Why or why not does .9 repeating equal 1?
HI! I am wondering if .999...=1? I mean if 1/3=.333 and 1/3+1/3+1/3=3/3 and 3/3=1. So wouldn't .333...+.333...+.333...=1? But it actually equals .999... not 1. I googled this question and I got many different answers
For example...(someone wrote):
I'm not really into math, but a friend brought something up to me today that really seemed very strange. (For the duration of this post, .999 will mean .9 repeating unless otherwise specified- just for the sake of ease)
10x - x = 9x
.999 = 1.
Someone else wrote:
Numbers are fake. They are a manifestation of our minds to describe something, similar to words. Just because we say "red" doesn't mean something is red. what is red? Languages and math are very similar. Math is universal...at least for our planet though. .999~ does not = 1. But what .999~ repeating represents, does in fact equal what 1 represents.
No-one will ever be able to comprehend infinity, os its time to stop trying. Think of space and the universe. IT IS GROWING. how can it continue to grow with no stop? what is there to contain it? WE need something to contain it in order for us to understand it. We need a stopping point, but there is none.
Another person wrote:
In your proof that .333...*3=.999... you forgot to include the fact that .3333... is NOT 1/3. 1/3 if not a number that can be turned into a decimal in any way. I thought someone might like to know this fact.
And the last person wrote:
Just a pedantic point the equation for differentiation is given by lim x->0 [f(x+h)-f(x)]/h, it has the minus sign. It is after all just telling you the slope and is no different really from doing simple trig using the tan function. Here you just take a really small triangle.
As to the 0.99~ thing, this is really just writing the supremum (spelling might be off) of the numbers less than 1. Just think of it as taking the smallest number 'n-word' than 0 away from 1. They are not identical for if they were we would not have a continuous number line, but rather a dashed one with lots of wholes in it. I could simply argue that 0.99~8 is just as close to 0.99~9 as 0.99~ is to 1. For those who really want to understand go and look up supremum numbers and the axioms of the real number line.
So who is correct? Do you agree that .999...=1? Or .999... does not equal 1?