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Transcendental Numbers

Updated 22 August 2003

Transcendental Numbers are in a word, profound.  For they excel, surpass and transcend human experience.  They are synonyms of peerless, incomparable, unequaled, matchless, unrivaled, unparalleled, unique, consummate, paramount, superior, surpassing, supreme, preeminent, sublime, excelling, superb, magnificent, marvelous, and... well, transcendental.  

Mathematically, they are, by definition, “not capable of being produced by the algebraic operations of addition, multiplication, and involution, or any of the inverse operations.”[1]  Philosophically, they are “existing apart from, not subject to the limitations of, the material universe”, not to mention: “1 transcendent 2a (in Kantian philosophy) presupposed in and necessary to experience; a priori. b (in Schelling’s philosophy) explaining matter and objective things as products of the subjective mind. c (esp. in Emerson’s philosophy) regarding the divine as the guiding principle in man. 3 a visionary; abstract b vague; obscure.”  That should about cover it.  

Accept for the fact that numbers are mathematical, and therefore anyone not inclined to mathematics who might hesitate to continue this particular excursion, might wish to first peruse A Non-Mathematical Digression, or simply skip the equations below, without the sense that you’re somehow missing the essential gems of the treatise.  You will not miss anything, but there are your mathematical compadrees who would fear slighted if there were no equations in this section.  Therefore, fear not, and scurry onward.

Meanwhile, back at the Transcendental Ranch, we should note that the three most intriguing transcendental numbers which readily come to mind are: p (pi), f (or F, phi), and e.


It should be noted that the term "transcendental" in mainstream mathematics requires that any transcendental number not be the root of a polynomial equation with integral coefficients.  A good reference on this view is http://mathworld.wolfram.com/TranscendentalNumber.html. 

However, such a definition by Math World and others is not necessarily the last word on the subject.  The non-root definition is, in fact, somewhat arbitrary.  By limiting what constitutes "transcendental" mathematically, Math World is imposing an unacceptable limitation.  This is not to suggest that the authors of Math World are alone in this regard, but mainstream science and mathematics is notorious for burying its collective head in the sand in order to avoid confronting anomalous and really interesting behavior. Math World, meanwhile, seems to have a singular lack of information on Sacred Geometry -- which is more the gist of this website.

To limit the word transcendental in the manner in which Math World has done, is equivalent to limiting Maxwell's Equations to his four equations later formulated by Oliver Heaviside (the old Ampere's Law, Faraday's Law, and so forth).  Much more importantly, to ignore the larger content of Maxwell's Quaterion Equations is worse yet.  In effect, Heaviside's version is a special case in which energy is conserved within a small, isolated system.  The fact that no system in Connective Physics is truly isolated ensures that such an attempted isolation is simply burying one's head in the sand.  For in the process of imposing conservation on a finite system, the mathematics and physics loses 99% of the true potential impact of Maxwell's work.

The same might be said of the lack of numerology, sacred mathematics, and so forth in mainstream mathematics.  Inasmuch as it is not within the purview of this website to accept such undue limitations, transcendental numbers will be viewed as those which cannot be precisely identified to the point where the next number in the sequence can be known purely from the previous numbers in the sequence.  This fact will become readily apparent when we encounter the Golden Mean, Phi, below -- which can be obtained from the polynomial equation:

x2 - x - 1 = 0

which has roots of (1 ± Ö5) / 2.  [This would suggest, of course, that the square root of 5 is also transcendental.]


The most well known of the transcendental numbers is p, which is typically defined as the ratio of the circumference of a circle to its diameter.  Fundamentally, p connects the profoundly distinct worlds of cycles (circles, orbits, and the like -- the essence of the feminine) with linear lines, such as diameters, radii, and distances (the essence of the masculine).  In such delights as The Joy of Pi [2], we learn that we can approximate the value of pi as:  

            p = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

                     58209 74944 59230 78164 06286 20899 86280 34825 34211 70679

                     82148 08651 32823 06647 09384 46095 50582 23172 53594 08128

                     48111 74502 84102 70193 85211 05559 64462 29489 54930 38196

                     44288 10975 66593 34461 28475 64823 37867 83165 27120 19091

                     45648 56692 34603 48610 45432 66482 13393 60726 02491 41273

                     72458 70066 0631...  (and so forth and so on)  

