Can the Infinite Be Randomized?

Is the infinite always the same? Is the infinite tomorrow the same as is today? How many infinities are there? Only one? How about if there are infinite infinities? How about if all of them are not the same, one to another?

How about if one of those infinities is not linear, without any kind of order, but random? Why not?

We have a naive and simple idea of the infinite. In the best of the cases, we think of the infinite as the unending series 1, 2, 3, ... In the worst scenario, we think of the infinite as the quantity of the grains of sand in all the deserts and beaches on the Earth.

It was Archimedes of Syracuse, more than two millennia into the past, who proved that it was impossible that the grains of sand were infinite because he was able to give a fair estimate of the grains of sand that could be placed in a sphere the size of the orbit of Saturn.

He knew that it was impossible to count grain by grain all the deserts and beaches. So it is impossible to prove that the grains of sand are finite by enumerating them one by one.

So, his approach was to establish an upper limit to the number of grains that can hold the planet Earth. This is an indirect proof that the sand cannot be infinite.

The Sand Reckoner.

Download the free EBook The Sand Reckoner, the book where he developed his proof.

But what if we have an infinite book in our hands, how big can it be? Infinite in size? Infinite in weight?. Infinite in volume?

Have you ever heard of The Book of Sand? It’s a short story about an infinite book with no beginning and no end. But this book has a finite amount of pages; how come?

The infinite is incredible and surprising! What is going to limit the limitless?

Read the article Three unexpected behaviors of the infinite and see three unforeseen aspects of the infinite.

Why Is It so Difficult to Reason About the Prime Numbers?

In the history of mathematics, the prime numbers are ancient as the invention of the natural numbers. To review a little, the natural numbers are the simple positive integers 1, 2, 3, ... —a series without end.

In this series of the natural numbers, when we are given some number, it is easy to tell what the next number is; just add 1 to the given number! (This explains why it is funny to ask elementary school kids for the biggest number he/she can think about, and then confront his answer by telling him/her to add 1 to the number he mentioned.)

On the other hand, the prime numbers are not so easy to visualize; their distribution is as if they were randomly distributed.

Statue of Euclid.

Let's review for a moment the standard definition of a prime number: a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the primes less than 100 are:

    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The number 1 is by definition not a prime number. Note how unusual is the beginning of this series: the first prime is even --the only even number that is prime; all the other following primes are odd. But not every odd number is prime. To the number 3 follows 5; to 5 follows 7, but to 7 does not follow the number 9.

The simple task of finding the prime number that follows another prime impossible. With the prime numbers it happens that given any prime number, there is no way to compute the next prime that follows—or precede—the given one.

However, the primes are very special in one aspect: every natural number is either a prime or it is the product of some primes. This is a far-reaching assertion because it surreptitiously states that the natural numbers are in one way or another all made of prime numbers.

Euclid (ca. 300 BC), and Eratosthenes (ca. 200 BC) were the first two Greek mathematicians to work extensively with the properties of the prime numbers. Euclid proved that the primes were infinite (no way to find the last prime), and Eratosthenes devised a method to sieve out the primes from the series of the natural numbers.

Since then, the primes numbers are a constant headache for number theorists and mathematicians in general.

Historical Quotations About Prime Numbers

However, mathematics is not a private property of mathematicians and philosophers of science. What about the public? They also have something to say. What does the layman have to say about the prime numbers? Do politicians have something to add to this arduous field of mathematics (they always have something to say!)?

Let's see how some imaginary examples of how some famous people would have reasoned about if the number 9 is prime or not.

Christopher Columbus: “3 is prime, 5 is prime, 7 is prime. According to some ancient manuscripts, 9 is not a prime number, but beyond the distant horizon of the oceans, in the New World that I am going to discover, there are surely lots of them.”

Dimitri Mendeleev: “3 is a prime, 5 is a prime, and 7 is a prime, but 9 is a noble prime that deserves a separate row in the periodic table of the primes.”

Charlie Chaplin: “3 is a prime, 5 is a prime, 7 is a prime, 9 is the next prime after 8.”

John F. Kennedy: “1 is not a prime number and 9 is not a prime number? Then ask not what the primes can do for you, ask what you can do for the primes.”

