The Truth Behind the Easy Façade
.99.. vs 1 a Real Problem
Algebraic proof (exploiting infinity ambiguity For example, infinity + 1 = infinity) . For these types of proofs, generally assume x = .99..., then claim 10*x = 9.99... (in between lines assuming ∞+ 1 = ∞ ). People 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.
Multiplying Infinite Decimals by 10
| Decimal Representation | Calculation Result | Observation |
|---|---|---|
| 0.a | 10 × 0.a = a | a ≠ a.a |
| 0.aa | 10 × 0.aa = a.a | a.a ≠ a.aa |
| 0.aaa | 10 × 0.aaa = a.aa | a.aa ≠ a.aaa |
| 0.aaaa | 10 × 0.aaaa = a.aaa | a.aaa ≠ a.aaaa |
| ... | ... | ... |
For example consider the induction below to illustrate the difference between 10× 0.99… and 9.99…
10 * 0.9=9 ≠ 9.9
10 * 0.99=9.9 ≠9.99
10 * 0.999 =9.99 ≠ 9.999
...
10*0.99... ≠ 9.99...
Thus they were never Equal,
Note that in mathematics, we don’t expect something to happen randomly or without reason( (which would be a violation of induction ) . The above inequality persists through countless steps, and it makes no mathematical sense for them to suddenly become equal without any logical reason. 10*0.99... = 9.99... contradicts some of the oldest and most fundamental principles of mathematical bijection and induction.
You can see that most people accept the algebraic proof below 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 described in the 3rd paragraph ) and is just an illusion to believe. Most people stop at this point and delve no further, making it inherently impossible to explain the more complex topic.
There are many flaws with this category of arguments for .99..=1 . The major flaw, is their contradictory nature. People assume that an infinite digit number is a real number without proof, which leads to contradictions. If we assume that infinite digit numbers are real numbers, then we must accept that π is a rational number because we can write it as the division of two infinite digit numbers. This eliminates the need for a set of real numbers because all real numbers can be considered rational numbers, making the concept of real numbers irrelevant. In other words, the moment we accept infinite digit numbers as real numbers, it proves that the definition of real numbers is flawed or unnecessary in the first place.
To put it differently 99..=1 highlights the paradoxical nature of real numbers definition. The logical paradox here is that we say the set of real numbers must include numbers with infinite decimal expansions (.9999.....) . This is correct because rational numbers do not include numbers with infinite decimal expansions or transcendental numbers.
However, if that’s true, one might argue that π is a rational number because it can be expressed as the ratio of two infinite whole numbers. This would make π a rational number, which challenges the need for the existence of the set of real numbers in the first place. This is similar to logical paradoxes, like the statement on the back of this card being both true and false. Please see this around timestamp 1:15: https://youtu.be/O4ndIDcDSGc?si=vCYGMjNrNWdl6zQ5&t=75.
In short, to include π as a real number, you need to allow for infinite digits to be real numbers. But if infinite-digit whole numbers exist, it would imply that π is a rational number.
Th bottom line is:
It is contradictory to assume that 0.999... is a real number in order to prove that it equals one.
Assume that 0.999... is part of the sequence 0.9, 0.99, 0.999, and so on, and therefore is a real number. In this case, by induction, it can be proven that 0.999... is not equal to 1.
Alternatively, if 0.999... is a number whose fractional part is an infinite string of 9s ( which is 999... is not a real number ), then a proof is required to show that such a number is a real number before making any claims like let x = 0.999..., then multiplying by 10 gives10x = 9.999..., and as you saw above, this leads to a contradiction.
Using the Archimedean Property to Show \( 999\ldots \) Is Not a Real Number
Let us consider the number \( 999\ldots \), where there are infinitely many 9s and no decimal point. That is, we can think of it informally as:
\( x = \lim_{n \to \infty} \underbrace{999\ldots9}_{n \text{ digits}} = \lim_{n \to \infty} (10^n - 1) \)
This value grows without bound, since for any real number \( R \), we can find a large enough \( n \) such that:
\( 10^n - 1 > R \)
Now, recall the Archimedean Property of real numbers:
For any real number \( x \), there exists a natural number \( n \in \mathbb{N} \) such that \( n > x \).
Suppose for contradiction that \( x = 999\ldots \in \mathbb{R} \).
Then, by the Archimedean Property, there must exist a natural number \( N \) such that:
\( N > x \)
Simply put: here’s the deal: if you say 0.999… is a number whose fractional part is an infinite army of 9s (and let’s be real—999... itself isn’t a real number, it’s more like a math unicorn), then you really need to prove that this thing is legit before you start doing math tricks.
People love to pull this classic move: “Let x = 0.999..., multiply both sides by 10, so 10x = 9.999..., now subtract and boom , magic happens!” But hold on , you just assumed x is a real number without any proof. That’s like saying, “Trust me, my invisible dragon exists because I can’t see it, but I can do tricks with it.”
As we all know from Archimedes himself: infinite processes can sneakily lead you into paradox land faster than you can shout ‘Eureka!’ So when some folks later claim 0.999… is real just because it’s equal to 1, that’s not math — that’s circular reasoning with a fancy mustache.
So, before you dive into infinite nines and claim victory, remember: infinity might be fun, but it sure loves to play tricks on us!
That said, we recognize that many people believe 0.999... is indeed a real number, even while accepting that 999... is not. While this may seem contradictory from a strict mathematical perspective, it’s understandable that such ideas can coexist in different interpretations or beliefs. It’s important to approach these subtle concepts with openness and respect for diverse viewpoints.
The last word : 999... is not a real number, and therefore 0.999... is not a real number either.
