Periodic number?

[Dale10101]

http://www.cut-the-knot.org/do_you_know/numbers.shtml#real

is a page from a very interesting math website, after an explanation of Cantor's definition of rational and irrational numbers in terms of periodic and non-periodic sequences the question is asked:

Is the number 1234567891011121314151617181920212223... periodic?

Well, nuts, on the one hand I see no repeating pattern that one can draw a bar over to represent the extent of a period, on the other hand, as a sequence of counting numbers, its extension is perfectly predictable and that is not characteristic of what I expect of a decimal representation of an irrational number.

I am going to guess periodic but really I cannot rationally make a choice so I guess I am being irrational, or is that just me being imaginary?

Going nuts here, any help please?

 

[stapel]

At what point in the eternal listing of un-like numbers do you expect the counting numbers to "repeat", that this would be a "periodic" expansion?

 

[Bob Brown MSEE]

http://www.cut-the-knot.org/do_you_know/numbers.shtml#real

Is the number 1234567891011121314151617181920212223... periodic?

Well, nuts, on the one hand I see no repeating pattern that one can draw a bar over to represent the extent of a period, on the other hand, as a sequence of counting numbers, its extension is perfectly predictable and that is not characteristic of what I expect of a decimal representation of an irrational number.

" draw a bar over to represent the extent of a period," is what they mean when they say periodic.
the number 1234567891011121314151617181920212223... IS NOT periodic! But it has a pattern that can be described as a rule. All irrational numbers that are 'interesting" enough to have a name, have rules that define them (like a series, solution to an expression, or a geometry). Seldom are you able to give a description of the digits directly.

Since it is not periodic, it is irrational.

 

[Dale10101]

Ah

OK, so M Stapel, no I did not expect the pattern to repeat any time before lunch or the extinction of the sun but I did think that the essence of a period was a state-able rule providing predictability which was what the sequence offered and hence I thought that the rule of periodicity might have been subsumed under a larger category, after all it was a tricky sort of question. I think I was associating rational with predictable and irrational with unpredictable.

I am glad I asked, this is precisely the sort of error that I could carry around for years otherwise. So irrational numbers might not have a predicable sequence like, pi, but they might have a predictable sequence, like the example above.

M. Bob, thanks for clarifying the difference between the particular type of pattern that denotes a period and the general notion of a pattern which "interesting" irrational numbers might exhibit.

This site is a great resource.

 

[HallsofIvy]

http://www.cut-the-knot.org/do_you_know/numbers.shtml#real

is a page from a very interesting math website, after an explanation of Cantor's definition of rational and irrational numbers in terms of periodic and non-periodic sequences the question is asked:

Is the number 1234567891011121314151617181920212223... periodic?

Well, nuts, on the one hand I see no repeating pattern that one can draw a bar over to represent the extent of a period, on the other hand, as a sequence of counting numbers, its extension is perfectly predictable and that is not characteristic of what I expect of a decimal representation of an irrational number.

I am going to guess periodic but really I cannot rationally make a choice so I guess I am being irrational, or is that just me being imaginary?

Going nuts here, any help please?

One can show that a number is rational if and only if it is "eventually repeating". But tht does NOT mean "perfectly predictable". A commonly used example of an irrational number is .101001000100001... which clearly is not "eventually repeating" because there is always an additional 0 between each pair of ones. But that is clearly "perfectly predictable". In fact saying that an infinite sequence of digits is the decimal expansion of a specific irrational number implies that those digits are "perfectly predictable". $\displaystyle \sqrt{2}$ is an irrational number but its digits are "perfectly predictable" by using a square root algorithm to find them. $\displaystyle \pi$ is an irrational number but there exist a number of algorithms for finding the decimal expansion- so its digits are "perfectly predictable".

 

[Dale10101]

What the ...

Grrrr … irrational numbers indeed! Do they even exist, or are they merely artifacts of algebraic machinations? I mean, by definition they cannot be measured right? They remind me of the invention of zero, another sort of fantasy figure never to be seen … grrr … minutes spent biting the floor.

OK, so counting numbers seem reasonable, tangible even, same with fractions, and then there is the problem of the hypotenuse of a right triangle with sides of 1 so that its hypotenuse is SQRT(2) by the Pythagorean Theorem. So what is the length SQRT(2)?

It is a simple proof by contradiction to show that the square root operation on 2 cannot yield a rational number. So what is to be done? Well first of all give such results a name, “the irrational ones”. Appropriate on more than one level.

