Think It's Easy? Think Again!
.99.. vs 1 a Real Problem
Circular Reasoning( Fractions).This kind of proof for 0.99... = 1 tries to show that a = b by starting with the assumption that a/n = b/n. Essentially, it assumes( no proof) a/n = b/n, multiplies both sides by n, and then cancels n to conclude that a = b.
For example 1/3=.33... => 1/3*3=.33...* 3 => 1 = .99...
You can see that most people accept the algebraic circular reasoning as fact, as evidenced by a simple google search and the overwhelming citations in 'The Naked Emperor' playlist. However, mathematically, this proof has no merit ( as apparent below ) and is just an illusion to believe. Many people stop at this point and delve no further, making it inherently impossible to explain the more complex topic.
Simple algebraic illustrations of equality are a subject of pedagogical discussion and critique. Byers (2007) discusses the argument that, in elementary school, one is taught that 1/3 = 0.333..., so, ignoring all essential subtleties, "multiplying" this identity by 3 gives 1 = 0.999.... He further says that this argument is unconvincing, because of an unresolved ambiguity over the meaning of the equals sign; a student might think, "It surely does not mean that the number 1 is identical to that which is meant by the notation 0.999...." Most undergraduate mathematics majors encountered by Byers feel that while 0.999... is "very close" to 1 on the strength of this argument, with some even saying that it is "infinitely close", they are not ready to say that it is equal to 1.[12] Richman (1999) discusses how "this argument gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking", but also suggests that the argument may lead skeptics to question this assumption.[13]
Interestingly, many who present this proof acknowledge its flaws but still claim it as a good proof because it often convinces people. It is sometimes used as a teaching shortcut—an attempt to “sell” a mathematical fact using familiar arithmetic steps. However, from a mathematical standpoint, this type of argument lacks rigor and serves as an example of circular reasoning rather than a valid proof.
The main issue is that it assumes what it sets out to prove. By starting with the claim that 1/3 = 0.333.., a fact that itself relies on an understanding of infinite decimal expansions, the argument simply rephrases the same assumption in a different form. Multiplying both sides by 3 does not prove that 1 = 0.999...; it merely transfers the original unproven equality into a new context.
Moreover, this approach can mislead students into thinking that mathematical proof is a matter of mechanical manipulation rather than logical reasoning grounded in rigor. It glosses over the essential math steps, which is required to rigorously define infinite decimals like 0.333... or 0.999.... Without a solid foundation in real analysis or a clear explanation of convergence, such proofs risk promoting superficial understanding and encouraging acceptance based on authority or pattern, rather than critical thinking.
In educational contexts, these kinds of shortcuts may do more harm than good. They reinforce the idea that mathematical truths can be shown by clever tricks rather than demonstrated through rigorous argumentation. Worse still, they can foster confusion when students later encounter the deeper ideas that such shortcuts obscure, such as the fact that decimal representation is not unique.
Therefore, while such informal arguments may serve as conversation starters or intuitive hooks, they should not be presented as legitimate proofs.
The above provides a summary of why this kind of algebraic reasoning is flawed, referencing renowned mathematicians who believe .99…=1, yet the authors clearly acknowledge that these kinds of proofs are not correct according to the well-known mathematicians' line of thinking. In short, the mathematicians who believe .99…=1 do not even consider this kind of proof as valid. For this reason and many others, no serious mathematicians accept this as valid proof. (No further discussion seems necessary to refute this type of argument).
The final word:If 999... (an endless string of nines without a decimal point ) is not a real number, then 0.999..., which depends on that same infinite string in its decimal expansion, cannot be a real number either. You can’t build a real number on top of something that isn’t real in the first place.
