.99.. vs 1 a Real Problem
An ongoing modern fairy tale of The Emperor's New Clothes
The documents below explore the question of when .99... is less than 1 and presents alternative interpretations of the symbol. It argues that the students' intuition can be rigorously justified and suggests not emphasizing the unital evaluation of .99... as 1 in a pre-limit teaching environment. The documents also discuss the use of Lightstone's semicolon notation, the possibility of choosing a canonical alternative interpretation, and the use of non-standard analysis and the construction of the hyperreal numbers. It addresses frequently asked questions, provides explanations and arguments for different viewpoints, and highlights the complexities and nuances involved in understanding the relationship between .999... and 1. Additionally, the document discusses various aspects of non-standard analysis, including the concept of infinitesimals and their role in calculus, as well as the historical context and philosophical implications of the subject. It also includes a glossary of terms and references for further reading.
Above documents samples of a common widespread misconception regarding 0.99... vs 1 at the time of the above publication and at the time of publishing The Naked Emperor playlist, we can see the misconception start with simply claim 1/3 = 0.33..., and as we go step by step forward, we discover the error and find a better and more apparat logically rigorous proof for .99..=1. In doing that, we ended up leaving a trail of misunderstanding for others. This means that for superficial observers who do not delve deeply, the erroneous repetitive false proofs are enough to confuse and accept, triggering a false positive feedback loop similar to the story of 'The Emperor's New Clothes,' hence the name 'The Naked Emperor.'
It doesn’t matter if we tell people a thousand times that a better definition is equivalent to a proof; it doesn’t change the fact that definitions are not proofs. The statement 0.99... = 1 is a contradiction and not very mathematical unless we are in the scientology church of definitions.
The proofs for 1 = .99... follows these categories:
Note that math is not a voting system; however, in this case, it seems to have become one. The right-hand side joke and funny meme illustrate why the voting system has failed us: a very small percentage of experts are being ignored in favor of sheer numbers and overwhelming opposition advocates.
1. Circular Reasoning( Fractions).This proof attempts to prove a = b by starting with the assumption that a = b. In reality, it assumes a/n = b/n and then says a * n = b * n, canceling n and thus concluding a = b.
For example 1/3=.33... => 1/3*3=.33...* 3 => 1 = .99...
Interestingly, many who present this proof acknowledge its flaws but still claim it as a good proof because it often convinces people.
2. Algebraic proof (exploiting infinity ambiguity) . For these types of proofs, generally assume x = .99..., then claim 10*x = 9.99... (In between lines assuming ∞+ 1 = ∞). They attempt to use 10* .99... to equal 9.99..., ignoring the fact that the goal was to show there is no tiny difference between .99... and 1. In doing this, they actually introduce that tiny mistake. The tiny mistake in this case is that 0.99…as an infinite number of nines after the decimal point, while 9.99... has an infinite number of nines after the decimal point plus one additional 9 at the beginning.
Moreover, consider the induction below to illustrate the difference between 10* 0.99…and 9.99…:
10 * .9 ≠ 9.9
10 * .99 ≠ 9.99
10 * .999 ≠ 9.999
...
10*.99... ≠ 9.99...
and thus they were never equal to it.
The statement 0.99…=10 is a contradiction and a Real Problem and not very mathematical unless you subscribe to a particular belief system regarding definitions.
The belief is the definitions are proof.
It’s even more hilarious because the most knowledgeable experts on this topic agree with the least experienced individuals. Middle-level semi-experts, fearing they’ll appear less competent and feeling embarrassed about their understanding, settle for incorrect ideas and accept them as facts (an ongoing modern fairy tale of ‘The Emperor's New Clothes’). They even resort to believing that 0.99... equals 1, assuming there will eventually be proof at a higher level that they cannot currently explain, and accepting definitions and conventions as proof.
Long story short, in our experience, after a few back-and-forth negotiations, almost everyone eventually acknowledges that 0.99... and 1 are not equal in the hyperreal system. Yet, they claim there is a ‘logic-free zone’ where many definitions exist for a set of numbers (real #) between the sets of rational and hyperreal numbers that completely defy the properties of these sets, ignoring the transfer principle and the fact that definitions alone are not proof.
This is a mathematical logic-free zone because it disregards the principles of mathematical logic, which is a branch of mathematics. It also ignores the fact that if a set is sandwiched between two sets, it will share properties with the larger set that also appear in the smaller set, making those properties both persistent and true in the smaller set.
3. Exploiting Limit Definition ( Non Newtonian Calculus). In this category, the assumption is that the value of a function exists because either the right-hand or left-hand limit exists. This leads to assuming that the value of a function or series exists because one of the limits exists. Additionally, it is assumed that the limit of a function, sequence, or series represents the exact value of that function.
To put it simplify, some mathematical generalists claim that 1/10^n eventually becomes zero. This ignores the fact that adding 1/10^n 10^n times always results in 1. If it were zero, adding zero would always result in zero.
4. Real Numbers Definition (Real Analysis): In this context, mathematical generalists claim that the completeness(*) of the real number set is a proven fact. They demonstrate, through several steps, that there are no "gaps" between real numbers because of other defined properties , thus establishing their equality. It is important not to accept unproven definitions as facts and to understand that proving the completeness of the real numbers depends on the assumption that all infinities are equal.
* Real Number Definition : In mathematical analysis, the definition of real numbers often involves the concept of completeness. Completeness means there are no "gaps" or "missing points" on the real number line, unlike the rational numbers, which have gaps at irrational values. In decimal notation, completeness implies that any infinite decimal string represents a real number. Completeness can be formulated as an axiom (the completeness axiom) . Two notable forms of completeness are Dedekind completeness and Cauchy completeness.
Dedekind Completeness: This means that every non-empty subset of real numbers that is bounded above has a least upper bound (supremum) within the real numbers. This property is not shared by the rational numbers.
Ironically, Dedekind using math created a "math-free zone" between the set of rational numbers and hyperreal numbers. This is used to explain the claim that there must be a set where we can include π, because it is not in the set of rational numbers and thus cannot be complete without real numbers. However, the same argument should not be applied to the existence hyperreal set. This double standard is uncanny and a definition of contradiction( flat argument) . If 0.99… is considered a real number, one could argue that π should be considered a rational number simply because one can imagine that a deviation of two arbitrarily long infinite(...) digits could yield π.
The idea that 0.99... equals 1 in the real number system is well-established in today's mathematics. However, arguments against this equality are compelling these days. One such argument is that while 1 x 1 x ... = 1, 0.99... x 0.99... doesn't converge to 1, therefore 0.99... cannot be equal to 1.