Stephen Hawking: “2, 3, 5 and 7 are prime numbers: 9 is not prime, but in the black holes, past beyond the event horizon, anything can happen.”

George W. Bush: “3 is prime, 5 is prime, 7 is prime, and 9 … well, any odd number can be prime as long as it is not 9.”


For more examples about how easily is to be misled when reasoning about the series of the prime numbers, then see the article:

Historical Quotations About Prime Numbers

Is Somebody out There? Is It Sensible to Send Messages to Other Stars?

Take a look at the quiet night sky. How quiet is it? Don't let the beautiful twinkling stars deceive you! Somebody may be standing and "listening" to us from one of the stars you are looking at.

Should we listen only from them, and at the same time send no message, like—"I am here"—to them? Isn't there a paradox in this attitude?

Alexander Zaitsev thinks so, and he called it the Paradox of the Great Silence.

Is there a way of detecting messages flying-by over our atmosphere? Do we have the technology to capture every conceivable way of detecting space messages? And more than that, do we have the knowledge and the required wisdom to answer them? Is radio messaging the only way of capturing and communicating our ideas to other stars in our galaxy?

The first coded message sent to outer space.

Well, astronomers seem to be short-patience people. They seem to work on a day-by-day basis: the technology that matters is the one we have "today".  The first attempt, on 1962, was a three-word message sent to Venus in an experiment from the Evpatoria (Eupatoria)  Ukraine, using a deep-space radio-telescope coded using the Morse code pattern.

Later, in 1974, another message was sent from the Arecibo radio-telescope. This one was an elaborate message with information about the chemical constituents of the organic life here on Earth, and even the double helix structure of the DNA molecule.

Is it fair that a few astronomers reveal the chemical structure of our organic life? Isn't this another case of scientific arrogance, like the one of Sir Arthur Eddington, when he said he knew exactly how many protons are in the universe?

Then, should we listen only and stop sending messages? This is the Paradox of the Great Silence that stated the Russian astrophysicist.

But the "text-messaging" to the outer space didn't stop there. Then next, on 1999, from Evpatoria --again-- another message was sent, this one in a layout similar to a 23-page book.

What is written (coded) in this message to the stars? You won't' believe your eyes!

A single page from a "binary book" sent to space.

But that's not all, almost yesterday, on 2001, was sent another message, this one with the participation of three teens. The Teen-Age-Message (TAM) is—without a doubt—the most sincere and original of all of them.

Why?

Books About the Fourth Dimension

Another Word,
or The Fourth Dimension.

Now you can download many controversial books from authors out the mainstream. Open your mind to other ideas, to other fields of knowledge.

These are carefully reformatted books for easy and joyful reading, and without password restriction for printing.

Flatland is a book about a journey of a character from a two-dimensional world that visits a one-dimensional kingdom—Lineland—and that is also visited by a character from the third dimension—Spaceland.

By analogy, since we are three-dimensional, what if we make a similar journey to Flatland, and then to the fourth dimension? Right now, how can we recognize visitors from the fourth dimension?

Selected Papers of Charles Hinton About the Fourth Dimension is a collection of essays from the pen of Charles Hinton, the classic writer that initiated an effort to put an order in the chaos of the thinking about other dimensions besides our three-dimensional world.

Some passages from the Bible
can be interpreted as from the fourth dimension.

Readings of the Fourth Dimension Simply Explained is another collection of articles, this time from different authors. Another classic in this subject, sponsored by the well-known magazine Scientific American. Many of the authors were teachers and experts in their field of knowledge.

Another World or the Fourth Dimension is a curious book about the presence of the fourth dimension in the Bible.  Is the presence of so many strange experiences narrated in the Bible evidence that the fourth dimension really exist?

The 4-D Doodler is the story of a man that is trapped between the edges of the Spaceland and the fourth dimension. Could this really happen in a future space-time travel? In 2012, or 2100, or 3500...? Is it happening now?

Download any of these books now!

The Trees That Came from the Moon

In the Apollo 14 mission on January 31, 1971, the astronaut Stuart Allen Roosa took with him about 500 hundred seeds to orbit the Moon with him. The seeds were from five different types of trees: Sycamore, Loblolly Pine, Sweetgum, Redwood, and Douglas Fir.