OK, then further investigation shows that while the division algorithm always produces a decimal representation with a repeating “pattern” of a certain period length, the square root algorithm when applied to certain integers like 2 and 5, produces a decimal extension that does not repeat. Nevertheless, thanks to M. H.I.’s comment (danke), one might say that by being the end product of a known algorithm both operations (division and square root-tation) produce a predictable result, a pattern of their own really --- There is a difference however, one pattern is periodic and the other is not.

Hmmm, having written that down it suddenly seems more significant, rational numbers are periodic in some sense, irrational numbers are aperiodic in some sense.

Does that mean anything?

I don’t know, I guess my question then, such as it is, boils down to this … is there any more fundamental data concerning rational and irrational numbers then the fact that they have irreconcilable decimal extensions? … that is, are they, black vs white, yin and yang, apples and oranges in the great salad of algebra …, or to put it oppositely, is there any additional principle which shows them to be commensurate in some fashion …

I am guessing not (maybe I can go 0 for 2), on the general observation that in nature not all things are commensurate, not all things fold neatly into one another …

hmmm, I recall an elementary demonstration of the Heisenberg uncertainty principle showing that the accuracy with which a wave's period can be measured as a function of the length of time over which it is measured. I think that would not be true if there were no such things as irrational quantities; there would be a minimum time (the reciprocal of the periods modulo times the period I think) but not an ever increasing accuracy with time.

Oops, another question enters begging on its knees … how long must one crank digits out the SQRT machine before one can guarantee that a repeating pattern does not appear?

(Which gives me a smarty pants reply to M Stapels wry question:

"At what point in the eternal listing of un-like numbers do you expect the counting numbers to "repeat", that this would be a "periodic" expansion?"

My response ... At the same point that you can assure me that no repeating pattern will appear ... yes I know about this stacking of the deck according to Peano's postulates, PMI et al ... but, perhaps what appears as the progression of integers is in fact not ... maybe the numbers shown are only suggestive of what is to follow, maybe the apparent pattern will not hold ... haw - ha, as I proudly preen )

Anyway, ignoring the digression, apparently this guarantee can be proved/computed otherwise the fundamental distinction between rational and irrational numbers falls apart … but then that argues for a further fundamental relationship between rational and irrational numbers doesn't it?

Shoot, I am feeling the compulsion to chew on the furniture again. I better go practice factoring polynomials for a while. Pardon my mania. Cheers.

(Actually, as I continue to think about it I think I am seeing where investigative computations led to the study of sequences and limits ... hours later, trying to move my fingers toward the send button ... must stop this ... need a break ... math as crack cocaine ... abandoned children, disheveled appearance, splinters in my tongue, fears that my dying thoughts will be about prime numbers ... send, submit, send, submit ....:wink

 

[HallsofIvy]

Grrrr … irrational numbers indeed! Do they even exist, or are they merely artifacts of algebraic machinations? I mean, by definition they cannot be measured right? They remind me of the invention of zero, another sort of fantasy figure never to be seen … grrr … minutes spent biting the floor.

That's right. It's all a plot by the evil mathematicians who are trying to take over the world! (Of course, they have been trying for a couple of thousand years now- kind of incompetent, aren't they?

OK, so counting numbers seem reasonable, tangible even, same with fractions, and then there is the problem of the hypotenuse of a right triangle with sides of 1 so that its hypotenuse is SQRT(2) by the Pythagorean Theorem. So what is the length SQRT(2)?

Uh, it's the length of the hypotenuse, of course!

It is a simple proof by contradiction to show that the square root operation on 2 cannot yield a rational number. So what is to be done? Well first of all give such results a name, “the irrational ones”. Appropriate on more than one level.

Absolutely. If you don't know what something is, give it a name!

OK, then further investigation shows that while the division algorithm always produces a decimal representation with a repeating “pattern” of a certain period length, the square root algorithm when applied to certain integers like 2 and 5, produces a decimal extension that does not repeat. Nevertheless, thanks to M. H.I.’s comment (danke), one might say that by being the end product of a known algorithm both operations (division and square root-tation) produce a predictable result, a pattern of their own really --- There is a difference however, one pattern is periodic and the other is not.

Okay, that's reasonable.

Hmmm, having written that down it suddenly seems more significant, rational numbers are periodic in some sense, irrational numbers are aperiodic in some sense.

Does that mean anything?

