When it comes to counting from 1-100 in Nepali, there are themes, and you can often make a good guess what a number will be, but you’re not always guaranteed to be right. The best presentation of them that I’ve seen is from the blog Learning Nepali. The Latin writing system isn’t a standardised one, but it gives you an idea of how the system is based on the smaller number coming first, and the tens number following (one-and-thirty, two-and-thirty) and some undercounting (one-from-eighty) with a later of sound change added on top to make it nice and difficult [original here].
So let’s talk about a problem with learning numbers in Japanese.
You start out doing great. These ones go the same as in English:
一（いち） 1
十（じゅう） 10
百（ひゃく）100
千（せん）1000
And then we get to 10000. At this point, English is like, okay we’re gonna slow down and not have a whole word for every single zero we add. This one is just “ten thousand.” No new words for a while.
But Japanese keeps up the fight for one more zero:
万（まん） 10,000 (or maybe you should think of it as 1,0000?)
Andthen it breaks down and starts saying things like “ten 万”…which is a hundred thousand. So anyone going between the two languages is kind of f*cked when it comes to thinking about big numbers, because they don’t line up nicely with each other.
How can you fix this? Honestly, I suggest learning/relearning some Big Number Facts and Statistics in Japanese. Knowing how many 万 people there are in your city, for example, can give you some landmarks so you don’t have to count up the whole way every time you see a number like that.
Hopefully these examples can help you get oriented a bit and get you thinking about some trivia you could use for this:
一万（いちまん） = 10 thousand
Dogs were domesticated maybe 1万4千 years ago. The Ice Age ended about 1万7千 years ago.
The median US household income is 6万3000 dollars.
Many towns, suburbs, and cities have a population in this range. What about the ones near you?
十万（じゅうまん）= 100 thousand
I grew up in a city of around 50万 people. That’s half a million!
Our species Homo sapiens appeared about 20万 years ago.
The average house price in the US is about 20-30万 dollars.
百万（ひゃくまん） = 1 million
The population of a large city:
Kyoto: 140万
London: 890万
Houston: 230万
Or a small to medium country:
Lithuania: 280万
Finland: 550万
Laos: 700万
Lucy, the famous Australopithecusfossil, lived 320万 years ago. Early hominids started using tools around 250万 years ago.
The median household income in Japan is about 400-500万 yen per year. The average car costs 170万 yen.
The top prize on the game show Who Wants to be a Millionaire is $100万.
千万（せんまん）= 10 million
The population of a very very large city:
Tokyo: 1300万
Mumbai: 1800万
Beijing: 2100万
Or a medium to large country:
Canada: 3500万
The Philippines: 9300万
South Korea: 4800万
United Kingdom: 6600万
The dinosaurs died 6500万 years ago.
The sun is 9300万 miles away.
一億（いちおく） = 100 million
In keeping with the “new word every 10,000″ pattern, there’s no such thing as a 万万. That’s an 億.
The population of a large country:
Japan: 1.2億
United States: 3.1億
The dinosaurs first appeared 2.4億 years ago.
The sun is 1.5億 kilometers away.
One Piece has sold 4.6億 volumes total.
The speed of light is about 3億 m/s.
十億（じゅうおく） = 1 billion
The biggest countries:
China: 13億
India: 12億
There were 77億 people in the entire world as of April 2019.
The earth is 45億 years old. Middle-aged, yeah? If you think of 1億 years as one earth year.
Bill Gates makes somewhere in the ballpark of 40億 dollars a year.
百億（ひゃくおく）= 10 billion
There’s debate about whether the Earth’s population will reach 100億, or whether it will reach a ceiling before then.
The universe is about 138億 years old.
I’ve only seen two episodes of Dr. Stone but 百億 sure seems to be Senku’s favorite number, doesn’t it.
千億（せんおく）= 100 billion
About 1070億 humans have ever existed, total.
There are at least 1000億 stars in the Milky Way, maybe as many as 4000億.
There are about 1700億 neurons in a human brain.
Jeff Bezos’s net worth is over 1200億 dollars.
The solar system is 2870億 km across.
The second largest number I’ve ever encountered in a manga was 6千億, which was how many freaking paper bombs whats-her-face used in that one battle in Naruto. Christ on a bike, girl, how many trees did you even have to kill? That’s more than half a 兆!
Speaking of which…
一兆（いっちょう）= 1 trillion
Japanese and English finally match up again!
Just in time to talk about…uh…the GDPs of major economic powers? The US national debt?
The number of cells in a human body? (37兆, also the largest number I’ve ever encountered in a manga, thanks Cells At Work!)
