Magic of 3?

You have that also in imperial?

No comprende...

Thanks Tornado, for the new 'gyaan' on this. And about 9 also. GA

If √2 is irrational, how does the hypotenuse of an isosceles right triangle with sides =1 exist in the real world?

My first impulse that was to ask sarcastically if you have ever taken an algebra class. However, the question may turn out to be philosophically interesting. It could well be that mathematical objects have a logically definable existence as concepts, but not as physical objects. For instance, if the universe is quantized, no irrational numbers would exist physically. (Except as synaptic structures?) Food for thought, perhaps.

Does such a thing exist? Can you be sure that the two x=1 sides are in fact exactly equal in length and equal to 1? Or is it purely a concept to be contemplated?

Friends,

That answer is simple--in spherical coordinates. It exists in the real world, if the unit side is equal to 1/4 the diameter of the sphere. All three sides are equal length and all three angles are 90-degrees. The area of that triangular surface is 1/8 the total surface area of the sphere.

--JMM

Or is this something known to every third-grade child?

Try to remember, you also learnt this in your third grade. Now, because of age you have forgotten it.

Now, henceforth, if you find something like this please contact some school and just confirm whether what you have found is already known to rest of the wolrd.

We in India have "Vedic Mathematics", very old ways of mathematical calculations, using which you can find answers to many great mathematical calculations. From same India, we find people like you, (who have travelled the world, yet) who do not know such a small things, as explained in post 2. I am surprised.

In older times bookkeepers were checking for transposed numbers by horizontally adding discovered difference numbers, and if result was greater than 9 then they would add result numbers same way until no numbers were left to add.

If final result was 9, they knew somewhere there is trans-positioned number.

For instance if original number was

15 and it become 51, difference was 36, so 3+6=9

16 and it become 61, difference was 45, so 4+5=9

17 and it become 71, difference was 54, so 5+4=9

18 and it become 81, difference was 63, so 6+3=9

19 and it become 91, difference was 72, so 7+2=9

..........

Of course, to know that there is difference, double Ledger system of bookkeeping should be used, where same number is used on 2 sides, one account for increasing and another for decreasing state of this account.

It is interesting that same >>test on 9<< can catch multiple transpositions also, since final sum of any transposition is nine, so in case there are 2 transpositions, 9+9=18

which again gives 9 since 1+8=9..........

I doubt that in our computer age such knowledge is still in use, but then it is harder to find cause of difference if reason is number transposition.

Maybe such horizontal summing of numbers can reveal whether number is divisible by some other number which must be smaller than resulting number and same type as original number (i.e. odd or even).

So in Your case 8511 gives horizontal product of 15, which gives result 6, and 6 is divisible by 3 since it is odd number, but not by 6 or 2 as those are even numbers...

It would be easy to test by making a program to do so, if someone has nothing else to do.......

>>All people are knowing all<< (is literal translation of proverb from my country, actually what we say that all people taken together posses knowledge about everything)......

Perhaps I should not have mentioned tests for divisibility, but sorry, I did not learn Number Theory in my school time........

Otherwise, I think >>irrational<< does not say numbers are crazy or impossible to reason about, they are just >>out of ratio<< because there is no whole number on the other side, i.e 1:3 is 1/3 so it is rational, while 1:3.14 is not.

But then, I am no mathematician so don't take my word for it, more likely than not, I am wrong there since I did not study this, ever.........

Yes, PI....... But PI itself is >>irrational<< number, is that not so?

Now, if I still recall what I have read about subject long ago, if You can write number as fraction of two integers, then it is rational number, else it is not.

It is true that result of division of 1 by 3.14 is same as division of 100 by 314, but 1/3.14 is irrational number while 100/314 is not, and this is not same kind of fraction.

Result of division can be considered to be rational number, but only in case when it is written as 100/314, according to the rule.

Perhaps we chanced on anomaly or there is no irrational numbers unless one side of fraction is irrational number?

Same way, 0.99999999999.... can never be equal to 1.0, unless we agree that at some point difference is so small that it could be considered insignificant and for practical reasons it should be considered to be non existing.

