,
,
,
,
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 
and 
.
and natural 
such that 
and 
,![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
and 
.
Is there any generalizations?
different icons.
are
Y by nikaaryu, Rg230403, meet18, Aritra12, p_square, NumberX, OlympusHero, sotpidot, bobjoe123
Betal marks 
points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any 
of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?
Note. A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
Proposed by Prithwijit De and Sutanay Bhattacharya
points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any 
of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?Note. A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
Proposed by Prithwijit De and Sutanay Bhattacharya
be three noncollinear points in the plane; then given either the midpoint of 
or the line through 
parallel to 
endpoints, so some two of the chosen segments have a common endpoint. Thus, we have a triangle 
and we are given the midpoints of 
and 
.
,
parallel to 
parallel to 
since they are parallel to intersecting lines. Now 
and we have the midpoint of 
.
.
odd points in the plane, and there are 
line segments joining some of the 
be a triangle with midpoints 
marked, then for any segment 
and a line through 
.
.
of 
.
.
is the midpoint of the segment 
and 
is parallel and midway between EF and BC. let this intersect DG at M, then if M intersects DG at K and K intersect EF at S, then SGCD is a parallelogram and so just extend DS to meet PB at its midpoint. Now if we connect everything to B, its all connected so boom.
![[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
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dot((10.87,4.65),dotstyle);
label("$P$", (9.85,4.55), NE * labelscalefactor);
dot((10.87,7.17),dotstyle);
label("$X$", (10.33,7.85), NE * labelscalefactor);
dot((12.756619588193846,4.975691380255226),dotstyle);
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dot((12.213615560640731,3.813844393592681),linewidth(4pt) + dotstyle);
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are given and we want to draw a line through 
to be a point on 
.
and let 
.
and as 
we have 
.
.
such that the midpoint of 
and midpoint of 
.
.
,
and let 
meet 
at 
.
is the desired line.
meet 
at 
meet 
at 
passes through the midpoint of 
be the midpoints of 
and 
.
and by claim 2 we can draw in 
meet 
and 
.
is the midpoint of 
.
draw in the midpoint of 
parallel to ![[asy]
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dot((3.12,0.12),linewidth(2.pt) + dotstyle);
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dot((8.48,0.12),linewidth(2.pt) + dotstyle);
label("$D$", (8.56,0.2), NE * labelscalefactor);
dot((5.8,0.12),linewidth(2.pt) + dotstyle);
label("$M$", (5.88,0.2), NE * labelscalefactor);
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[/asy]](http://latex.artofproblemsolving.com/0/f/2/0f2f87cc45da2532e2843323e5202ec465159438.png)
,
.
bisects 
segments each with end point from the 
segments parallel to a particular direction.![[asy]
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not parallel to 
be the midpoint of 
.
,
.
.
.![[asy]
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.
,
[Since
,
through 
(Old Bengali Cartoon 
).Next time Nonte and fonte pls or maybe motu patlu. 
on the IMO
and 
with midpoints 
and 
,
.
.
we can mark the midpoint 
of 
intersects 
at it's midpoint 
.
on 
,
.
and 
with their midpoints 
,
can be constructed using a straight-edge.
