AoPS Academy
, 
,
, 
, why 
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.. It is usually regarded that 
, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.? The issue here is that 
isn't even rigorously defined in this expression. What exactly do we mean by ? Unless the example in question is put in context in a formal manner, then we say that is meaningless.? Suppose that 
. Then we would have 
, absurd. A more rigorous treatment of the idea is that 
does not exist in the first place, although you will see why in a calculus course. So the point is that is undefined.
? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
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 is an indeterminate form.
, 
or , obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](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 
. On the other hand, 
is undefined because dividing by is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context. is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](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 it is perfectly OK to say that,![\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 is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](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 is allowed here! In fact, we have![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and are left undefined. We also have
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 , by defining them as![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by also being allowed with the same indeterminate forms (in addition to some other ones).
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.
is a connected graph with 
edges. Prove that it is possible to label the edges 
in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.
belongs to at most one edge. The graph 
is connected if for each pair of distinct vertices 
there is some sequence of vertices 
such that each pair 
is joined by an edge of 
.
. I couldn't solve this and was wondering if any of you can find the pattern
[hide=P#]a1b2c3[/hide] [hide=S#]a1b2c3[/hide]
to 
have the digit 
and are divisible by 
?
. Adding to its area the area of 
we get:![$[CRB]+[CRD]=[BCD]=R^2\ (\ 2\ )$](http://latex.artofproblemsolving.com/9/d/0/9d09cf84b632d93d68eb266f0c47d1d75612d4b9.png)
. Adding to ![$[RDS]$](http://latex.artofproblemsolving.com/2/7/e/27e6a50c0abfcae32df9fcf2b2a8d0925998515b.png)
the same ![$[CDR]$](http://latex.artofproblemsolving.com/6/4/6/6462e2fd63e60e317abb0d4831f987ba2da5dbff.png)
we get ![$[RDS]+[CDR]=[CDSR]$](http://latex.artofproblemsolving.com/3/c/6/3c68ba02871db35c8fdf828286901404cd364a77.png)
. 
having perpendicular diagonals, its area is ![$[CDSR]=\frac{CS\cdot DR}{2}\ (\ 3\ )$](http://latex.artofproblemsolving.com/6/3/9/6396ef4e79fe87988d731e007023e2e4f68aff81.png)
, so we need 
. The last relation we get from 
, done.
is the circumradius the square.
be the center of 
and let 
be the point at infinity of 
(also point at infinity of the tangent of 
at D).
![$\frac{OB}{OD} \cdot \frac{RD}{RB}=\frac{OA}{OS} \Longrightarrow OS \cdot RD=OA \cdot RB \Longrightarrow [DSR]=[ARB].$](http://latex.artofproblemsolving.com/1/1/0/1109c08461d9b93da58cc7ded47e73461e3fddfa.png)
, then 
too. Also let 
be the length of the radius of the circle.
we have 
. Consequently 
is: ![$$[ARB] = \frac{1}{2} \cdot AB \cdot BR \cdot \sin(\angle ABR)=\frac{1}{2} r\sqrt{2}\cdot r(1-tan\phi) \frac{\sqrt{2}}{2}=\frac{1}{2}r^2(1-tan\phi)$$](http://latex.artofproblemsolving.com/a/e/2/ae251524382847a6702441fef1efe00c1eb2ffe8.png)
On the other hand, in the right-angled triangle 
we have: 
, so 
is: ![$$[DSR]=\frac{1}{2}\cdot OS \cdot DR=\frac{1}{2}rtan(45^\circ-\phi)r(1+tan\phi) = \frac{1}{2}r^2tan(45^\circ-\phi)(1+tan\phi)$$](http://latex.artofproblemsolving.com/8/4/c/84cce5c5a32ebf85639c1bae46edc6dd2cc39fb6.png)
Finally, using the trigonometrical formula:
So the area of the triangle is:![$$[DSR]=\frac{1}{2}r^2tan(45^\circ-\phi)(1+tan\phi)=\frac{1}{2}r^2\frac{1-tan\phi}{1+tan\phi}(1+tan\phi)=\frac{1}{2}r^2(1-tan\phi)$$](http://latex.artofproblemsolving.com/9/d/2/9d21642f91837863d2e8444737cf6b80395a72fa.png)
Thus the areas of and are equal.
be the intersection point of lines 
and 
and let 
be the projection of onto line . be the circumcenter of circle . Now 
, assume that side length of square = 1.
= 
= 
= 
, 
is cyclic.
= 
= 
, implying that 
.
area of 
, 
is an isosceles right triangle. So 
= 
and 
above, ![$[DSR]=[ARB].$](http://latex.artofproblemsolving.com/a/1/c/a1cf1807f2fdd7a23d58ae91bc83e1b38dd7d710.png)
and 
. Prove that
Where 
and 
. Prove that
Where 
:o
so 
and since 
is just 
therefore 
so 
be a point on the shorter arc 
and 
.
,
.
.![\[\frac {DA}{DS}=\frac {\sin \angle ASD}{\sin \angle SAD}=\frac {\sin \angle OSD}{\sin \angle SAD}=\frac {\sin \angle PBR}{\sin \angle BPR}=\frac {RP}{RB}\,.\]](http://latex.artofproblemsolving.com/4/0/5/405fb8d8f8b8eb64a33716fbe5c92b863c68420b.png)
Calculating the power of 
.
and 
.
,
and 
have equal areas. But 
.
is tangent to 
.
the claim follows.
which is true by the claim, we are done.![[asy]
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pair O,A,B,C,D,P,R,S;
O=(0,0);B=dir(45);A=dir(135);D=dir(225);C=dir(315);path w=circumcircle(A,B,C);P=dir(100);R=extension(P,C,D,B);S=extension(P,D,A,C);
draw(A--B--C--D--cycle,deep);draw(w,deep);draw(A--C,deep);draw(B--D,deep);draw(P--D,deep);draw(P--C,deep);draw(D--S--R--cycle,org);draw(A--R--B--cycle,light);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$D$",D,dir(D));
dot("$P$",P,dir(P));
dot("$R$",R,dir(R));
dot("$S$",S,dir(S));
[/asy]](http://latex.artofproblemsolving.com/d/e/6/de65c126817b1946dcb5e31f18111a80e915f020.png)
.
.![$[PAB]=[PSR]$](http://latex.artofproblemsolving.com/e/7/f/e7f7b0ce26638596ff2035e06f35a3d64441509c.png)
.![$[DSRC]=[PDC]-[PSR]=[PDC]-[PAB]=[DBC]$](http://latex.artofproblemsolving.com/8/7/4/874118f361646a6781f453351857dc5cd25c0ac5.png)
.![$[DSR]=[CRB]$](http://latex.artofproblemsolving.com/2/2/e/22e13fa7d195107bf8b935959ea5147e32c4e0b5.png)
,