Some Sycamore trees alive today
are from seeds that traveled to the Moon.

Upon return to Earth, the seeds that Roosa took with him to the Moon were sent to the Southern Forest Service Station in Mississippi and to the US Forest Service Western Station in California in an attempt germinate them. Surprisingly, nearly all the seeds germinated successfully.

One beautiful example of the Moon Trees is this Sycamore tree in Lincoln State Park, Lincoln City, Indiana.

But are there more trees like this one? And what about the existence of trees in the Moon in the early science fiction literature?

The answer to the above questions are in this more extensive article: What do you know about the Moon Trees?

Perspective Drawing

Perspective is everywhere.

The Essentials of Perspective is a simple book about how to draw true-perspective landscapes.

The book, by the late Prof. L. W. Miller, is the perfect acquisition for the artist of the pen and ink media.

The book is not a complicated and hard-to-follow textbook in the somewhat-abstract field of projective geometry. Its the opposite: an easy to follow book suitable for high school and college students.

Shadows also show perspective.

The Essentials of Perspective is a fully illustrated book, with more than one hundred figures.

To drawing or paint a correct perspective we have to take into account the always present illusion of the vanishing point at the far horizon.

The book not only covers the correct depiction of the converging lines in a true perspective but also shows how to take the always present shadow of the objects.

Want to unleash the artist within you? Do you want to start drawing simple objects from your surroundings? Don't want to buy expensive art materials? Then, this book and a simple sketchbook are all you need!

Essentials of Perspective.

Drawing shadows of rounded objects is not easy.

Complex Numbers: How Complex Are They?

The "history" of the integer numbers is a simple one. From the natural numbers 1, 2, 3, ... we move to the positive integers 0, 1, 2, 3, ... then we add the negative integers ... -3, -2, -1, 0, 1, 2, 3, ....

Then we escalate to fractions and decimals and non-terminating decimals (although historically was not in this order). The ladder continues to the irrational numbers and to the algebraic and transcendental numbers. This is the "world" or universe of the real numbers.

Every real number has a specific place on the Number Line.

But mathematics is a product of our minds so this "universe" or field can be further expanded to suit our needs.

The next heaven after the real numbers field is the imaginary numbers; numbers that in combination with the reals make the complex numbers field.

But how complex are the complex numbers? Curiously, they are as simple as the "preceding" ones.

The negative numbers haunted the mathematicians and philosophers for many centuries; no wonder the misnomer "negative". Even the number zero took a long time before it was accepted in the kingdom of the mathematics (in Europe, where it was later accepted.) It was unacceptable to count "backward".

The imaginary numbers suffered the same fate: no wonder the epithet of "imaginary". The square root of minus 1 was impossible to compute because no number times itself is equal to minus 1.

Take a read at this article: "The imaginary numbers are not so imaginary and the complex Numbers are not so complex" and you will see how easily and beautifully the complex numbers emerge out of the real numbers.

The Deluge Revisited ... Once More

The story of The Deluge has many interpretations.

The story of the Deluge is a narrative that is mythical for many, however historically true for millions.

The Deluge In The Light Of Modern Science, by William Denton is a critical analysis of this story as it is found in the Christian Bible.

A few extracts of his commentaries are sufficient to grasp Denton's style:

"Noah, his family, and the animals, went in seven days before this time, and left the ark the six hundred and first year of Noah’s life, the second month, and the twenty-seventh day of the month. They were therefore in the ark for one year and seventeen days.What a quantity of hay would be required, the material most easily obtained!"

"An elephant eats four hundred pounds of hay in twenty-four hours. Since there are two species of elephants, the African and the Indian, there must have been four elephants in the ark; and, supposing them to live upon hay, they would require three hundred tons."

"Many animals live upon insects; and this must have been the most difficult part of the provision to procure. There are nineteen species of goatsuckers; and there must have been in the ark two hundred and sixtysix individuals. These birds feed upon flies, moths, beetles, and other insects. What an innumerable multitude must have been provided for the goatsuckers alone! But there are a hundred and thirty-seven species of fly-catchers; and Noah must have had a fly-catcher family of nineteen hundred and eighteen individuals to supply with appropriate food. There are thirty-seven species of bee-eaters; and there must have been five hundred and eighteen of these birds to supply with bees. A very large apiary would be required to supply their needs."

