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).
board, some squares are painted red. Let 
be the number of rows that have an odd number of red squares and let 
be the number of columns that have an odd number of red squares. Find all possible values of 
. For each value found, give a example of how the board can be painted.
sides are marked on the blackboard. Ana and Beto play alternately, Ana begins. Each player, in turn, must do the following:
join two vertices with a segment, without cutting another already marked segment; or delete a vertex that does not belong to any marked segment.
be the set of positive real numbers. Determine all functions 
such that ![\[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]](http://latex.artofproblemsolving.com/e/7/1/e71b6e0d0b858eb46b81149f1e6be8c41e13d301.png)
for every 
.
and 
are numbers that work, but 
and 
don't work.)
feet and area ft^2. What is the arithmetic mean of all possible values of ?[/quote]
and 
are real numbers such that 
and 
, what is the value of 
? Express your answer as a common fraction.
be the expected number of flips to get HTT after getting an H. = (case TT) 1/3 * 1/3 * 2 + (case TH) 1/3 * 2/3 * (+2) + (case H) 2/3 * (+1)= 12) + (case T) 1/3*(1+E)
be a triangle with 
its orthocenter. The circle with diameter ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
cuts the circumcircle of triangle 
at 
. Prove that the point of intersection of the lines 
and 
is the midpoint of the segment ![$[BH]$](http://latex.artofproblemsolving.com/a/9/f/a9fd89d48264ff1c3f9b77f62ef66f916978d99d.png)
the foot of 
and the midpoint of ; we have 
but 
is the circumcenter of 
thus 
cyclic so 
in the other hand 
hence 
are collinear 
denote the foot of the altitude from 
and 
denote the foot of the altitude from . Then 
and 
are cyclic. From these cyclic figures we get 
and 
. This means that the circumcircle of 
is tangent to at and that the circumcircle of 
is tangent to at 
. In these two circles 
is a common chord and is a common tangent. The result follows, since the radical axis of two circles (the common chord) bisects a common tangent.![[asy]
import olympiad;
size(8cm);
pair A=(3,6), B=(0,0), C=(9,0), D, E, F, H, K,M;
H=orthocenter(A,B,C);
D=foot(A,B,C);
F=foot(C,A,B);
M=(B+H)/2;
K=intersectionpoint(C--M,circumcircle(A,C,D),0);
draw(A--B--C--cycle);
draw(circumcircle(A,D,C),dotted);
draw(circumcircle(A,B,H),dotted);
draw(C--F);
draw(C--M);
draw(B--H);
draw(A--D);
draw(B--K--A);
draw(K--H);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,NE);
label("$K$",K,S);
label("$H$",H,NE);
label("$D$",D,S);
label("$F$",F,W);
[/asy]](http://latex.artofproblemsolving.com/3/8/5/3856251c321c9593956a01322db13186f2b94267.png)
be a triangle with its orthocenter. The circle with diameter cuts the circumcircle of triangle at . Prove that the point of intersection of the lines and is the midpoint of the segment 
vertexed HM point of 
which clearly 
median of the 
be the midpoint of and 
be the 
altitude.Notice that 
.It is well known that 
.Hence,
are concyclic 
.So,we conclude that 
are collinear.
,
. It is clear that 
and 
are cyclic so,
is tangent with .
is also tangent to and we get the desired result.
b the intersection of lines and ,
Proof:
proof:
is right angle triangle and
so
hence we are done 
such that 
now we need to prove is indeed the midpoint of line 
is a cyclic quad. and radius 
note that 
which means 
hence 
also note that 
must be the midpoint of since it lies on 
and 
intersect at the point 
.
,
.
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
.
,
lies on diagonal 
such that 
.
.
has 
,
,
.
.
and note that the midpoint of 
is the intersection of the tangents at 
and 
to the circle with diameter 
.
.
,
and 
is the midpoint of