,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
positive integers have no single-digit prime factors?
,
or 
?
Y by dantx5, cerberus88, awesomeguy2, gaygaygaygay, nbute, jkyman, yrushi, zmyshatlp, DSL13, sachinpgupte, TsunamiStorm08, Adventure10, and 31 other users
Recently, many responses to threads have little or no explanation, with only an answer provided. These will not contribute to understanding and following the question and defeats the purpose of the math threads.
Here are some suggestions to improve the quality of your responses.
• If you are introducing a topic that would be unfamiliar to the general public of the forum, explain what it is and how it is applied to the specific problem.
• If you are using a non-standard approach to the problem, explain why it is a more efficient approach than the normal methods and algorithms, and elaborate upon why your method works in the particular case.
• If you are dealing with complex mathematical symbols, surround them with dollar signs ($$), and the website will automatically transfer the code to
. Learning codes is easy and makes examining a math question a lot easier. For example, one is definitely more likely to be satisfied by 
than 120=-x^5+3x^4+23x^3-27x^2-166x. Note that this is just a simple (polynomial) example, and the more complex notations would call for even more, like summation.
• When finished, judge your own solution, imagining if you didn't understand how to solve the problem whether you would have been helped by the solution or not. Although it may be hard to transfer your mind to the state of this hypothetical other person, it is a good skill to have and can really help improve the overall quality of forum posts.
• It is okay to state that you made a guess or carried out a dubious process, or to make a move that carries a mistake, as long as you point it out as a guess or an unsure solution. Mistakes are a defining characteristic of the human species, average thinkers and geniuses alike, so you should not be afraid to announce that you think you might have done something incorrectly in your solution—if you really don't know what the question is talking about though, don't answer it, as you would then be leading the unfortunate question-asker astray into an nonexistent dimension.
• Don't be afraid to use colors and graphics to explain. In fact, many people find that these make the solution much clearer as well as more interesting. External links to informational websites also help, but don't just post that link and call it a day for counting as your explanation; at least append a label describing the content on the other side of the link.
Of course, as with any set of rules, these are just standards and suggestions, and need not be strictly and dogmatically followed. When unsure, common sense is your best friend, and if you must defer beyond that, just follow the rules. Always keep in mind that the person reading your responses and explanations is most likely a high school ninth, tenth, or eleventh grader.
Here is an example of an excellent solution. The only improvement I would make to it is to explain exactly what it is a definition of. Note how the poster of that explanation showed an analogous case of committees to explain the situation that could possibly be confusing to some people, and that he even went above and beyond the standards by leaving a little for the original poster to solve for himself or herself, as well as hiding the solution in case a viewer wanted to try to solve the problem without accidentally looking at the solution. Note that it shows that you don't have to be an adult to be able to write good solutions. If you find a solution you particularly enjoyed or learned significantly from, rate it 5 or 6.
If you have spent the time to read this, thank you very much, and if you have given some thought as to how you compare to such standards, another thanks. As a last note, feel free to point out anything you notice or want to mention or suggest an alteration to the standards listed above. Moreover, questions are always welcome, and it is much better to first clarify something here than to have ruined a few threads because of a grave misunderstanding.
Here are some suggestions to improve the quality of your responses.
• If you are introducing a topic that would be unfamiliar to the general public of the forum, explain what it is and how it is applied to the specific problem.
• If you are using a non-standard approach to the problem, explain why it is a more efficient approach than the normal methods and algorithms, and elaborate upon why your method works in the particular case.
• If you are dealing with complex mathematical symbols, surround them with dollar signs ($$), and the website will automatically transfer the code to
. Learning codes is easy and makes examining a math question a lot easier. For example, one is definitely more likely to be satisfied by 
than 120=-x^5+3x^4+23x^3-27x^2-166x. Note that this is just a simple (polynomial) example, and the more complex notations would call for even more, like summation. • When finished, judge your own solution, imagining if you didn't understand how to solve the problem whether you would have been helped by the solution or not. Although it may be hard to transfer your mind to the state of this hypothetical other person, it is a good skill to have and can really help improve the overall quality of forum posts.
• It is okay to state that you made a guess or carried out a dubious process, or to make a move that carries a mistake, as long as you point it out as a guess or an unsure solution. Mistakes are a defining characteristic of the human species, average thinkers and geniuses alike, so you should not be afraid to announce that you think you might have done something incorrectly in your solution—if you really don't know what the question is talking about though, don't answer it, as you would then be leading the unfortunate question-asker astray into an nonexistent dimension.
• Don't be afraid to use colors and graphics to explain. In fact, many people find that these make the solution much clearer as well as more interesting. External links to informational websites also help, but don't just post that link and call it a day for counting as your explanation; at least append a label describing the content on the other side of the link.
Of course, as with any set of rules, these are just standards and suggestions, and need not be strictly and dogmatically followed. When unsure, common sense is your best friend, and if you must defer beyond that, just follow the rules. Always keep in mind that the person reading your responses and explanations is most likely a high school ninth, tenth, or eleventh grader.
Here is an example of an excellent solution. The only improvement I would make to it is to explain exactly what it is a definition of. Note how the poster of that explanation showed an analogous case of committees to explain the situation that could possibly be confusing to some people, and that he even went above and beyond the standards by leaving a little for the original poster to solve for himself or herself, as well as hiding the solution in case a viewer wanted to try to solve the problem without accidentally looking at the solution. Note that it shows that you don't have to be an adult to be able to write good solutions. If you find a solution you particularly enjoyed or learned significantly from, rate it 5 or 6.
If you have spent the time to read this, thank you very much, and if you have given some thought as to how you compare to such standards, another thanks. As a last note, feel free to point out anything you notice or want to mention or suggest an alteration to the standards listed above. Moreover, questions are always welcome, and it is much better to first clarify something here than to have ruined a few threads because of a grave misunderstanding.
This post has been edited 3 times. Last edited by phiReKaLk6781, Mar 28, 2010, 10:55 PM
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