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0.9999999999999…
let x = 0.99999999…
10x = 9.999999999…
9x = 9
9x ÷ 9 = 9 ÷ 9
x = 1
0.9 = 1
Yes, although unlike the image, this is actually true. Assuming that you still meant 0.9 recurring in the final line, not just 0.9.
The “9x = 9” is false. It should be “9x = 8.99999999(cont)”, which kinda kills your proof.
OK, this has actually been a can of mass-internet-anger worms for years so I probably shouldn’t start up this debate. But general consensus is that the proof is true. It is really a matter of notation rather than a matter of number theory, because of the fact that the notational concept of “recurring” means “repeat to infinity”, and infinity is a strange and often misunderstood concept. Personally though it seems clear to me that 0.999… is simply another way of writing 1, and similarly 8.999… is another way of writing 9. Numerically they are the same, they only differ in how they are written.
By the way, to specifically address your point: your statement is true, and I presume that you derived by simply multiplying 0.999… by 9. If you keep multiplying each digit by 9 and adding it to the result, it will endlessly add 9s to the end of the number, seemingly forever.
Now consider this:
x = 0.999…
10x = 9.999…
10x – x = 9
9x = 9
This part is the key, so it is unfortunate that it is missing from rotatebilly’s post. You can see from the LHS that this is correct and should also be identical to your result.
than queue
The fallacy is in the final line, since (a^2 – ab) is always equal to zero, for all a and b. Therefore the final step involves division by zero, which is undefined.
Now do this with a=a so I can fuck with some objectivists.
The equation becomes invalid when you divide by zero. Using this, any number can equal any number.
In general, there is leniency in mathematics.
.9999… does not equal 1, ever. They aren’t the same thing.
However, we relate them in taking a limit. That is to say, we define parameters so that we can use a number that runs on to infinity.
Ok, here we go.
You can express the same number as a fraction or a decimal.
if:
1/3=.333…
then:
3(1/3)=1
which can also be written
3(.333…)=.9999
Same number. Not “close enough.” Same. Nothing to do with limits, it’s the identity of a real number.
Q.E.D.
“However, we relate them in taking a limit. That is to say, we define parameters so that we can use a number that runs on to infinity.”. Yes, and this is how 0.999… is defined. (where the … indicates a recurring decimal). This is why 0.999… = 1. As I state above, this is a property of the notation, not so much of the numbers themselves.
You are anonymous. If you would like the luxury of referring back to previous statements, please get a UID and use it. Are you quoting yourself? In that case you are contradicting yourself. Are you disagreeing with someone else? You both have the same name. You are anonymous.
Please clarify your identity. If you can’t do that, why should I trust your ability to identify a mathematical value?
Because this is a image board I casually browse go one when I’m bored and am done masturbating, not a conference on number theory. I have a UID but I don’t usually care enough about my online identity to bother logging in. It’s pretty clear from context which anon comment I’m referring to anyway, because it’s the one where I said the thing that I wrote after “as I state above”.
“.9999… does not equal 1, ever. They aren’t the same thing.”
“This is why 0.999… = 1.”
Forgive me if it wasn’t obvious. Now can you please reconcile your two opposing statements? If you’re going to respond casually, maybe you shouldn’t bring up complex mathematics.
there. i just died of boredom. thanks a lot.
Godel. Look it up.
I use 0.9…=0 to fuck with my students heads when I tutor Chemistry. After they see it they actually want to learn.