I see already an issue with your claim that the square root of 2 "pertains to a point of indefinite measure". The measure of a point on the line is well defined: it is zero. (Reason why: for every d>0 there exists an open interval of diameter d which contains the point.) And although we can identify a real number with a point on a line, this is not really what is done for the square root of 2; rather, square root of 2 is the length of the side of a square of area 2.

Coming to the main point, I think there is a confusion between the number and its representation in base 10. It is the latter, not the former, which can be or not be "repeating" (meaning: ultimately periodic). And here, I think, comes the critical issue in your argument: multiplication between real numbers is not "defined as repeated addition". It is multiplication between natural numbers (nonnegative integers) which can be defined recursively by m * 0 = 0 and m * (n+1) = (m*n) + m for every m and n natural numbers; from this one can define multiplication of integer numbers and rational numbers. On the other hand, if a and b are positive reals, then a * b is the least upper bound of the set of the products of the form x * y where x and y are positive rational numbers, x is not larger than a, and y is not larger than b; and although this could be used for the definition, it is not a recursive process.

Ever think that this may be an artifact of the special number system you are used to working in?

There are an infinite number of possible number system, where the basis may be uniform or not.

I think there is nothing special about sqrt(2) compared to 2. Both just points of the real axis.

Assume the phrase "Root 2 is fundamentally a number that is non-repetitive and unending" describes decimal representation of number sqrt(2). Also, assume the phrase "geometric point's perspective" describes other (finite) way of description of the same number, e.g. via some geometric figure.

The problem with real numbers is the mankind did not yet invented a way to represent them in precise and finite way. The decimal representation is only finite and precise for integer numbers. For rational numbers it is infinite but precise, e.g. 1/3 = 0.(3) = 0.33333.... For real numbers this cannot be achieved. There is another problem: The same number in decimal representation can be presented in several ways, e.g. 4.(9) = 4.9999999..... = 5.000000..... = 5.(0). Here, the equal sign is precise equality, not approximate.

So, for real numbers, when no exact and finite decimal representation can be used, the approximate decimal value can be used, e.g. sqrt(2) = 1.4142135. Or, instead of actual value, a way to compute this value can be used. The token "sqrt(2)" is exactly this: a finite algorithm of how to compute the number with desired precision. Or a named constant can be used, e.g. π is a common symbol used for value 3.14159265...

Speaking about computational, or algorithmic, finite and precise representation of real numbers, the continued fraction is often used (https://en.wikipedia.org/wiki/Continued_fraction). The same number sqrt(2) can be presented as continued fraction [1; (2)] = [1; 2, 2, 2, ....]. Of course, the geometric mains are also used. If we denote the Pythagorean sum by symbol ++ (x++y = sqrt(x^2 + y^2)), then sqrt(2) = 1++1, which has a well defined geometric meaning.

Irrational numbers, such as the square root of 2, pi, and e, have interesting properties and can lead to geometric "absurdities" or counterintuitive results.

For example, the discovery that the diagonal of a square with sides of length 1 (unit square) cannot be expressed as a rational number led to the realization that the square root of 2 is irrational. This means that there's no exact fraction that can represent the length of the diagonal. At this point, I choose to agree with Silvio Capobianco, stating that your claim that the square root of 2 "pertains to a point of indefinite measure" is not true for the reason he has rightly pointed out.

Additionally, the number pi is irrational as well. Its decimal expansion never repeats or terminates, which implies that the exact circumference or diameter ratio of any circle cannot be expressed as a simple fraction.

These irrational numbers can lead to geometric paradoxes and unexpected results. For instance, there are various geometric constructions that appear to be valid but can't actually be realized using only a compass and straightedge due to the limitations imposed by the nature of these irrational numbers.

Overall, the presence of irrational numbers in geometry highlights the rich and sometimes counterintuitive connections between mathematics and the physical world.

Silvio Capobianco sir, firstly thank you for actually taking your time out for answering my question and clearing my doubt.

Now, the measure of a point doesn't make sense but the measure of a line does and that is what I meant in the first place. Also, if the recursive definition of multiplication holds only for Natural numbers, then I still have hard time comprehending about how would we define multiplication of real numbers specifically irrationals? Your statement " If a and b are positive reals, then a * b is the least upper bound of the set of the products of the form x * y where x and y are positive rational numbers, x is not larger than a, and y is not larger than b." is a bit unclear. I mean we're still using the term product here and the kind of operation being performed here is still fundamentally probably the same multiplication that we're doing for natural numbers, but with an added notion of an upper bound or a limit, which has been introduced basically to account for the unending nature of reals. So, at the heart of it we're essentially doing the same kind of operation that we were doing for natural numbers.

Also, sir I would like to apologize if I am being wrong about my idea, because sometimes my ideas are a bit too far-fetched and I sometimes miss the intricacies and details in definitions. If there's something that is wrong with my follow-up answer then I would like to know still sir.

Thank you

Addition (subtraction) takes place inside the set. Multiplication (division) takes place between the sets, producing a new set (unit).

–> More-times-addition generates the same result (in value) as multiplication does.

–> No `more-times-addition´ for genuine multiplication.

Acting in math on complete quantities (ABC) shelters for such problems which here are under examination.

A := counter [a number (cardinal) which counts how often the measure is taken]

B := measure [a defined part of the unit, to take in square-brackets]

C := unit [name of the phenomenon, by defining symbol or word / explanation]

1[1m] length + 1[1m] length = 2[1m] length

1[1m] length x 1[1m] width = 1[1m2] area

(1[1m] length x 1[1m] width) + (1[1m] width x 1[1m] length)

=> 2[1m2] area

=> long side of the triangle: 2[1m2] = 1[2m2]

=> 1[√2m] distance

Math is without the measure `m´. So only on numbers. And √2 is nothing else than a special measure! Be open for the Reform.

The nimber 2 is also in the need

2.0000000....