But while such endlessly repeated decimal number 9 could be close to 1 enough to be able to be represented as fraction of two equal/same integers, (but here we have problem of choosing which fraction it is as there is infinite number of such fractions), 0.9 or any number that do not satisfy requirement of minimal or less than minimal difference size to whole number/integer should not be considered to be rational.

But then any decimal number can be represented as whole number if multiplied by 10 raised on number of decimal places it has and divided by same number, so there are no irrational numbers?

I could understand that square root of 2, 3 or 5 can be considered to be irrational number because there are is no integer that squared gives such result, but so should be case with any prime number?

Anyhow, what is big deal with irrational numbers?

I have read claims that there is more of them then rational numbers, which seems only logical because even if there is infinite number of integers (whole numbers), there should be at least twice as much decimal numbers because they have both integer and decimal part.......

Of course, since even concept of infinity is hard to grasp to some, then to say something is in number two times infinity is not possible to be comprehended, therefore it must be irrational as there can be only one infinity, right?

I'm sorry, but most of that is quite wrong. For one thing, π ≠ 3.14; only approximately, π ≈ 3.14.

Absolutely! ...er, so to speak. Which brings me back to my earlier triangle conjecture except this time with reference to pi. In a circle c=πd. If d=1 does the circle ever complete? But then if in this same circle we arbitrarily set the circumference to 1 based on a different measuring rod; from that frame of reference d becomes irrational. This begins to take on the character or Relativity and Quantum physics or is the defect in our numbering system?

I know. So it is very rough approximation, true. It has little influemce on rest of my discussion.

Same way, π ≠ 3.14 it is much less 314 You turned it into.........

I think you, and others, are confusing precise mathematical definitions with various irrelevant psychological concepts. There is nothing crazy about irrational numbers, nor fictitious about imaginary numbers (which just so happen to be the best way to understand certain electrical concepts).

1/3.14 is exactly equal to 100/314, which is a ratio of integers, and hence a rational number. Neither pi nor 1/pi is rational, but pi is not equal to 3.14; it is only approximate. [I am now using "pi" because "π" is in a dopey font that looks the letter "n". That's an Editor Crankshaft thing..., and "δ" is equally bad.]

No informed person is hung up on the rationality or irrationality of various numbers. Instead, the problem is with persons who do not understand the concepts correctly.

I need to paste into here some other items that seem to cancel this post, so I will continue shortly....

Must be You confused me with someone else.

I first gave explanation that >>irrational<< here means >>our of ratio<<, not crazy......

As I told You before, result of division of 1 by 3.14 IS equal to result of division of 100 by 314, but so is same as result from division of 1000 by 3140 and other greater number that is on different scale of 10 on xth.

My reasoning that it must be considered to be irrational is because You cannot restore original 1:3.14 ratio.

What I do not understand is importance of >>irrational<< numbers in practice, or conversely, importance of >>rational<< numbers.........

What is difference in engineering, for instance?

It seems to me that only difficulty is getting final value if there is irrational number used, therefore closest approximation has to be used.......

What on earth are you talking about that I can't restore the original ratio of 1:3.14?

For 100:314, I simply divide by 100 to get back to 1:3.14. For 1000:3140, I divide by 1000. Or it could be the reduced fraction 50:157, and I could still divide by 50 to get 1:3.14 again. And so on ad infinitum. Once again, I'm sorry, but you are really confused. And even worse, you do not listen to genuine information, nor digest it.

It Seems to me that we talk like blind and deaf persons.

If You have just 0,3184713375796178 and do not know it was result of 1:3.14, it could have been result from many different scaled ratios, as You have pointed out. How would You know by which number You should divide it to get original ratio?

That is why I say that it cannot be restored, as rule implicit integers on both sides, not fractions......

Anyhow, De gustibus non est disputandum..........

I asked what is practical importance and practical difference between rational and irrational numbers in engineering?

Because you yourself gave the original ratios of 1:3.14, 100:314, 1000:3140, etc. So any fool can easily restore the original ratio, or create infinitely many other equal ratios. It is trivial by inspection to know what number to divide by in order to do this.

You are correct to observe that beyond 2 or 3 significant digits, this would not normally matter in engineering calculations. (Partly because of various uncertainties in practical measurement.)