Well, it’s exactly the same as a trillion, we probably don’t need THAT many examples.
There are number words higher than 兆 but, well…you see 京（けい） probably even less often you see someone say “ten quadrillion” in English. Usually people just go for scientific notation for things that can’t be expressed in 兆.
My goodness, at this precise moment my dear little collection’s esteemed audience stands at an astounding eight thousand followers! I never even imagined it might make eight hundred, I was honestly pleased with achieving but eighty, and now look at you all! I thank each and every one for your interest, your sharing and showing your admiration of the photographs, and all the kind messages of appreciation; and I fervently hope you’ve all found plenty to enjoy in what you’ve seen here over the last couple of years. I have been unable to find you a photograph of eight men together, to mark today’s total the way I did some other significant figures, so allow me to cheat just a little and present eight exposures of the same man, in a Marey Wheel photograph by Thomas Eakins, circa 1884.
To learn how to pronounce the Sino-Korean numbers 1 through 99, you only need to memorize how to pronounce numbers 1 through 10. Study those numbers below, then practice with the rotating graphics above. 행운을 빕니다!
If you pay attention on your daily life, you can see and feel the universe giving you signals. This set of numbers are an example. Keep it always with you or take notes of the numbers you see and when.
Can a number system be both the new kid on the block and older than written history?
The real number system as it exists today has been with us for a few centuries. In foundation it is monovalent, monophasic, and sequential.
The probable number system dates to prehistory but was lost in the mists of time until recently rediscovered and resurrected. In contrast to the real number system it is foundationally bivalent, biphasic, and cyclic.
The probable number system has considerably more structure than the real number system and is therefore more robust. In this sense, it is similar to the complex number system.
In contrast to the complex number system, the probable number system in its foundation presupposes that numbers can assume wavelike forms capable of constructive and destructive interference operationally through the compositing of higher to lower dimension.
By means of compositing of dimension probable numbers are able to distribute throughout the entire mandalic unit vector cube (which is structurally a superposition of the 6-dimensional unit vector hypercube on the 3-dimensional unit vector cube) a function analogous in important ways to that performed in the complex number system by the centralized imaginary unit i.
Another important way in which the probable number system differs from both the real number system and the complex number system is the absence of nothingness and the zero representing it. In its place we find the concepts of balance and equilibrium. Nullification still exists in form of annihilation and its opposite in the form of creation. But the Cartesian coordinate system of ordered pairs and ordered triads is transformed by this approach to handling number and dimension from a ring into a field of hyperdimensional numbers over real numbers in three dimensions.
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Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
If we describe a Cartesian ordered triad by x,y,z we can describe an analogous 6-dimensional ordered sextuplet or 6-tuple by xa,ya,za,xb,yb,zb
The definitions that translate a 6-dimensional ordered sextuplet (hexagram in Taoist terminology) into a 3-dimensional ordered triad (trigram in Taoist terminology) are:[1]
(xa + xb) / 2 = x
(ya + yb) / 2 = y
(za + zb) / 2 = z
I think the methodology will work for all scalar quantities. But as currently formulated, mandalic geometry (MG) is a discrete geometry based entirely on unit vectors. We are talking about the line segments between -1 and +1 in the various dimensions and only points -1, 0, and +1 in each line segment in Cartesian terms.
In essence we are not yet particularly concerned with scalars here but only with vectors : -, +, and neutral (0).
Mathematically √−1 is important because by adding it to the real number field, as we have done, we create the algebraically complete field of complex numbers. In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. The real numbers and complex numbers are both complete fields. Cartesian coordinates- - - ordered pairs and ordered triads- - - although based on real numbers, do not form a field. This has important implications, implications which can be ignored only at peril to the particular conceptual system involved..
The definitions above all give three possible results in Cartesian terms: -1, 0, +1. Remember though MG hybridizes six dimensions with three dimensions and represents them superimposed. Wherever one or more zeros occurs in Cartesian coordinates we have also corresponding 6-dimensional forms, composed of just +1s and -1s, of which there are always two for each Cartesian zero. A Cartesian ordered triad with one zero is associated with two such 6-dimensional forms; an ordered triad with two zeros, with four; an ordered triad with three zeros (the origin), with eight. An ordered triad without zeros will have only one associated 6-dimensional form. This constitutes the mandalic pattern, which is an essential feature of the 6D/3D formulation of this geometric system and isomorphism naturally comes into play here as well.
