,
,
,
,
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 
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 
.
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
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.)
and 
are real numbers such that 
and 
,
?
be the expected number of flips to get HTT after getting an H.
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 
,
,
.
.
Y by Pluto1708, YS_PARKGONG
For a positive integer 
, let 
denote the number of positive divisors of . Determine whether there exists a positive integer triple 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation
![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](//latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.
, let 
denote the number of positive divisors of . Determine whether there exists a positive integer triple 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](http://latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.This post has been edited 2 times. Last edited by MNJ2357, Aug 12, 2024, 10:04 AM
Reason: Added version
Reason: Added version
is never a perfect square.
,
then as 
is QR so product must be NQR, if 
,
is perfect square, which means 
leading to descent contradiction!.
is odd only when the input is perfect square, and now Claim 1 proves that we can't construct odd 
,
the construction is 
which gives 
;
take 
for 
giving all even numbers, now as number of even numbers before 
works. We make use of the following claim to avoid odd numbers:
,
for which 
is a perfect square.
,
to be 
.
or 
,
or 
.
are all even, so we can trigger infinite descent.
works since 
which is also prime. For 
,
,
.
