In my first university calculus class, the professor wrote on the board:
0.999... infinity
1
And asked us if they were different numbers. Per her argument, they are not, since there's no number in between 0.999... and 1. Ergo, they're the same. Or, more technically,
If you know its a wiki....oh never mind.
Yea, never mind
Burden is on YOU to proof they are not the same after proofs proving they are the same have been established
Also, feel free to look up all the references on the topic the wiki conviently provides proving they ARE The same by the way
I'm having a great discussion with a friend about the following...
1. if x=1.999...infinity, then 10x=19.999...infinity. Subtracting 10x from x gives 9x=18. Dividing both sides results in x=2.
2. Using your calculators on this one. 1/9=0.111...?. 2/9=0.222....?. 3/9=0.333...?. 4,5,6,7,8 follow the same pattern. What, then is 9/9? It is 0.999...infinity.
Do you think those are valid? I think they are and thinking in infinite terms 1.999...infinity is equal to 2.
My friend thinks he can add any number to infinity because he just can.
Food for thought. Any you want to add?
How so? maybe a section of the cone seems like a cylinder but there is still a slight angle--perceptibly yes it is for all practical purposes a cylinder, But as part of the whole not the section at some point as it is being infinitely stretched it is still a cone.Think about this in the form of conics (using cones) When a cone is stretched to infinity it becomes a cylinder...
Reference something juried for the proofs, that's all I'd like to see for a change Wikipedia is notoriously chaotic. But that doesn't necessarily make it true either are the references primary? secondary?....?
k don't know if anyone said this.
BUT INFINITY IS NOT A NUMBER
my teacher has a shirt that says this.
o and repeating digits has a vinculum on top of it, not to the infinity.
Doesn't matter. That Wiki page references a number of simple, easy to follow proofs (we'll call 'em proofs for now) that demonstrate that 0.(9) = 1.
This is not a debatable subject. By mathematical definition, 0.(9) = 1.
And no, 0.9~ is not an irrational number. Because it = 1.
There is no last 9. Your gut is telling you that there is a last 9 in the sequence, because infinity is a difficult concept to understand on an intuitive level. What's happening here is your intuition is overriding your intellect.
You said you took calculus. Review your definition of a limit, then infinity. The maths are there, as plain as day, yelling at you that 0.(9) = 1. You're just not listening because you're looking for the last 9.
Then is .8888infinity the same as .99999infinity? therefore .8888888infinity equals .99999999infinity?
I think 0.888 (repeating) is (apparently) equal to 0.9, not 0.99999 (repeating).
But, Doctor Q, care to explain how D=0, when you said that D>0? I mean, does D stand for a specific unit in your polygon problem, or just any variable?
Like, would it be the same if I said X>0, and you proved me wrong by saying that X=0?
Fireshot, think about it this way, which also addresses an error in your post.
What positive numerical value fits between 0.(9) and 1?
What positive, numerical value fits between 0.(8) and 9? It's greater than 0, therefore, the two are not equal. 0.8(9) = 0.(9), not 0.(8)
I think 0.888 (repeating) is (apparently) equal to 0.9, not 0.99999 (repeating).
True or false? - "A regular polygon with an infinite number of sides is a circle."
Here's why it's true:Consider the original question in this thread. When we say that 1=0.9... or that 1 is 0.9..., the definitions of "=" and "is" are "there is no difference between the values".
One way we know that there is no difference is that if you tell me any number D>0, I can show that the absolute value of the difference is smaller than D. In other words, I can show you that you are wrong if you pick any positive value. Therefore, the difference D can't be more than zero, so it must be zero.
In the same way, if you tell me any number D, I can make an N-sided regular polygon inscribed in a unit circle so that the difference in area between the circle and the polygon is less than D. All I have to do is make N big enough. Therefore, by making the number of sides infinite, I can arrange that polygon=circle.
Here's why it's false:By definition, a regular polygon is a figure in a plane consisting of a finite number of end-to-end line segments that have equal lengths and form equal angles. (There are many other ways to say this.)So apparently the statement is false even though you can prove that it's true.
Therefore, an infinite polygon can't exist by definition.![]()
Sets have "last ones"? You learn something new every day!That's horse puckey, man.
Looked at simply, how large is the set of all even numbers? How large is the set of all numbers divisible by 43564?
How large is the set of all regular polygons? Take the last one in the set. That's your circle![]()
Sets have "last ones"? You learn something new every day!![]()
Looked at simply, how large is the set of all even numbers? How large is the set of all numbers divisible by 43564?
Those two sets have the same cardinality, that is, they have the same size. The reason is that there is a bijective map from one set to the other, namely 2x gets mapped to 42564x.
It doesn't matter that one set is a subset of the other. Quite weird.
What if anything anyone has said leads you to an assumption that people are looking for a "last number" in a decimal that clearly is on going (eternal, forever, infinitely, whatever the term for endless in general you want to nitpick about concerning definitions) and still not so clearly equal to one _ pretty damn close by a constantly narrowing margin but still not 1 this .9 thing is a bit of mystery otherwise until someone figures out a way to express it accurately. You speak as if all the rules of numbers have been established and there is nothing that's going to change them. This is a case of an anomaly where no solid rule has been figured out so for sheer laziness on the part of mathematicians for the time being 1=.9(and so on) for practical purposes.This goes back to the thing about there being no last 9. Most people don't have a good grasp on the concept of of infinity and this is one of those issues that trips them up.
I know plenty of engineers that can't grasp that the set of all even numbers is the same size as the set of all whole numbers. To me, this is a good mathematical shibboleth on whether a person actually fundamentally grasps calculus or if he is just a technician with a degree who does what the numbers tell him to do.