I don’t know, I guess my question then, such as it is, boils down to this … is there any more fundamental data concerning rational and irrational numbers then the fact that they have irreconcilable decimal extensions? … that is, are they, black vs white, yin and yang, apples and oranges in the great salad of algebra …, or to put it oppositely, is there any additional principle which shows them to be commensurate in some fashion …

Okay, now you are starting to confuse me! (Not that that is hard to do.) But basically, what you have said is "all there is". "Rational numbers" are numbers that can be written as a fraction: an integer over a non-zero integer. "Irrational numbers" are all other real numbers. (I'm not sure this is what you are talking about but there are other ways of dividing up the real numbers. The "algebraic numbers" are all numbers that satisfy some polynomial equation with integer coefficients- that includes all rational numbers and some irrational numbers. The "transcendental numbers" are all real numbers that are not algebraic.)

I am guessing not (maybe I can go 0 for 2), on the general observation that in nature not all things are commensurate, not all things fold neatly into one another …

hmmm, I recall an elementary demonstration of the Heisenberg uncertainty principle showing that the accuracy with which a wave's period can be measured as a function of the length of time over which it is measured. I think that would not be true if there were no such things as irrational quantities; there would be a minimum time (the reciprocal of the periods modulo times the period I think) but not an ever increasing accuracy with time.

Physics is NOT mathematics. No physical law says anything about mathematics.

Oops, another question enters begging on its knees … how long must one crank digits out the SQRT machine before one can guarantee that a repeating pattern does not appear?

There is no finite answer. But one can prove, without having to do the calculation, that rational numbers always are "eventually repeating", irrational numbers are not. If you are asking "how do you determine whether a given number is rational nor irrational", certainly not by 'cranking out digits'. Of course, to 'crank out digits', you must have been given some specific definition of the number. It would be easier to use that definition to determine whether or not it can be written as a fraction.

Apparently this can be proved/computed otherwise the fundamental distinction between rational and irrational numbers falls apart … but then that argues for a further fundamental relationship between rational and irrational numbers doesn't it?

Shoot, I am feeling the compulsion to chew on the furniture again.

I had a dog with that problem. We had him neutered.

I better go practice factoring polynomials for a while. Pardon my mania. Cheers.

(Actually, as I continue to think about it I think I am seeing where investigative computations led to the study of sequences and limits ... hours later, trying to move my fingers toward the send button ... must stop this ... need a break ... math as crack cocaine ... abandoned children, disheveled appearance, splinters in my tongue, fears that my dying thoughts will be about prime numbers ... send, submit, send, submit ....:wink

Try putting "think" somewhere in that sequence!

 

[JeffM]

Grrrr … irrational numbers indeed! Do they even exist, or are they merely artifacts of algebraic machinations? I mean, by definition they cannot be measured right?

Correct. So if you take a radically empirical approach, irrational numbers do not exist. On that basis, moreover, you can dispense with rational numbers because, for any given problem, there is always a unit of measurement that will allow everything else to be described in terms of an integer number of those units.

If you want, you can say that rational numbers, real numbers, complex numbers are idealizations created by the human mind and never observed in physical reality. I have seen a pie, but have I ever seen precisely one half of a given pie and if I did, how would I know? The problem with that position is that much of physical science is far easier to describe utilizing concepts like real numbers. The idealizations (if that is what they are) make solving empirical problems much much easier. So the idealizations have great practical utility.

They remind me of the invention of zero, another sort of fantasy figure never to be seen

If I tell you to count how many coats are in a closet, one possible answer is none. If I elect to call "none" "zero" I have merely changed the name of an experience quite often observed in the physical world.

is there any more fundamental data concerning rational and irrational numbers then the fact that they have irreconcilable decimal extensions? … that is, are they, black vs white, yin and yang, apples and oranges in the great salad of algebra …, or to put it oppositely, is there any additional principle which shows them to be commensurate in some fashion …

There is a famous proof by Cantor that shows that rationals and irrationals are not commensurate.

 

[Bob Brown MSEE]

Oops, another question enters begging on its knees … how long must one crank digits out the SQRT machine before one can guarantee that a repeating pattern does not appear?

(Which gives me a smarty pants reply to M Stapels wry question:

"At what point in the eternal listing of un-like numbers do you expect the counting numbers to "repeat", that this would be a "periodic" expansion?"

My response ... At the same point that you can assure me that no repeating pattern will appear ... yes I know about this stacking of the deck according to Peano's postulates, PMI et al ... but, perhaps what appears as the progression of integers is in fact not ... maybe the numbers shown are only suggestive of what is to follow, maybe the apparent pattern will not hold ... haw - ha, as I proudly preen )

Anyway, ignoring the digression, apparently this guarantee can be proved/computed otherwise the fundamental distinction between rational and irrational numbers falls apart … but then that argues for a further fundamental relationship between rational and irrational numbers doesn't it?

