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).
, such that 
. 
lie on a moving circle 
. intersects 
at 
and 
at 
. 
intersect at 
. 
intersect at 
. 
is the perpendicular image of with respect to 
. Prove that 
and the orthocenter 
are collinear.
be a positive integer. Let 
be the number of pairs of integers 
satisfying 
for odd values of 
. Let 
be the number of pairs of integers satisfying . Prove that 
.
and 
be positive integers and let 
be a set of points in the plane such that are collinear, and
of there are at least points of equidistant from 
![\[ k < \frac {1}{2} + \sqrt {2 \cdot n}
\]](http://latex.artofproblemsolving.com/d/a/5/da56146bf1ad38f257e203e4a0231bd604060a7a.png)
. Currently he will score a on 
. He can expend 
units of effort to gain 
points on the 
, 
units of effort to gain 
points on the 
, and units of effort to gain point on the 
. He will need to get at least 
points on 
and combined and get at least 
points on 
to make . What is the minimum amount of effort he can expend to make ?
.
be a regular tetrahedron with center 
. Then choose the rays 
through so that they make very acute angles with 
. Similarly define the rays 
for 
. Then for each ray among these rays there are exactly two which make with it acute angles. For example for the ray 
only 
and 
make acute angles with it where 
.
be the set of the given rays. For any ray 
define 
as the subset of the rays which make acute angle with 
. According to the problem condition 
for any . Let be the center of the rays. For any ray from draw the plane 
through perpendicular to and denote the half-spaces containing and not containing by 
and 
, respectively, so that 
and 
.
. Take any ray 
. Then exactly two other rays of lie in 
. Choose another ray 
of . Again exactly two other rays of lie in 
. Then choose another ray 
of . Exactly two other rays of lie in 
. At last take another ray 
. Exactly two other rays of lie in 
. Thus, up to this moment we have counted rays.
be the ray we have not yet come across. Clearly 
. Then one of 
is the complement ray of 
. Indeed, suppose that none of 
complements . It is not difficult to see that if not all the rays among lie in 
then 
. Thus all of lie in . In this case the intersection 
is a ray which should coincide with 
. Clearly 
. In the other case, i.e. when one of is the complement of we can simply swap its notation with . So we can safely assume from now on that the complement of is and 
. three lie in 
and three lie in so the remaining at least 
should lie in . Let 
be the set of these at least rays. Let 
and 
.
exactly one ray of 
lies in . Indeed, consider 
as . 
thus no more than one ray of 
may lie in . Suppose that 
. But then all the rays of lie in the closed half-plane 
. Indeed, if we divide this half-plane into two quadrants then one of these quadrants (with its borders) will contain at least 
rays of . Thus one of them will make acute angles with three others — contradiction. Consequently 
.
and 
for some 
. Particularly 
. So 
and it can be checked that the intersection of and 
is an infinite closed angle (with its border lines) with degree no more than 
(to see this, for example, notice that the projection of 
on does not lie in 
). But this angle contains 
rays so one of them makes acute angles with at least three (even four) of them. This contradiction finishes the proof.
lie in then 
. Thus all of lie in .
are 
from the beginning of your solution, then they don't lie in 
, why this isn't the case?
. lie in then " but now I noted that there is a slight problem with this claim so edited this paragraph a little bit. Now the solution should be correct, I guess.
lie in this half-plane. But then, if we divide this half-plane into two quadrants (for example one closed and one open) then one of these quadrants will contain a ray which makes acute angles with three rays -- contradiction.
for each .
, then 
, but 
, which is a contradiction, and same with 
, so I don't see why you need the last paragraph
lie in this half-plane. But then, if we divide this half-plane into two quadrants (for example one closed and one open) then one of these quadrants will contain a ray which makes acute angles with three rays -- contradiction. for each ., then , but , which is a contradiction, and same with , so I don't see why you need the last paragraph
. I used this to repair the solution. Now I hope everything is correct.