But this thread is not about engineering; it is about mathematics, in which greater analytical exactitude is required, and can be attained. You have to start by knowing the correct definitions of various terms. You really need to take some valid classes in the subject, or absorb better what is taught.

Okay, I'm back.

For the number 0.9999..., you must realize that the ellipsis (the 3 dots) means that the 0.9999... never stops. Of course, 0.9999... can never be greater than 1. But for any number less than 1, you can extend the sequence 0.9999... to be greater than that number. Now, if 0.9999... is greater than any number less than 1, and less than any number greater than 1, then it must be equal to 1. (Contrary to what you said.)

For √2, √3, and √5 to be irrational, it does not suffice to prove that they are not some other integer. You have to prove that they are not any possible fraction, either. Euclid did this a long time ago, as already mentioned. For √n to be irrational, it is not necessary for n to be prime; it is only necessary for n not to be a perfect square. For example, √6, √8, √10, and gazillions of others, are irrational.

You will need to study up on Georg Cantor to understand how the rational and algebraic numbers are only countably infinite, but the irrational (and transcendental) numbers are uncountably infinite.

No, there is not only one infinity. Some varieties of infinity can be shown to be different from others. It is even possible to choose whether or not there is an infinity between that of countable (the rational or algebraic numbers) versus continuum (the real numbers). That gets pretty deep, though.

To understand all of this, one must get certain agreed-upon mathematical defintions correct, not psychological definitions.

Thanks for explanations, now I am much more enlightened :-))

Still I cannot agree that 0.9999999999..... is 1 as it would be always little less than 1, even if this is number closest to 1.

If what You say would be correct, then nobody would write 0,999....., they would always write 1.

I am no mathematician, so forgive my ignorance.

I also have no hobby exploring infinitive systems. Since nobody can handle anything with infinitive dimensions, I am usually not concerned with such matters.

What I have told about decimal numbers being twice as numerous than whole numbers stems from fact that each whole number can have also decimal places, but thinking back difference is not just double as each number can have infinite number of decimal places and combinations of decimals, therefore decimal numbers vastly outnumber infinite number of integers.........

That said, I shall not discuss infinite systems any more, as I have other things to do, sorry.......

Once again, you are not familiar with the concepts you are trying to discuss. The "numbers" or "cardinalities" of the integers, the terminating or repeating decimal fractions, the rational numbers, and the algebraic numbers are all identical, because each of these other sets can be placed in one-to-one correspondence with the integers. These are all countably infinite (enumerable) and have cardinality aleph-null. (Aleph is a Hebrew letter that looks like an ornate N, but I don't know how to access here the font to show it; so I will just say N₀.) The irrational numbers have cardinality c (continuum); in crude terms, there are more of them.

Any finite expansion of 0.9999...9 is less than 1, but the three unterminated dots of 0.9999... are not a finite expansion. The whole point is that the expansion continues forever, and it must equal 1, because it certainly isn't larger than one; and for any chosen number less than 1, the expansion can be continued (finitely at that) to exceed the chosen number. For a limitlessly extended expansion, the concept "always" does not fit. There is no such thing as the "number closest to 1". Whatever number you might pick, it is possible to find an even closer number. (Just split the difference between the chosen number and 1.)

Since you admit to not understanding infinity, why are you arguing about it? One of the beauties of algebra is precisely that it can comprehend infinity.

Now, if 0.9999... is greater than any number less than 1, and less than any number greater than 1, then it must be equal to 1.

but, it is also less than any number equal to 1, then it can't be 1.

Sorry, but that is incorrect. Each successive finite sequence of 9's is less than 1, but the three unterminated dots express an infinite sequence. Please take a calculus class. Or, if you didn't get it the first time, take it again. Etc., ad infinitum if need be...

Or, if you didn't get it the first time, take it again. Etc., ad infinitum if need be...

LOL!! only if you're the professor ...

Never took calculus. I know that there are infinite sequences but will a baseball player ever attain a 1.000 (known in the sport as a one thousand batting average) if he makes out his first at-bat, even if he hit successfully thereafter ad infinitum?

Only in an infinite time!