Very insightful! You can be sure that SQRT(2) never repeats.
As Hallsofivy pointed out, there exists an easy proof that a decimal representation that repeats must be rational.

But a second proof that SQRT(2) is irrational is necessary to prove that SQRT(2) never repeats. Click Here

PS: You may enjoy this class on the foundations of mathematics: Click Here
Be sure to watch class #MF80 in this series about SQRT(2): Click Here

 

[Dale10101]

Well

Please Doc Halls of Ivy, lighten up, I think you have a deaf funny bone. I could make a withering come back to your neutering comment that would concern dogs as a metaphor for students, but it is difficult to read intent over the internet. I am sorry that my questions and style seem to disturb you, but you know, take a pill.

Having said that I do respect that you are here, that you are sincere, that you imparting high quality knowledge. Let me say then that I neither disrespect mathematics as a human endeavor nor teachers, quite the opposite.

I think you took my hyperbole too seriously (maybe I am doing the same thing); I was caricaturizing math passion, a mixed blessing considering my very average abilities. I mean, surely you too must find yourself occasionally self-sequestering, being late for appointments, declining social engagements because some vague enticing math inspiration has come your way, usually fool’s gold … ah but the moment of excitement and the occasional insight however mean.

BTW, my comment to M. or Dr (?) Staple was naturally another example of caricaturization, obviously his knowledge is leagues beyond mine and could in a battle of math wit squash me like a bug on fool’s parade … but again, and one must continually self-remember that intent is not so easily conveyed over the faceless, voiceless medium of the internet. I suppose the safe bet is to indeed neuter ones sense of humor and persona … or not, and trust over time a correct balance of sincerity, irony, and genuine disdain (not my style) is reached.

You did accord me the respect of a detailed response and I will look at it closely because I will almost assuredly learn something. I do think I failed to express what is bothering me, and perhaps will try again … honestly, I do find this type of thinking intoxicating and find it difficult to unplug.

Meanwhile I would say this, that you are in the middle of your life with it’s successes and failures and I in mine with it’s successes and failures … I am mindful of that. I hope that occasionally you are rewarded not only by the gratitude of those you help with your knowledge but by insights of your own gleaned from the odd reflections of other's confusions. Cheers. Dale

 

[Dale10101]

Thank you

Collectively the information shared gives me what I sought, a much clearer understanding of the nature of irrational numbers and the role they play in algebra. Truly interesting, all help appreciated.

 

[Dale10101]

point to point, no, not a diatribe, too long, so, part 1.

Originally Posted by Dale10101


Grrrr … irrational numbers indeed! Do they even exist, or are they merely artifacts of algebraic machinations? I mean, by definition they cannot be measured right? They remind me of the invention of zero, another sort of fantasy figure never to be seen … grrr … minutes spent biting the floor.

That's right. It's all a plot by the evil mathematicians who are trying to take over the world! (Of course, they have been trying for a couple of thousand years now- kind of incompetent, aren't they?

OK, I apologize, I was being theatrical. How were you to know. I should have at least set off the comments in quotes and better yet given the character a name like:

MadMathMax: Grrr … etc

Should I even being doing this in the context of a math club? Well, wasn’t it the great pianist Ignacy Paderewski who, in reply to the Queen of the England’s effusion “ Ignacy, you are a genius” reportedly replied, “Yes madam, but first I was a drudge.” My point being, math is incredibly interesting but like every disciple it is also a drudge. Why not be a bit theatrical, within bounds, if it helps one think and stay motivated and perhaps illustrates a point … after did not Schrodinger speaks of cat coffins and Maxwell of trap doors and demons.

Mathematicians are not evil, quite the opposite, they are those most likely to notice Emperors without clothes, and be obstinate about reporting it ala Galileo.

Again, apologizes, my bad.


OK, so counting numbers seem reasonable, tangible even, same with fractions, and then there is the problem of the hypotenuse of a right triangle with sides of 1 so that its hypotenuse is SQRT(2) by the Pythagorean Theorem. So what is the length SQRT(2)?

Uh, it's the length of the hypotenuse, of course!

Yes, exactly so, set a miter saw to 45 degrees and lop the corner off a square ended board, that is what we are talking about, a right triangle with equals sides … the hypotenuse exists with a definite length.

