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.