To put it more technically, if n is the number of at-bats, at each stage this guy's average is (n-1)/n, which also equals 1-1/n. The resulting sequence is 0, 1/2, 2/3, 3/4...99/100...999/1000...999999/1000000. At any FINITE stage, you don't reach 1 yet. Maybe it will help to think of this process as projecting beyond where you are at any given time.

More technically, this would be written as lim_n→∞ [(n-1)/n] = 1. No one is saying that any individual member of the sequence, say 9999999999/10000000000, is itself equal to 1. But projecting beyond any finite stage to the INFINITE, you get 1, and not anything else. Some these colloquial explanations may be helpful, I hope.

The (∞-1)/∞ way of looking at it is good, I think. It's pretty intuitive and easy to understand. (There might be a puristic question about this, but I am more interested in the basic concept so long as no important details are violated.

A quick look at open and closed sets wouldn't go amiss here.

Well, have it Your way, for practical use it is so close to 1 as it could be......

I have been explained that there would always be some difference infinitesimally small, and that it can never become 1.

Since this explanation is logical to me, I shall continue to think of it this way.

Still You did not answer my questions about practical use of rational and irrational numbers, save that one knows it is not possible to get final result of calculation with irrational numbers, so we must determine in advance how precise result we need in each case.......

An apt explanation, GA.

Thanks!

Actually, I find Sqrt(2) or Sqrt(3) or pi far more informative than 1.414... or 1.732... or 3.14159... Not only are they easier to write, they can provide insight into the processes being studied (such as the use of Sqrt(3) in voltage/amperage relationships in 3-phase power, or Squrt(2) in analysis of quadratic equations). Pi is handy for frequency analysis, not only for drawing circles- which helps one underdant the cyclical nature of a process, and so on...Note also that Sqrt(2) etc are EXACT values, while 1.414... etc. are approximate values.

I recently came across a problem expressed something like (I don't remember the exact problem):

.33x + .67y = H

.67x - .33y = K

A quick matrix analysis gives a quick solution (assuming one has values for two of the apparent unknowns above). But by multiplying the two equations by 3, rather than working with the decimal fractions, the relationships become obvious from visual inspection, without having to resort to exotic math. This is a simple case, and I have encountered such cases before.

I do both, depending on context. I majored in math and minored in education, but I wound up doing other things like crab fishing and refrigeration design/fabrication/operation. Spreadsheets and engineering tables are typically in decimals to a few places, but it often helps to recognize the common constants and square roots for what they really are.

I agree totally with your emphasis that √2, √3, and π are exact; whereas 1.414, 1.732, and 3.1416 are only approximate. The distinction has been blurred here and there in this thread.

Right. Now Pi radians equals 180 degrees and is a whole number. So, (number of degrees in a radian), i assume, is another never-ending number, its product with Pi being a perfect number...yes? How do we know ?

(i hope this won't bring another stinging reprimand)

Interesting, interesting..........

Are they just curiosity or there is practical use?

I know some crystals form 6 sided prisms ( SiO2) and honeybee nest is made out of 6 sided cells, so I used such prismatic shapes for my invention structure, and now that I was searching for some such greater number, I think I shall use 28 next :-))

I have idea for making moire lenses of Fresnel type that would be able to catch and focus sunlight from different directions and focus it on same place all day long, which would come handy (if possible at all) in accumulation of Solar heat energy, so maybe it is possible to make 28 circles all passing trough center of 29th circle of same size, where each circle contain 28 concentric circles inside like original Fresnel lens.....

If this would work (I need simple light focusing feature, no need for optical clarity at all) then much more energy could be harvested from Sun without need for heliostat devices......

No moving parts means no energy is expended and there is no wear and tear, therefore no need for repairs and replacements, maintenance and the rest.......

I hope You may tell me if I am on right track, as I am no engineer nor Physicist, and certainly no mathematician.........

I would name it TORNADO Lenses in Your honour as You inspired me with this post about perfect numbers :-))

Excellent Henrik. i will keenly watch your progress and may you succeed beyond your expectations...more power to you

I appreciate your generosity in proposing the "Tornado lenses," but I would respectfully decline such an honor.

