– why is 0.999… equal to 1?

Question

I know the Arithmetic proof and everything but how to explain this practically to a kid who just started understanding the numbers?

In: 3202

Answer 1

The weird feeling we get only arises because we usually dont think about what 0.999 … actually IS. “It just has infinitely many 9’s”. What does that actually mean?

If you write 0.999 … down, does it get more 9’s as we speak? In that case, any equation containing it is wrong because its value changes all the time. You cant work with that. Its like saying “This section of the river has 10 fish”. That statement can never be right for long because the amount of fish changes all the time, so eventually, there may be more fish than 10.

So its a fixed amount of 9’s? No, thats nonsense. We cant say that “infinitely many 9’s” means that there is a fixed amount of 9’s.

So the notion of “infinitely many 9’s” doesnt actually make sense. No matter how we define it, we get clear logical issues. If we want to do math with it, we need to assign it a value that stays *fixed* and which doesnt “change as we speak”. There are 2 important observations for this task:

(1) 0.999 … is *always* less than or equal to 1.
(2) 0.999 … is bigger than *any* number below 1 (because it surpasses 0.9, 0.99, 0.999, 0.9999 etc.)

So IF 0.999 … is equal to any *fixed* number, the best candidate would be 1. Thats why mathematicians defined 0.999 … = 1.