,
,
,
,
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 
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
is cut along the planes 
,
.
,
is a nonnegative integer. Call a number gyatt if it has 
or 
factors. What is the sum of all skibidi numbers less than 
that are also gyatt numbers?
unit
,
and 
are integers. What is the value of 
?
and 
for all 
in the domain of 
?
.
.![\[\frac{a_{2}}{a_{3}}= p_{2}, \frac{1}{p_{2}}\]](http://latex.artofproblemsolving.com/1/4/4/144336317908fb86e62f631515136ad40b27650e.png)
![\[\frac{a_{3}}{a_{4}}= p_{3}, \frac{1}{p_{3}}\]](http://latex.artofproblemsolving.com/7/3/a/73a92db1b04b01c8e144443b9ceecea7fc8e1c0b.png)
![\[... \]](http://latex.artofproblemsolving.com/d/6/f/d6fa8b1ace24d35a1e1d2b2339d7808768035c64.png)
![\[\frac{a_{1995}}{a_{1}}= p_{1995}, \frac{1}{p_{1995}}\]](http://latex.artofproblemsolving.com/f/c/6/fc694daf21f909344ec7e2e2a69c8e86805f0d95.png)
![\[\frac{a_{2}}{a_{1}}= \frac{\prod p_{k}}{\prod p_{j}}= p_{1}\]](http://latex.artofproblemsolving.com/9/9/7/997b2f05a2443aaccf3ef9b34f00418473908561.png)
is some product of the elements in a subset of 
.![\[p_{1}\prod p_{j}= \prod p_{k}\]](http://latex.artofproblemsolving.com/8/4/e/84e8a3410f83c0f8207a22eeee7f5f57862eed64.png)
is equal to the set 
.
Y by Adventure10
Is it possible to place 
different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime?
I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it:
different natural numbers along a circle so that for any two of these numbers, the ratio of the greatest to the least is a prime?I feel that my solution's wording and notation is awkward (and perhaps unnecessarily complicated), so please feel free to critique it:
,
,
.
,
,
.
and 
are primes, their product cannot be, so as long as you have n different natural numbers along a circle, where n >= 3, this problem is impossible.
,
are 
distinct natural numbers around the circle, indexed by 
in 
(so 
)
are primes, and 
.
,
.
the number of the prime factors of 
(including multiplicities), we have 
and 
,
,
.
,
prime.