The Deluge In The Light Of Modern Science.

Denton concentrates his analysis of the narrative of the Deluge on the many difficulties Noah had when collecting the animals for his Ark:

"How could the ostriches of Africa, the emus of Australia, and the rheas of South America, get there,–birds that never fly? There are three species of the rhea, or South-American ostrich; and forty-two of these would have a journey of eight thousand miles before them, by the shortest route: but how could they cross the Atlantic?"

However, some questions arise concerning all the water that fell during all those forty days of heavy continuous rain:

"It is as great a difficulty to discover where the water went when the flood was over. We are told that the fountains of the deep and the windows of heaven were stopped, and the rain was restrained. But this could do nothing towards diminishing the water".

Transcomplex Numbers

Foundations of Transcomplex Numbers.

Integer numbers, negative numbers, fractions, real numbers, transcendental numbers, irrational numbers, and imaginary numbers are a few of the number types we usually find in mathematics.

Is there no end to this? Is there no "final" type of numbers?

From the standpoint of number fields, all of them can be encompassed into one type called the complex number field.

Wikipedia, in a short background, mentions how the complex numbers emerged:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

Complex numbers can also be understood and developed from the standpoint of view of ordered pairs. Today, developing the complex number system from the foundations of set theory and the concept of ordered pairs is possibly the most intuitive approach we have at hand.

For a rigorous development of the complex number system download the free EBook: Foundations Of Transcomplex Numbers: An Extension Of The Complex Number System To Four Dimensions.

This mathematics book is about a way of extending the complex numbers system to four-coordinate variables, maintaining the usual operations attributed to the complex numbers.

Foundations ... is a fully illustrated EBook. See --and Click-- for example, the following figure about how to multiply two ordered pairs:

Transcomplex numbers are an extension of
the common complex numbers. 

Complex numbers are usually plotted using the familiar plane Cartesian coordinate system, but transcomplex numbers are four-entry ordered pairs, also called 4-tuples, so they belong to a four-dimensioned space.

In a nutshell, transcomplex numbers are complex numbers whose elements are ordered pairs.

In the following simple illustration, also taken from Foundations ... we can see that out of a four-entry complex number system we can extract four 3-dimensional spaces like "ours".

The transcomplex numbers need a 4-dimensional
coordinate system to be represented.

The chapters of the book are divided as follows:

• Ordered Pairs. The whole theory of transcomplex functions is based on the ordered pair concept: from the two-dimension plane up to the four-dimension space.
• Complex Numbers. The complex numbers system is derived from the ordered pair's concept.
• Transcomplex Numbers. Here starts the extension of the complex numbers into ordered pairs of complex numbers, arriving at the concept of transcomplexs.
• The Coordinate System S4. This chapter is devoted to deriving a suitable coordinate system to plot transcomplex functions.
• Transcomplex Functions. Functions of complex variables evolve to make space for functions of four-entries ordered pairs.
• Transcomplex Surfaces. A radical and totally new perception of surfaces generated by complex variables.
• Theorem Proofs. This chapter collects all the proofs of the theorems stated along the book.

Big Numbers: Can We Really Understand Their Meaning?

Is it possible to count all the protons of the universe?

I recently wrote an article about the 'unreality' of the Eddington number. The article title is: The Eddington number: a case for scientific arrogance?

Arthur Eddington (1882-1944) was an all-time advocate of the emerging Theory of Relativity since its introduction by Albert Einstein.

The 'Eddington number' is an extremely big number that supposedly represents the exact quantity of proton in the visible universe. This number, sometimes abbreviated as N_Edd, needs 83 digits to describe it fully:

N_Edd = 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296.

The 'problem' with this number is that it is to difficult to 'digest'. How it is possible that with the incipient science methods we have, the 'unelaborated' artifacts, instruments, and appliances we have to explore our surroundings, can anybody come to tell us the exact count of protons in the universe?

Read the full article for more info and rants about how other scientists and writers have dealt with the problem of big numbers.

