,
,
,
,
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 
in radians, between 
and draws the angle on the unit circle with red marker. Bob extends Billy's line until it reaches 
Joe (with blue marker) draws a line from the point that Billy and Bob's line meets the circle to a point on 
where Joe's line is perpendicular to the x-axis. Joe then continues with another line from the end of Billy and Bob's line to 
Billy sees the drawing, and colors in the area below using a green marker:
is divided by 
, the remainder is
:o
so 
and since 
is just 
therefore 
so 
![\[\frac{(2+1)(2^2+1)(2^4+1)\dots(2^{32}+1)}{2^{64}-1} \]](http://latex.artofproblemsolving.com/f/8/d/f8d7e505481f8c8cf9654a15deb52b346e79e522.png)
of the questions wrong, and Pablo got 
answers right. Furthermore, the number of questions where both got the answer right is 
of the total. How many questions did Pedro get right?
has integer sides. Furthermore, the numerical value of its perimeter is equal to that of its area. Find the sum of all possible values for this area.
Deeds' pencil wears out just as he finishes writing the hundredth row. What is the sum of the numbers in that row?
of positive integers such that 
are the sides of a right triangle whose area is 
.
positive integers is equal to the sum of the remaining 
.
be real numbers such that 
and 
.
.
Y by ImSh95, HWenslawski
Hello!
Recently, in my forum, Xarcade!, I made a poll "Vote Ten Times", where if you vote for all ten options, you win a prize of XB, our forum currency.
In order to claim the XB, you need to prove using a photo that you voted for all ten options.
I was the first one to claim the XB.
However, bpan
was the second one to claim the XB.
Weirdly enough, the AoPS poll said that only one person voted the tenth option.
His screenshot is below.
My screen said that I voted for the option, and so did his.
When I unvoted the tenth option to figure the problem out, it said that nobody voted for it.
This is a weird poll glitch which I don't understand.
Would anybody mind explaining?
Thank you!
Recently, in my forum, Xarcade!, I made a poll "Vote Ten Times", where if you vote for all ten options, you win a prize of XB, our forum currency.
In order to claim the XB, you need to prove using a photo that you voted for all ten options.
I was the first one to claim the XB.
However, bpan
was the second one to claim the XB.Weirdly enough, the AoPS poll said that only one person voted the tenth option.
His screenshot is below.
My screen said that I voted for the option, and so did his.
When I unvoted the tenth option to figure the problem out, it said that nobody voted for it.
This is a weird poll glitch which I don't understand.
Would anybody mind explaining?
Thank you!
