theory to trigonometry to the concept of infinity, derives not from immutable properties of the universe but from the evolutionary history of the human brain and body." Arthur Markman, Professor of Psychology at the University of Texas. And Angeline Lillard, Psychology Professor at University of Virginia, suggests that if we reconsider the "Montessori Method" of early child education, developed by Maria Montessori in the early twentieth century, we see evidence that children learn faster, and possibly better, if they are moving or using their hands while learning. This is true not only for learning technical skills, but also language skills.
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