The rub is apparently (and of course I am speaking as one trying to comb out sophomoric ignorance, always that) that, symbolically, formulating a number that captures what is now exists before us is not so easy as say one might have supposed, not as easy as expressing a sum, or proportion.

As you have stated below, “Physics is NOT mathematics. No physical law says anything about mathematics.” One might say then that the hypotenuse does not at that point exist in the ‘world of math’ since nothing in the math world really exists until it can be formulated in terms of fundamental truths/axioms … didn’t O. Heavyside take heavy heat using Laplace transforms . Naturally this was remedied by investigations into solutions for such expressions such as the SQRT(2) and doing so, as I understand it, brought to light/invented/discovered/gave credence to, irrational numbers, but prior to that … in the math world … what was the length of the hypotenuse … it was the SQRT(2), essentially an undefined point on the number line. That is what I was thinking.

Again, how were you to know. It is difficult to ask conceptual questions I think, certainly more difficult then asked how manipulate an expression to a desired result and even more difficult then how to approach a word problem. I say that without deprecation.

It is a simple proof by contradiction to show that the square root operation on 2 cannot yield a rational number. So what is to be done? Well first of all give such results a name, “the irrational ones”. Appropriate on more than one level.
Absolutely. If you don't know what something is, give it a name!

Of course, when you give something a name you can refer to it, discuss it. I was making light of the meaning of “irrational” both within and outside of math while at the same time thinking about their intersection, but perhaps it did not come out right.

OK, then further investigation shows that while the division algorithm always produces a decimal representation with a repeating “pattern” of a certain period length, the square root algorithm when applied to certain integers like 2 and 5, produces a decimal extension that does not repeat. Nevertheless, thanks to M. H.I.’s comment (danke), one might say that by being the end product of a known algorithm both operations (division and square root-tation) produce a predictable result, a pattern of their own really --- There is a difference however, one pattern is periodic and the other is not.
Okay, that's reasonable.

Thank the Prime Mover, I was hoping I had that much straight.

 

[Dale10101]

point to point, part 2

Hmmm, having written that down it suddenly seems more significant, rational numbers are periodic in some sense, irrational numbers are aperiodic in some sense.

Does that mean anything?

I don’t know, I guess my question then, such as it is, boils down to this … is there any more fundamental data concerning rational and irrational numbers then the fact that they have irreconcilable decimal extensions? … that is, are they, black vs white, yin and yang, apples and oranges in the great salad of algebra …, or to put it oppositely, is there any additional principle which shows them to be commensurate in some fashion …

Okay, now you are starting to confuse me! (Not that that is hard to do.) But basically, what you have said is "all there is". "Rational numbers" are numbers that can be written as a fraction: an integer over a non-zero integer. "Irrational numbers" are all other real numbers. (I'm not sure this is what you are talking about but there are other ways of dividing up the real numbers. The "algebraic numbers" are all numbers that satisfy some polynomialequation with integer coefficients- that includes all rational numbers and some irrational numbers. The "transcendental numbers" are all real numbers that are not algebraic.)

Okay then. Good. Yes, I discovered the difference between algebraic numbers and transcendental numbers a while back and am still excited by the distinction. In the first place it seemed a change of perspective to see types of numbers defined by the equations they can solve rather than the reverse, the usual pedagogic introduction of numbers then equations. (That might be vague.) Also, to realize that Transcendental numbers as a type are distinct because they cannot be expressed as polynomials. That shed some light on what one can and cannot do with algebraic manipulation but I am still learning the limits.

The somewhat literary license, ying, yang etc was at heart, as I see it now, recognition, it seems to me, of a duality in the math world. In a sense it seems odd to me that it is the nature of things that irrational numbers exist. They sort of remind me of the discovery of radiation while we slept complacent in the world of classical physics, or the discovery black holes and dark matter just when science was about to become a branch of accounting. Sorry, but odd, very odd, and in both cases, the natural world, and the math world, were vastly expanded in our realization of their potentiality

But to refocus on the math. Yes rational numbers can be expressed as a fractions, irrational numbers cannot. The fundamental difference is that the former are periodic, the latter are not, meaning that every rational number will eventually find a unit mate somewhere along the number line if you tumble it exactly along enough times while the latter never will. That’s it, the fundamental difference, beyond that everything else is description and implication, seemingly huge implication, no doubt deep implication.

Dare I ask at this point if that is right? Probably not. I know, I a simpleton, one excited by simple things, which are not so easy to find, usually I am wrong many times before I am right.