In the context of pure math, 28 has the role of being the third perfect number, and no doubt some other nice properties as well.

However, it possesses no particular magic as to the number of rings in a basically planar Fresnel lens, and even less so in a cylindrical Fresnel lens. (Ship indicator lights and lighthouses are examples of cylindrical Fresnel lenses, but they employ much fewer than 28 "ribs").

Yes, usually there is much less than 28 ribs, but I would say this is because usually there is some strong light source close behind lenses and their purpose is to make light rays go out in parallel.

I would like to reverse their function to focus basically parallel rays of Sun......

Only, where light source in use of usual Fresnel lenses is fixed in position to lenses, I have moving light source and stationary lenses, so lenses should be able to catch and focus light of Sun regardless of its position in the sky during the day, and regardless of Sun declination during the year.

If I would have some formulas to calculate refraction of light on prisms, I would be able to write simulation program to create such virtual lenses and see if such moire lenses would really focus sunlight continuously to one stationary point underneath such lenses, and under which configuration it would do so....

I have seen Fresnel Lenses applied on surface of cylinder which contains reservoir for heating water painted black underneath, which is also supposed to refract light regardless of Sun declination and position during the day, but not on same side of cylinder while Sun apparently moves from East to South and then to West.

My idea is to eliminate Heliostat mechanism from use of Solar Thermal concentrators, thus making such systems cheaper and more reliable, as there would be less wear and tear possible, so there would be no need for maintenance and repair, and such plates could be made from recycled poured glass, also usable in desalination plants that use water distillation process.

I guess this would be more appropriate problem for mathematicians like You to solve, if my idea is right in first place, of course.........

Therefore, can You tell me if this would be at least theoretically possible or not?

I am just asking for Your opinion, mind You, I think it would be qualified one......

In case there is some other CR4 member with experience specialized to Fresnel Lenses use, I invite them to tell me what they think of such idea, also!

Refraction is governed by Snell's Law: sin θ₁/sin θ₂ = v₁/v₂ = n₁/n₂. (When light passes from medium 1 to medium 2; the sines of the angles of travel, the velocities, and the indices of refraction are all in the same ratio.) As a consequence of this, if direction of the light source changes, so will the direction of the light's path in the second medium. I think this rules out your objective. The Wikipedia articles on Snell's Law and Fresnel lenses are good summaries.

True, but each set of circles should have different angles cut, that would be proper for one Sun position during the day, so there would be 14 Sun positions covered for 10 hours during the day when there is enough sunlight to be useful, and 2 instances of 14 sets should cover Summer and Winter declinations of Sun trajectory.

What I would need to have is formula that show what slant other surface of denser media has to have so light ray coming vertically on first surface would be refracted at required angle to act as lens and so concentrate light in a point below this Fresnel Lens. That I think could be taken from standard Fresnel Lens that have just one set of concentric circles. Then 14 Fresnel Lenses should be cut for unique Sun position in the sky during the day and declination for Summer, while another 14 would be cut for same positions and Winter declination. Fresnel Lenses are basically collection of prisms with differently slanted top surface, that is how they can appear to be flat and still act as standard lenses. Unfortunately such informations are not available to me.

Sunlight need not be concentrated in very small focus (like one square millimeter) that would be on exactly same place all the time, also this focus can wander inside wider circle and still be useful if purpose is channeling this concentrated light heat on some receiving surface of thermal device like Pelletier plate or into thermal reservoir to melt salt and thus accumulate heat.

Now, do You think it is possible?

This all depends on whether you want to focus the insolation accurately onto a small area, or if the focusing needs only to be approximate. There are some other CR4 threads by "gaiatechnician" that are working on a similar idea for solar cookers without tracking. A linear parabolic reflecting trough, if aimed in the ecliptic plane, will do pretty well. If it can swivel perpendicular to the sun, it will perform even better, which can be done by a simple 24-hour clock drive.

I haven't matured the idea yet, but I have a concept that combines all the concepts mentioned thus far in a way I think could be promising. The key variable is the area of the solar collector, as is always the case.

Such Solar cookers are already produced in India, 5000 per month.....

You can buy one with or without Solar tracker...........

