Proving 0.99999... = 1: A Wikipedia Insight

[nazgjunk]

0.99999... = 1?

I found the following proof on Wikipedia. It looks fine, but I can hardly imagine it to be right...

Another kind of proof adapts to any repeating decimal. When a fraction in decimal notation is multiplied by 10, the digits do not change but the decimal separator moves one place to the right. Thus 10 × 0.9999… equals 9.9999…, which is 9 more than the original number. Subtracting the smaller number from the larger can proceed digit by digit; the result is 9 − 9, which is 0, in each of the digits after the decimal separator. But trailing zeros do not change a number, so the difference is exactly 9. The final step uses algebra. Let the decimal number in question, 0.9999…, be called c. Then 10c − c = 9. This is the same as 9c = 9. Dividing both sides by 9 completes the proof: c = 1.

You can find the full article http://en.wikipedia.org/wiki/Proof_that_0.999..._equals_1" .

 

[HallsofIvy]

"I do not understand it, therefore it must be wrong!"

It assumes some technical facts but it is, in fact, completely valid. Why do you say "I can hardly imagine it to be right..."??

Would you have a problem with this: if x= 0.3333... then 10x= 3.3333333... so, subtracting, 9x= 3 and x= 1/3. Would it surprise you to learn that 1/3= 0.3333...? If not why does it bother you to learn that
3/3= 0.99999...?

This thread has been locked since it has all been said before. Look in the archives:
https://www.physicsforums.com/showthread.php?t=5513

 

[tacman]

The proof provided on Wikipedia is a valid and widely accepted proof that 0.999... is equal to 1. It may seem counterintuitive at first, but it is a result of the way we define and understand decimal notation. In decimal notation, the number 0.999... is defined as the limit of the infinite sequence 0.9, 0.99, 0.999, 0.9999, ... As the number of 9s increases, the number gets closer and closer to 1. In fact, it can be shown using the concept of limits in calculus that the limit of this sequence is exactly 1.

The proof provided on Wikipedia uses a different approach, by showing that 0.999... is equal to 1 by using algebraic manipulation. This proof is also valid and has been rigorously verified by mathematicians. It is important to note that both approaches are equally valid and lead to the same conclusion – that 0.999... is equal to 1.

It is understandable that the concept of an infinite decimal sequence can be difficult to grasp, but it is a fundamental concept in mathematics and has been extensively studied and proven. So while it may seem counterintuitive, the proof provided on Wikipedia is indeed correct and accepted by the mathematical community.