,
,
,
,
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 
with circumcircle 
,
on 
,
,
be points on lines 
,
,
,
,
,
concur at a point 
.
be a point on line 
be a point on line 
be a point on line 
,
,
concur at a single point.
,
the sum of its digits. Find all integers 
such that there exist 
with 
for all 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
be non-negative real numbers such that 
Prove that
:o
so 
and since 
is just 
therefore 
so 
and 
intersect at the point 
.
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
,
,
is a 
is 
,
is a 
is also prime
Y by mathematicsy, Adventure10, NicoN9
Suppose one is given 
real numbers, not all zero, such that their sum is zero. Prove that one can label these numbers 
in such a manner that
![\[a_1a_2 + a_2a_3 +\cdots + a_{n-1}a_n + a_na_1 < 0\]](//latex.artofproblemsolving.com/6/9/4/6949824017301ebae714606e15eb1af825cbac33.png)
Can we multiply expectation? In other words, is
valid? I feel that this is valid for independent events 
and 
but may not be for dependent events. In this case would 
? If that is the case the problem becomes trivialized.
real numbers, not all zero, such that their sum is zero. Prove that one can label these numbers 
in such a manner that![\[a_1a_2 + a_2a_3 +\cdots + a_{n-1}a_n + a_na_1 < 0\]](http://latex.artofproblemsolving.com/6/9/4/6949824017301ebae714606e15eb1af825cbac33.png)
Can we multiply expectation? In other words, is
valid? I feel that this is valid for independent events 
and 
but may not be for dependent events. In this case would 
? If that is the case the problem becomes trivialized.
and there is no 
such that 
.
where 
and 
.
such that 
,
will yield a labeling that satisfies the problem. We will now show that 
is impossible. Assume for the sake of contradiction that this is possible. Then this, along with the fact that the sum of the 
,
![\[ \sum_{sym} a_1 a_2 = \frac{(a_1+a_2+\dots+a_n)^2 - a_1^2 - a_2^2 - \dots - a_n^2}{2} = - \frac{a_1^2 + a_2^2 + \dots + a_n^2}{2} < 0.\]](http://latex.artofproblemsolving.com/9/8/5/98559bbe77a3334853ea6f926ee7f6a5211e0689.png)
Hence ![$\mathbb{E}[a_1a_2] = \mathbb{E}[a_2a_3] = \dots = \mathbb{E}[a_na_1] < 0$](http://latex.artofproblemsolving.com/8/6/c/86c18e04e219c4f78f35c4c382495151665438b5.png)
,![\[ \mathbb{E}[a_1a_2 + a_2a_3 + \dots + a_na_1] = \mathbb{E}[a_1a_2] + \mathbb{E}[a_2a_3] + \dots + \mathbb{E}[a_na_1] < 0.\]](http://latex.artofproblemsolving.com/2/2/f/22fa6869b5e432734094a459b4c355db4eee9308.png)
Therefore there must exist some 
![\[a_{b_1}a_{b_2} + \cdots + a_{b_n}a_{b_1} \ge 0\]](http://latex.artofproblemsolving.com/e/4/b/e4b835e36eda441ed1c10c7d92775c2d52b12731.png)
where 
is some permutation of 
.
is counted exactly 
times for all 
.
.
![\[-K(a_1^2 + a_2^2 + \cdots + a_n^2)\]](http://latex.artofproblemsolving.com/d/c/c/dcc9a59b6a2eac491c4f93739bbb2615e151e8ee.png)
'
.
Then 
because 
Hence there must exist one 
,
,
![$n \cdot \mathbb{E}[a_1a_2] < 0$](http://latex.artofproblemsolving.com/b/0/b/b0b6c73eb1b55eb306d7a1153458ed14800a62b5.png)
.![$$\mathbb{E}[a_1, a_2] = \frac{(a_1+\dots+a_n)(a_1+\dots+a_n)-a_1^2-a_2^2-\dots-a_n^2}{n(n-1)} < 0$$](http://latex.artofproblemsolving.com/3/7/5/375a6a2ebbbce0fd0fdfc9a3c1ba53e79aea2480.png)
so we are done.
,
of numbers 
On the one hand, the sum of all 
.
.