I am guessing not (maybe I can go 0 for 2), on the general observation that in nature not all things are commensurate, not all things fold neatly into one another …

hmmm, I recall an elementary demonstration of the Heisenberg uncertainty principle showing that the accuracy with which a wave's period can be measured as afunction of the length of time over which it is measured. I think that would not be true if there were no such things as irrational quantities; there would be a minimum time (the reciprocal of the periods modulo times the period I think) but not an ever increasing accuracy with time.

Physics is NOT mathematics. No physical law says anything about mathematics.

Dealt with above although I wonder about the symmetry of math and science, certainly they influence each other’s direction of investigation but beyond I wonder if, given any set of mathematical axioms the ensuing implications/relations (especially what is not possible) might be in fact the same or an analog of those encountered in nature. Well I don’t really know what I am talking about on that point … so I will just say I am keeping an open mind (a fig leaf?).


Oops, another question enters begging on its knees … how long must one crank digits out the SQRT machine before one can guarantee that a repeating pattern does not appear?

There is no finite answer. But one can prove, without having to do the calculation, that rational numbers always are "eventually repeating", irrational numbers are not. If you are asking "how do you determine whether a given number is rational nor irrational", certainly not by 'cranking out digits'. Of course, to 'crank out digits', you must have been given some specific definition of the number. It would be easier to use that definition to determine whether or not it can be written as a fraction.

Yes. Good. I can see that rational numbers will eventually repeat … whenever the accumulated remainder adds up an even unit … I think.

Yes, if you have a computable sequence, then work with the algorithm rather than the sequence it produces … plausible to me without knowing enough to make use of the idea.

The only other point that I was thinking about is this, given a sequence that magically grows an additional digit every day, can you ever know if the sequence will be rational? Well, obviously not, I think. Any guess could be defeated by a next digit … hmmm, which I suppose is the reality of all existence …. how about another beer bartender.


Apparently this can be proved/computed otherwise the fundamental distinction between rational and irrational numbers falls apart … but then that argues for a further fundamental relationship between rational and irrational numbers doesn't it?

Shoot, I am feeling the compulsion to chew on the furniture again.

I had a dog with that problem. We had him neutered.

Ouch, but enough said about that.

I better go practice factoring polynomials for a while. Pardon my mania. Cheers.

(Actually, as I continue to think about it I think I am seeing where investigative computations led to the study of sequences and limits ... hours later, trying to move my fingers toward the send button ... must stop this ... need a break ... math as crack cocaine ... abandoned children, disheveled appearance, splinters in my tongue, fears that my dying thoughts will be about prime numbers ... send, submit, send, submit ....

)

Try putting "think" somewhere in that sequence!

Maybe, but I am not sure the rational mind is ultimately in control, very powerful for sure, especially when it secures ones needs in advance, but even then … an untoward opportunity, allure, misunderstanding ...

 

[Bob Brown MSEE]

Digits of an irrational number (phi)

You noticed that for an irrational number to have meaning (and a name), then a description of that number must be given. Some descriptions are...
1) Geometrical
2) Limit
3) Infinite sequence
4) Infinite Nested functions
etc.

What makes algebraic irrational numbers so special?
Because the meaning is provided by a RATIONAL expression.

We can guess and immediately test the next digit in an algebraic irrational number. That is because we are only using and getting rational numbers. Let's try to get the digits of the golden ratio.
The best way for me to show that the golden ratio is algebraic, is to give you the polynomial that has the golden ratio as a root.
Here: y(x) = x² -x -1

Now we try all possible first digits for x = {0,1,2,3,4,5,6,7,8,9} in order, the value of y(x) will go from negative to positive. The lower value of x is the first digit.

We then work on the second digit, third digit ... etc.
Here is a table to demonstrate

Digit	Lower value {x,y}	Higher value {x,y}
1	{1, -1}	{2,1}
2	{1.6, -0.04}	{1.7, 0.19}
3	{1.61, -0.0179}	{1.62, 0.0044}
4	{1.618, -0.000076}	{1.619, 0.00216}

Each x approximation is a rational number (terminating decimal), so each y(x) is also. Also notice that this simple process of "guess and try" will work to obtain an irrational root of any polynomial with rational coefficients (WLOG integers).

 

[Dale10101]

Amen

We're all here because we're not all there...

M. Denis, I think we are talking common divisor here. (... except, alas YOU may be beyond the golden mean)

M. Bob. You have raised a number of interesting points in your last two posts, it will take a while to digest them. I will probably start a new post because I think many here, like me, are are onlookers and the focus is shifting somewhat. Thanks