,
,
,
,
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 
Prove that
denote the sum of the sum of the factors of all positive 
,
is the sum of the factors of 
added to the sum of the factors of 
and so on until the sum of the factors of 
.
denote 
.
or find a counterexample
such that
,
,
such that 
,
in terms of 
and 
,
.
are natural numbers satisfy that: 
and 
are prime numbers. Prove that 
is a perfect squared number
?
MathJuice bottles and Bob starts with 
MathJuice bottles. An unfair coin is then flipped, with probability 
of landing heads. If the coin lands heads, Alice gives Bob a bottle; otherwise, Bob gives Alice a bottle. This process repeats until someone runs out of bottles.
Y by anantmudgal09, Ankoganit, Tawan, Taha1381, Adventure10, Mango247
Let 
be fixed integers. There are 
people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of 
distinct players 
such that 
beat 
for each 
, player 
also beat 
. Prove that the players can be numbered 
in such a way that, whenever 
, player 
beat player 
.
Gabriel Carroll.
be fixed integers. There are 
people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of 
distinct players 
such that 
beat 
for each 
, player 
also beat 
. Prove that the players can be numbered 
in such a way that, whenever 
, player 
beat player 
.Gabriel Carroll.
such that every edges between them are directed to 
.
into some cycles 
such that if 
,
are directed to 
,
and each points in the same cycle by the reverse order of original cycle.
-
-
be that cycle. Remove one point 
in the 
,
,
.
.
,
and 
.
if player 
beats player 
.
.
.
cycle. We now split up the proof into two cases.
,
,
not in 
for some 
,![\[
w>v_i>v_{i+1}>\cdots >v_{i-1}
\]](http://latex.artofproblemsolving.com/3/6/9/369c075ca49da6b20ca32038f43ff7842f7a4f0d.png)
This is a cycle of length 
.
it easily follows that if 
,
,
for all 
,
beats all members of 
,
,
and 
are the sizes of 
,
.
to 
to 
to 
player 
implies 
![\[ v_1 \to v_2 \to \dots \to v_k \to v_1. \]](http://latex.artofproblemsolving.com/e/0/4/e04a3565af96040e19a0f054ca31e53da652a64f.png)
By maximality, this means we cannot have 
for any 
.
for every 
for every 
is a path). Since ![[asy] size(6cm); dotfactor *= 1.5;
pair A = dir(90); pair B = dir(210); pair C = dir(330);
dot("$v_1$", A, A, red); dot("$v_2$", B, B, red); dot("$v_3$", C, C, red); draw(A--B, red, EndArrow, Margin(2,2)); draw(B--C, red, EndArrow, Margin(2,2)); draw(C--A, red, EndArrow, Margin(2,2));
draw(ellipse( (2.6,0.3), 0.6, 1.6), blue); draw(ellipse( (-2.6,0.3), 0.6, 1.6), blue); draw((1,0.3)--(2,0.3), blue+1, EndArrow, Margin(2,2)); draw((-2,0.3)--(-1,0.3), blue+1, EndArrow, Margin(2,2)); draw((-2.3,-1.2)--(2.3,-1.2), blue+dashed, EndArrow, Margin(2,2));
dot((2.8,0.7), blue); dot((2.7,1.2), blue); dot((2.5,0.1), blue); dot((-2.8,0.7), blue); dot((-2.7,-1.0), blue); dot((-2.3,0.1), blue); [/asy]](http://latex.artofproblemsolving.com/5/5/4/554bf46373abcd06e04036ba34b6a3e99e11385e.png)
.
,
so that 
are edges for all 
respectively. The condition ensures that 
is an edge for all 
.
by induction on 
by 
,
is the smallest integer with this property.
with the maximum graph theoretic distance. Then 
;
then we have a 
cycle. If 
then for any chain 
we have 
contradicting the minimality of 
,
.
if we can find 
with 
as edges then it is possible to append 
all edges between 
is an edge, then we have a directed path 
for 
as a chain of 
,
to vertices in 
.
auch that each player in 
.
.
in 
and a path of length 
in the other 
vertices. Hence the path (length 
path of length 
.
.
then there is a cycle of length 
.
for all 
.
is odd. We will deal with the case 
.
,
,
works. Notice how 
is not in the cycle.
,
with 
where 
and sort them in that order.
,
where 
.