Andthat is how and why all numbers in this coordinate system based on higher-dimensional extensions of the real numbers “square” to numbers which can be either positive or negative and then reduce or "collapse" to corresponding Cartesian forms that preserve the same sign. This is a necessary result of the fact that a primary “zero form” in 6-dimensional terms is lacking, only +1s and -1s exist. These can then interfere constructively and destructively as number waves, to produce a "secondary zero" by destructive interference whenever linked forms differ in sign in one or more paired dimensions. Since the two linked 6-dimensional numbers are always inverse to one another, any Cartesian zero then can be substituted with two such 6-dimensional forms. This is the process that makes imaginary numbers unnecessary, replacing them with two inversely related probable numbers which behave in most ways like real numbers and are distributed throughout the entire geometric system.
“Hybridization” is probably not the best term here but will be used until I can think of a better descriptor. What I intend is not actual joining and unification, but rather a superposition and conceptual commingling in three-dimensional terms. Such a representational mapping substitutes for all Cartesian forms "equivalent" forms containing only 1s and -1s, no zeros. In so doing, it effectively converts the Cartesian coordinate system from just a ring to a field as well, properly interpreted. Basically then, the probable numbers do for the real numbers much the same as the complex numbers do, but with even greater and more utilitarian results which are also more easily managed.
In operational terms, complex numbers perform two rather simple binary operations: a scaling and a rotation. Scaling capability is clearly inherited through its real number lineage; rotational capacity, from its imaginary number lineage. Together, scaling and rotation combine to augment or diminish an axis of growth and produce vector ambulation in a circular path about a central origin point of reference. The scaling factor could be said to detemine the radius of revolution; the rotation factor, the angle of revolution. And that’s pretty much all there is to the “great mystery” of complex numbers. Their importance resides in the great number of fields of endeavor where the combination of these two superpowers is necessary and/or convenient.
Nature uses this combination of scaling and rotation in many of its processes. Atomic and subatomic proceedings are probably not among these. How then did it come about that quantum mechanics arrived at the notion that rotation and scaling could be applicable to modeling of discontinuos states of being? Both refer to changes through continuous space. I think it was an accident of history. In 1925, Erwin Schrödinger, in his search for a way to explain certain mysteries then perplexing the greatest physicists of the day, hit upon his eponymous equation which appeared to do the trick. So well, in fact, that quantum mechanics has been justly considered the single most successful description of reality ever devised. And the equation that basically accomplished this success involves the imaginary number i and complex numbers.[2]
An important aspect of the operation of rotation, one which may have bearing on the Schrödinger equation and its huge success, has been largely overlooked. The result of a rotation can often mimic the result of inversion (reflection through a point), making the two indistinguishable by measurement alone. To someone wearing a blindfold there is no way to tell whether i has by the operations of squaring and rotation changed itself into -1 or -1, the inversion element of multiplication, has simply reflected +1, the identity element of multiplication, through the origin point to -1. Explaining away a 90° rotation with a right angle reflection will no doubt prove more difficult but let’s not just yet deny that it might be doable.
Could there be a way to reformulate the Schrödinger equation then so it contains no imaginary or complex numbers? Many have tried to do that very thing and failed. No one has succeeded in nearly a century. Still, we might wonder if the time is ripe now to remove the blindfold. Perhaps we might do well to inquire whether quantum physics is, in some manner we don’t quite understand, a victim of its own success.
In theory, circumventing use of complex numbers in a defining equation of quantum mechanics should be possible. On what basis do I say this? The equation we have now relies on complex numbers. These in turn derive an ability to produce rotation from the imaginary number √−1 . But there are other mathematical means to accomplish the same. Trigonometry comes most immediately to mind. The circle and cyclicity it models have a very long and distinguished history. Complex numbers as we’ve noted can also produce scaling. But so can real numbers. And close examination reveals that complex numbers inherit their ability to scale from the two real numbers they contain. The hard truth ultimately is there is nothing all that special about complex numbers or complex plane. Possibly it is their utilitarian ease of use that positions them as an attractive methodology. Other routes to ease of use exist as well. There is always more than one way to skin the proverbial cat (even a cat residing only in the mind of a physicist named Schrödinger.)
Consider also, how great is the actual need for scaling in quantum mechanics? The distance from centermost part of the atom to the outer reaches of electron orbital space is in fact quite small. Furthermore, the elements of this universe of discourse are quantized, so actual distances involved are moot. In the extreme, the question persists as to whether “distance” is a concept even applicable in this context of quantum logic. Quantum numbers themselves range between 0 and 2. I can count the allowed values on the fingers of one hand.