Well... You get the idea.  Currently, the record [2] for calculating p is 51 billion digits, a record which may prove beyond a shadow of a doubt that some people have entirely too much time on their hands.  Blatner [2], for example, has thoughtfully noted the sequence, 3333333, which appears at the 710,100th digit (and again at the 3,204,765th digit) in the value of p, but the numbering sequence is ultimately not repeating.  Whew, that was close!  

But there is a curiosity about the number that is tantalizing to say the least.  For in all the billions of digits calculated, there is never a repeating sequence or an end to the number... It just goes on and on, as signified by the three dots, i.e. “...”.  There is never a period or an end, as there is to a sentence or definitive statement.  

And therein lies the first clue in the transcendental world of these numbers.  They cannot be calculated to the point of knowing a priori what comes next in the sequence!  There is always a lack of total accuracy or precognition.  This is in spite of the fact that the definitions of p and the other transcendental numbers seem to work extraordinarily well.  The “ratio of the circumference of a circle to the diameter” is even aesthetically pleasing.   

Or perhaps, the approximate formula which can be used to calculate p, one based on an Infinite Series:  

arctan x » x - x3 /3 + x5 /5 - x7 /7 + x9 /9 -/+ ...

= 3(-1)n  x2n+1 /(2n+1)  

where 3 is the summation of all terms from n = 0 to n = 4 .  Inasmuch as the arc tan 45°  = arc tan (p/4) = 1, we can calculate the elusive p from:  

p/4 » 1 - 1/3 + 1/5 - 1/7 + 1/9 ...

The “...” in the formula denotes that the series of numbers is infinite.  Considering that all the subsequent numbers are “odd”, this formula may indeed seem an odd statement.  <mischievous grin>  

There is also the classic form of calculating p from The Joy of Pi [2]:  

p / 2 = [2x2x4x4x6x6x8x8...] / [1x1x3x3x5x5x7x7...]

which is curiously aesthetic.  

A second charmer in the realm of Transcendental Numbers is the base of the natural logarithm, denoted as e.  This is also known as the exponential -- the stuff of increases in population, taxes, and crooked politicians.  The value of e is given by [3]:  

            e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996 ...  

Like p, e can be calculated from an infinite series of the form:  

e = 2 + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7!  + 1/8! + 1/9! + ...  

where the “!” notation is a number factorial, e.g. 6! = 6x5x4x3x2x1 = 720.  For example,

            e =       2.5000000000000...

                        0.1666666666666...                1/6

                        0.0416666666667...                1/24

                        0.0083333333333...                1/120

                        0.0013888888889...                1/720

                        0.0001984126984...                1/5040

                        0.0000248015873...                1/40,320

                        0.0000027557319...                1/362,880

                        0.0000002755732...                1/3,628,800

                        0.0000000250521...                1/39,916,800

                        0.0000000020877...                1/479,001,600

                        0.0000000001606...                1/6,227,020,800

                        0.0000000000115...                1/87,178,291,200

                        0.0000000000008...                1/1,307,674,368,000

                        0.0000000000001...                1/20,922,789,888,000


Note:    1)  The last number is off slightly, due to round off error in the last digit.

            2)  ALL factorials (starting with 6) reduce, as per Numerology to 9. 

                  E.g. 3+9+9+1+6+8+0+0 = 36/9.  Nines seem unexpectedly important!  

The great mathematician, Euler -- who based his work on a discovery by De Moivre -- went to the trouble to define a relationship between these two charmers.  His famous, elegant, concise, and almost mystical equation is:  

eip  + 1 = 0  

where “i” is the Ö-1 (i.e. the square root of minus one), and known as an imaginary number.  From the trigonometric identity:  

ex + iy  = ex  (cos y + i sin y)

we can conclude that:

eiy  =  cos y + i sin y  

the latter, also by Euler, being a special case of the previous equation for x = 0, since the cosine p = -1, the sine p = 0 and,  e° = 1.  Aren’t you glad you know this?  

p and e are notorious in physics, engineering, and science, appearing everywhere, and thus leading us to suspect they might be important.  Which, of course, they are!  Duh!  