Alice in Wonderland

Alice In Wonderland is based on real characters.

Is the girl shown here in this oil paint Alice Liddell, the main character of Alice in Wonderland, the all-time hit in the children and adult's literature?

(Picture: One Summer's Day by William Brymner - 1884)

Maybe not, but the picture title and the walking little girl are evocative of the dedication that Lewis Carroll made for Alice in the book he wrote for Alice Liddell, which at this time he titled: Alice's Adventures Under Ground.

In fact, Lewis Carroll wrote the following dedication in the manuscript he handled to Alicia in November of 1864:

A Christmas Gift
to Dear Child
in Memory
of a Summer Day

Now you can download Alice in Wonderland in EBook format from Datum.

Alice's Adventures In Wonderland.

This masterpiece of the children's literature is more than a fantastic narrative from the imagination of a prolific writer, photographer, and mathematician. Alice in Wonderland is also the product of an epoch in the search for new horizons in the science.

Previous to Alice in Wonderland, Ludvig Holberg, near a century before, in 1741, wrote an adventure story about a character that goes down a cave to explore the underground world: Niel’s Klim’s Underground Travels. And in 1692, Edmund Halley, a British astronomer, and mathematician put forth the idea of a hollow Earth when he tried to explain the deviations of the magnetic field of the Earth,

Not to mention that Athanasius Kircher, in 1664 published a geological and geographical investigation that culminated with his Mundus Subterraneus (Subterranean World) in which he suggested that the ocean tides were caused by water moving to and from a subterranean ocean.

Niels Klim's Journey Under The Ground.

Almost simultaneously with Alice’s Adventures Under Ground, in France, Jules Verne, in 1864, published his Journey to the Interior of the Earth.

So, the idea of a “habitable” underground was not new to the fiction and fantastic literature writers, and possibly Lewis Carroll was related with some of those published works.

Do Random Numbers Really Exist?

One Million Random Digits.

While I was collecting information for this month's post, I was also looking for new material for a new free E-Book to compile for my readers. The idea of the so-called random numbers sprung into my mind, so I began to search for this topic.

I found an interesting book review titled: A Million Random Digits with 100,000 Normal Deviates. The book was originally published in 1955 by the RAND Corporation, so the "review" was a little late, but its OK; the author was "reviewing" one of the oldest books in his library. This book can be found and read at Google Book Search.

Doing a deeper search I also found another article and an E-Mail by Mr. Nathan Kennedy complaining to the RAND Corp about their stand that the One Million Random Digits table was of their property and that it cannot be redistributed on the Internet. By a great coincidence, my intention was the same as Mr. Kennedy's, and his intention of putting the table on the Internet was the same as mine. However, and in a great unselfish gesture, Mr. Kennedy generated his own table of random digits and proceeded to place it on the Internet for free as a text file on the same page.

Since he authorized the use of his million digits table, I reformatted the text file as a PDF file, designed a cover page, wrote a small introduction for it, and made an E-Book, to be distributed also for free.

I am not a statistician, so maybe I will never find a practical use for this kind of numerical table, however, random numbers are of interest for me, and probably for many others, for the degree of strangeness they bear.

Are there really random numbers, or there are random events?

Can we really speak of random "numbers"? Isn't it more appropriate to speak of random "events"? Can we make at least some arithmetic operations with them? Can we add two RNs and still say that the sum is also "random"? Can we multiply them to obtain -without a doubt- that the result is also random?

Note that the RNs are not obtained by any formula, or equation, or matrix, or any predefined mathematical operation; they are mainly obtained by algorithms fed by some "physical" events like atmospheric variations, radioactive decay, thermal processes, or the like. Hence, what we are doing is using unpredictable physical events, assign to each "event" a number, and say that this is an RN. Another nonphysical source of random sequences of digits (but sometimes questioned) is by selecting digits or portions of digits of the decimal expression of irrational or transcendental numbers.

But without physical events, can we still generate RNs? There are some rudimentary approaches, but they are mostly mere mathematical curiosities.

The interested reader can find more authoritative articles at Random.org where he/she can obtain instant (real-time) random numbers for the lottery, cards, passwords, etc.