Regarding rotation, where exactly does that come into play in the quantum realm? Electrons do not orbit the nucleus of the atom. They jump from orbital to orbital by discretized changes in energy involving photon exchange. In the nucleus it seems such discretized instanteous changes take place as well, obviating any need for rotation. Obviously physics misguided here by labeling one of the quantum numbers “spin”. Sometimes a rose is best referred to as a rose. The problem here is that we don’t really know what it is that “spin” refers to.
The quintessential equation of quantum mechanics was formulated by a physicist, not a mathematician. It is not a simple algebraic equation, but in general a linear partial differential equation, describing the time-evolution of the system’s wave function (“state function”). “Derivations” of the Schrödinger equation do generally demonstrate its mathematical plausibility for describing wave-particle duality. To date, however, there are no universally accepted derivations of Schrödinger’s equation from appropriate axioms. Nor is there any general agreement as to what the equation actually signifies. Moreover, some authors have demonstrated that certain properties emerging from Schrödinger’s equation can even be deduced from symmetry principles alone. This would appear to be a worthwhile direction of investigation to pursue. Quantum mechanics is most fundamentally about symmetry. Let’s make Emmy Noether proud by giving her the recognition she deserves.
Finally, it was not without considerabledifficulty that Schrödinger developed his equation. In the end, it almost seems he pulled it out of a hat, as a magician might a rabbit.[3] Part of the Zeitgeist of the physics community in the early 1920s revolved around the peculiar notion that particles behaved as waves. Schrödinger decided to follow this direction of thought and find an appropriate 3-dimensional wave equation for the electron. His equation succeeded beyond his wildest dreams. Adopted in the canon of the new physics, it became the cornerstone of that radically different physics, changed forever. Physics has never looked back since.
Still, one startling and haunting fact persists: nowhere else in all of physics has it ever been found necessary to invoke complex numbers.
Once, quite a long time ago, I believed imaginary numbers were wrong. I was the one that was wrong. Later, having grown a little more clever, I came to think that √−1 was a necessary evil- - -correct but not validly applicable to quantum physics. Wrong again. Currently it is my belief that imaginary numbers are guilty of an even worse offense: both true from the mathematical standpoint and partly applicable to physics. The worst of both worlds. Yielding results that are in large part correct, imaginary and complex numbers have managed to lead us all down the garden path for the better part of a century. Have we then gone past the point of no return? My contention is that it is possible to complete the ring that Cartesian coordinates present and transform it to a field over the real numbers, with appeal only to higher-dimensional analogues of the reals and no need for imaginary or complex numbers, an approach which, if actually possible, would offer certain undeniable advantages.[4]
Essentially the method of composite dimension does away with i and complex numbers by distributing an operation analogous to that of i throughout six dimensions or three in Cartesian terms and then working with same by means of reflections (inversions) only. So an algebra based on the system necessitates use of only the real numbers and their higher dimension extensions that I have called probable numbers. Only simple addition and multiplication are required. For those in the audience who are "sufficiently mad”, there is the added bonus that a kind of division by zero becomes possible. We’ll find out soon enough whether you qualify.
A few additional explanatory remarks are in order here:
Depending on the variant, Cartesian geometry (CG), represents space in two or three dimensions. Points in the former are referenced to two pairwise perpendicular axes; in the latter, to three.
Because Descartes assumes as axiomatic a 1:1 correspondence of number to spatial location each of his three axes becomes a facsimile of the number line, only in different dimensions.
Mandalic geometry (MG) approaches representation of space differently, using a hybrid coordinate system which relates a higher dimension space to a lower dimension space with a 2:1 correlation.
Itcan be represented entirely commensurate with CG, but in so doing a “glass slipper effect” occurs. Just as Cinderella’s stepsisters can manage to force a too fat foot into her glass slipper, the results leave something to be desired. In our context here, the "something to be desired" is a clear and full understanding of six-dimensional reality in its own right. We end up interpreting it in time-sharing terms of probabilities and randomness.
What Descartes refers to as an ordered pair requires two higher dimension ordered pairs to represent in MG; a Cartesian ordered triad requires three higher dimension ordered pairs to represent in MG.
In Taoist terminology the notational equivalent of a Cartesian ordered pair is a "bigram", a two-line symbol, each line of which can take one of two values. As a result there are four types of bigram. Two bigrams make up a tetragram; three, a hexagram.
Descartes views a point as having only two essential characteristics:
It is dimensionless.
It is just a location in space which can be uniquely represented by a single ordered pairorordered triad.