Contrary to this fact is the total lack of mention in mainstream science and engineering of the third transcendental number which is profusely manifest in Sacred Geometry.  This third charmer is denoted by the Greek letter, phi, f (or F), and represents the Golden Mean. It’s also called the Golden Ratio, the Golden Section, the Golden Cut, and the division of a line into two unequal parts such that the lesser is to the greater as the greater is to the whole.  Whatever it is, it’s golden!  If not a little weird.  

It can be demonstrated by considering a point on the line AB, marked C:  


such that:

                                            AB/AC = AC/BC = 1.6180339875...  

This is the Golden Mean, and for clarity can be assumed to be either one of two values, given by [3]:  

F = 1.61803 39887 49894 84820 45868 34365 63811 77203 09180...


f = 0.61803 39887 49894 84820 45868 34365 63811 77203 09180...  

And just like our other two transcendies, f can be represented by an infinite series, in this case, the very strange equation:  

                         f = 1 + 1/{1 + 1/[1 + 1/(1 + 1/{1 + 1/[1 + 1/(1 + ...)]})]}  

[It is possible to go blind, trying to understand this equation.  But for the foolhardy, a hint would be to go for the definition of brackets in sequence.]  

In summary, we might now note that we have the values for all three of the transcendental numbers, but that for purposes of having an intimate knowledge of the them, it is not necessary to commit more than about ten decimal places to memory, i.e.  

e   =     2.7182818285...

p      3.1415926536...

f       0.6180339887...


F      1.6180339887...  

We’ll wait while you do the memorization.  

Back already?  

These three Transcendental Numbers cannot be written as a finite sequence of numbers (or a repeating sequence of such numbers), but require an Infinite Series of terms, and where increasing accuracy continually refines the number.  This use of an infinite series to define f, p, and e is one example of a transcendental quality.  And while all three are heavily involved in mathematics, physics, and the natural sciences, they are also applicable to ancient historical monuments, esoteric traditions and virtually everything else  

In Nature, for example, phi or the Golden Mean virtually defines the dimensions of the human body, the shapes and numbers associated with plants and animals, and along with pi the orbits of various planets and natural satellites (i.e. moons) of those same planets.  Man, meanwhile, has used the transcendental numbers in a host of ways.  

To demonstrate this, we first approximate the relationship between e and F by noting that:  

e = 2.718...  » 10 x (ÖF - 1) = 2.720...  

The value of 2.720... varies from 2.718... by a percentage difference (an “error”) of roughly 0.07%.  This represents reasonable agreement to three decimal places, and within the accuracy required for many endeavors, a virtual equality.  For example, if we applied this F-e equality to large scale construction using stonework, the accuracies involved would be sufficient to justify the argument that the structures were built in such a way as to conform to these transcendental numbers.

A portion of the justification for this argument arises when the value of 2.72 is applied to the ancient monolithic structures which proliferate the hills and valleys of the British Isles.  The dimension is so prevalent in the construction of these monolithic structures that archaeologists working in the region have begun calling the dimension of 2.72 feet the megalithic yard -- due to its approximate equality to a yard of three feet, and to the fact that it shows up in the large scale ancient structures in such profusion.  

The important inference, of course, is that our megalithic ancestors seem to have been well acquainted with both F and/or e, else why would they be using the 2.72 dimension in such a prolific manner!  (This might even account for the Fee, Phi, Foe, Fum we keep hearing about.)  We are now encountering the fact that not only does Nature seem to have a fondness for transcendental numbers, but that our long-departed ancestors -- being nature lovers, supposedly -- may have had an similar inclination themselves.             