In regard to my idea, yes, it has to be only in approximate position as there could be light-pipe opening in size of maximum variance from some point where light would be focused at noon at time of equinox, so to encompass most Southern and Northern declination and Sun positions on East or West during the day......

I believe that use of such moire Lens would concentrate light most efficiently, and size of course depend on how much thermal energy has to be produced, where 1m2 receive 1 KW of light energy. Therefore for 1 MW one need 1000m2 or 34x34 meters square.

Check this out for a value of Pi.

indiana pi bill

Talk about being proud of your fellow citizens.

With Pi being so much in the discussions here, i have a doubt here .. is it 3.1415626535... or is it closer to 2 ? A circle of diameter 2 will have a half-circumference of Pi .. when we sub-divide the arc as shown ad infinitum... will Pi tend to 2 ???? (The pi/2, pi/4 do not show up at this size, but you can guess)

Please, this may be juvenile, please don't kick me

Sorry for the typo of Pi value ...

pi is the ratio of whole circumference to it's diameter, not half-circumference to diameter.

Well... please check ... what i said is that the half-circumference is Pi. The diameter is 2, therefore the full circumference is 2xPi therefore the half-circumference is Pi .. OK?

Ok, but each subdivision will have it's unique circumference and it's unique diameter which ratio will always be 22/7

If you compute the total length of the semicircular arcs for each successive subdivision, it always comes out as pi, even if the individual arcs are subatomically small. This sum never departs from pi, so it can't approach 2. The arcs soon become so small that collectively they look like a line segment, but with a powerful enough microscope, they would still be seen as semicircles.

At the first step, there is one arc of length pi;

At the second step, there are two arcs each of length pi/2; total length pi.

Etc.;

At the nth step, there are 2^(n-1) arcs, each of length pi/2^(n-1). In every case, 2^(n-1) × pi/^2(n-1) = pi.

[In Eudlidean geometry, that is; in black holes, pi → 1; where → means approaches.]

Despite some differences of perspective, which I hope will prove transitory, this thread has raised some genuinely interesting issues. Corelite's question seems to be based on a nonmathematical usage of the term "irrational," which may be naive, but it opens up an intriguing exploration into just what sort of reality is possessed by mathematical concepts. I do not know the answer to this, but I have some ideas on it. Maybe this angle won't attract wide interest, but I would regret to see the thread closed down. Rather, I would hope for a variety of contributions that might actually break new ground.

Corelite's question seems to be based on a nonmathematical usage of the term "irrational," which may be naive, but it opens up an intriguing exploration into just what sort of reality is possessed by mathematical concepts.

If it may be more of a philosophical concept

I don't know of any test for 7: I always thought there was none.

But I suppose if the number is ....EDCBA

then if A +3B +2C -D +4E (or -3E) etc. is a multiple of 7 then the original number is, but that seems a bit contrived and not a lot of help really.

GA. This certainly added to my knowledge. (Thanks shridhar for opening the thread).

But the rule you have given seems to apply only after certain number. I do not from where it starts applying. For example: 147 is divisible by 7, but your rule doesn't apply. But, next divisible number 154, 161 fits in the rule. Again 168 is out of the rule.

It would be interesting to find from which smallest number rule applies 100%.

Please recheck; the rule does work for all the given examples. The rule is a consequence of 7 dividing evenly into 98.

I said" But the rule you have given seems to apply only after certain number. I do not from where it starts applying. For example: 147 is divisible by 7, but your rule doesn't apply. But, next divisible number 154, 161 fits in the rule. Again 168 is out of the rule."

According to rule

147 separated in 14 & 7 2 X 7 = 14 14 - 14= 0...... Rule doesn't work

154 separated in 15 & 4 2 X 4 = 8 15 - 8 = 7 .... divisible by 7 rule works

161 separated in 16 & 1 2 X 1 = 2 16 - 2 = 14 ....divisible by 7, rule works

168 separated in 16 & 8 2 X 8 = 16 16 - 16 = 0 Rule doesn't work

Yes, they all do work. All integers, including 7, divide evenly into zero, with a quotient of zero and a remainder of zero (it is the remainder that matters). Please don't forget who invented the zero!