Mandalic geometry rejects both of these axioms. It regards a point, or a particle so represented, as an evanescent entity emerging from interaction of two higher dimensions expressed in our world of three dimensions in such limited manner.
Thiscan be represented in context of Cartesian space but in making mandalic coordinates commensurate with Cartesian coordinates it is no longer possible to represent every “point” in space uniquely with a single mapping of number to location. What results instead is the probabilistic distribution pattern of the mandala, which we, from our limited vantage in spacetime, misinterpret as something it is not.
MG is a discrete geometry. The result of the mapping formula used is a mandalic configuration in which the 3-dimensional cube composed of unit vectors in Cartesian space becomes a "probability distribution" in combined mandalic space.
I have placed the quotation marksaroundprobability distribution because this is a perspective that arises from our inability to see all that is involved accurately. I suspect this has repercussions pertinent to a full comprehension or grokking of quantum mechanics and possibly of string theory as well.
Since the 64 discrete “points” of the unit vector hypercube of six dimensions represented by the hexagrams cannot “fit” simultaneously in the 27 discrete points of the 3-dimensional unit vector cube by any representational method available to our inherited bio-psychocultural mechanism, a sort of time-sharing process occurs in observations and measurements of reality which we interpret in terms of probability.
What has been described here occurs at enormous velocities close to that of light, and likely refers only to processes in the subatomic quantum realm. For MG, which is also a hybridization of mathematics and physics, context is always of the essence.
There is much more to be said in explanation of mandalic geometry. I see, though, this post has already run rather long, so we will end it here. Enough has already been said in way of introduction of basic material.
Notes
[1] Since the coordinate system is describing a cube with an n-hypercube superimposed, there is an additional constraint placed on all coordinates in the 6-tuples. All scalar values must be identical for x, y and z values. That constraint assures that all vectors though they may differ in sign (direction) maintain equal magnitudes.
When the 6-tuples are dimensionally reduced to 3-tuples by the method I’ve called “compositing of dimension” the resulting geometric figure consists of four different dimensional amplitudes of 6-tuples collapsed. The amplitudes of dimension correspond in spatial terms to the vertices, edge centers, face centers and cube center. The pattern that emerges is that of a mandala. This is a highly symmetric pattern though all symmetries aren’t necessarily apparent immediately, even using Taoist notation. The probability distribution of the 6-tuples allots the hexagrams in the following manner: one to each vertex; two to each edge center; four to each face center; one to the cube center. The result is placement of 64 6-tuples in 27 positions of discrete 3-tuples in the specific mandalic distribution pattern described.
Think here of the analogy of a hydrogen atom confined within a cubic space of specified side length determined by the nuclear and atomic force fields. The single electron, existing in such quantized energy levels that are possible, can assume various different locations in different orbital shells, but every location in a given orbital must be equidistant from the nuclear proton. Once reduced by dimensional compositing the 6-tuples described here fill four distinct shells that have different radii or distances from the center. From center to periphery these distances can be described as zero; one (or square root one); square root 2; and square root 3. (Pythagorean theorem)
[2] Schrödinger was not entirely comfortable with the implications of quantum theory. About the probability interpretation of quantum mechanics that came out of Solvay ‘27 he wrote: "I don’t like it, and I’m sorry I ever had anything to do with it.“ ["A Quantum Sampler”. The New York Times. 26 December 2005.]
[3] In later years another great physicist, Richard Feynman, would remark, “Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger.”
[4] A different approach to avoiding the need for complex numbers from the one I am suggesting is described here. To my mind it offers little of value other than an interesting alternative explanation of what complex numbers are and do. A similar conclusion seems to have been reached by the author.
Please note: The content and/or format of this post may not be in finalized form. Reblog as a TEXT post will contain this caveat alerting readers to refer to the current version in the source blog. A LINK post will itself do the same. :)
Scroll to bottom for links to Previous / Next pages (if existent). This blog builds on what came before so the best way to follow it is chronologically. Tumblr doesn’t make that easy to do. Since the most recent page is reckoned as Page 1 the number of the actual Page 1 continually changes as new posts are added. To determine the number currently needed to locate Page 1 go to the most recent post which is here. The current total number of pages in the blog will be found at the bottom. The true Page 1 can be reached by changing the web address mandalicgeometry.tumblr.com to mandalicgeometry.tumblr.com/page/x, exchanging my current page number for x and entering. To find a different true page(p) subtract p from x+1 to get the number(n) to use. Place n in the URL instead of x (mandalicgeometry.tumblr.com/page/n) where n = x + 1 - p. :)