This, of course, is in direct contradiction to the concept that anyone older than yourself is clearly and obviously dumber, more obtuse, and clearly lacking in any sense of what is really going on in the universe.  This intuitively obvious reality is based on the logic that if your parents are dumber than you are and your parents thought their parents were dumber than they were, then taking the tack of logical extrapolation back to the time of our ancient ancestors, it becomes clear that these people must have been incredibly ignorant!  One can easily imagine them wandering about running into trees and falling into pits.  

Unfortunately for this otherwise, unassailable concept, our really stupid ancestors seemed to have been intimately acquainted with fundamental concepts, some of which, you as a modern reader may only now be encountering for the first time.  To paraphrase the great philosopher, Pogo Possum (but with a different slant), “there are only two possibilities.  Either the ancients were a lot dumber than we are, or the ancients were a lot smarter than we are.  Either way, it’s a mighty sobering thought.”           

But the plot is much thicker than this.  For just as we have found a connection of sorts between F and e, there is also p to consider. The relationship between the transcendental numbers, F and p, for example, can be written as [4, and 5]]:  

p @ (6/5) F2         and/or            p » 4/ÖF  

and/or (F - 1) (p) (7)3 @ 666

Again, the equation(s) relating two of the transcendental numbers are not exact, as denoted by the symbols » (approximately equal), and @ (i.e. very nearly equal).  But this should not cause us consternation.  This is because the very nature of the numbers implies that any two transcendental numbers will never have an exact relationship.  This ultimate lack of precision is noteworthy, and becomes, potentially, even more important when one delves into Sacred Geometry, and thereafter ventures even into the rarefied atmosphere of such fields as Hyperdimensional Physics.  Suffice it to say for the moment that:  

The inexactitude of transcendental numbers and other factors may be essential to the structure, evolution, and maintenance of the universe!   Wow!           

Now.  Given this wholly unsupported supposition (i.e. wild guess), we might now look a bit closer into the 6/5th multiple connecting F and p.  This rectified multiple may seen a bit arbitrary, but it is worth noting that the number 6 is traditionally associated with inanimate objects and 5 is associated with animate objects.  The 5-sided pentagon is thus an important symbol of life (as is the five-pointed star), while the six-sided hexagon (or six-pointed star) represents structural, physical relationships (such as in the honeycomb -- a very strong, stable structure).  In effect, F and p can be thought of as connecting the animate and inanimate worlds of nature.   And just as F and e were connected to the ancient monolithic structures of the ancient British Isles, F and p, in combination, are also profoundly connected to all manner of ancient structures.  Two of the best examples are The Great Pyramids of Egypt (Giza) and Mexico (Teotihuacan).   But of all the connections of Transcendental Numbers, one of the best is the limerick that should stir in the heart of every mathematical poet among us.  This wonder of rare beauty is in words:

“The integral of z squared d z, from one to the cube root of three, times the cosine, of three pi over nine, equals the log of the cube root of e.”  

and in symbols:   

1 ò 31/3 [ z2 dz ] cos (3p/9)  =  ln (e1/3)  

Such mathematical beauty is astounding.  But before you begin to wax poetic yourself, note that your homework assignment is to prove the limerick is mathematically accurate.   (8/22/05) You can also check out other mathematical limericks at Limericks Alive!

Alternatively -- there are always choices -- one can return to Sacred Mathematics, press on to Infinite Series, Magic Squares (they really are magic!), Nines, Sacred Geometry, or even Music and the Harmony of the Spheres.  Or perhaps Connective Physics, Money, Scapegoatology, or Ha Qabala. Whatever.  

Sacred Mathematics         A Non-Mathematical Digression

Forward to:

Infinite Series         Magic Squares          Nines         Fibonacci Numbers

432 and/or 666

__________________________ References: [1] Reader’s Digest Oxford Complete Wordfinder, Oxford University Press, Inc, Reader’s Digest, Pleasantville, New  York, 1996. [2]  Blatner, David, The Joy of Pi, Walker Publishing, Inc. USA, 1997.

[3] Handbook of Chemistry and Physics, 56th Edition, CRC Press, 1975-1976.

[4] The twisted mind of the author, 2002.

[5] (6/6/05) http://users.pandora.be/kenneshugo.



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