,
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
[/quote]
,
and height 
The volume of the space where the sphere overlaps with the pyramid can be expressed in the form 
Find 
.
.
.
of 
is said to be invariant if for all 
we have 
.
we define the number of invariant subsets 
.
.
there exists a function 
.
and 
for 
.
be a positive integer. Suppose that for every positive integer 
there exists a positive integer 
such that 
.
satisfying following conditions.
for all 
.
with 
,
holds.
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
.
and the circle of radius 
cross 
and 
,
and circle 
are concurrent.
of functions from the set of real numbers to itself that satisfy ![\[g(f(x+y)) = f(x) + (2x + y)g(y)\]](http://latex.artofproblemsolving.com/0/a/3/0a3bc25299a0bcbd3f3602a0e6881d68fce750c5.png)
for all real numbers 
and 
.
such that for all 
be given pairwise coprime positive integers where 
.
be positive integers. We call 
of primes (greater than 5) such that : 
of positive integers such that:
and incircle touching 
at 
be the antipode of 
is chosen on the internal angle bisector of 
such that 
.
be the midpoint of arc 
,
be the midpoint of 
.
Y by megarnie, ibmo0907, asimov, NoSignOfTheta, mahaler, Critical, juliankuang, violin21, ruthbaderginsburg, ANAIYER, TheKnittedFairy, IMUKAT, Jwenslawski, Bluejay24, son7, eagles2018, rayfish, babyhamster, jhu08, fuzimiao2013, suvamkonar, angrybird_r, hh99754539, wamofan, DramaticFlossKid, StopSine, bingo2016, MathNinja234, je100, cloudybook, the_mathmagician, vaporwave, G8L18M, kante314, sanaops9, Math_Geometry, ChromeRaptor777, mod_x, chloetseng, kundusne000, OliverA, asdf334, Ericorange, Rock22, math31415926535, A380, eggybonk, smartatmath, Lamboreghini, Robomania_534, eibc, DankBasher619, ilovepizza2020, Toinfinity, Bimikel, cj13609517288, HWenslawski, inoxb198, rg_ryse, Ansh2020, ConfidentKoala4, Inventor6, Nickelslordm, Voldemort101, Taco12, rmac, megahertz13, aayr, tigeryong, pikapika007, fluff_E, Mingyu_Chen, ShyamR, skyss, ImSh95, A_Crabby_Crab, russellk, CyclicISLscelesTrapezoid, TsunamiStorm08, mc21s, mathdragon2000, Alex100, mathking999, FalconMaster, Jc426, Arrowhead575, thodupunuri, ab2024, YaoAOPS, moab33, jmiao, stuffedmath, happymathEZ, AlphaBetaGammaOmega, RoyalCactus, RedFlame2112, Cygnet, goodskate, sdash314, oval, peelybonehead, gnidoc, aidan0626, Doudou_Chen, wxl18, Testking, veranikaia, RedFireTruck, boing123, MathWizard10, RaymondZhu, Rudra1101001, peppapig_, hdrcure, rohan.sp, Peregrine11, DepressedSucculent, s12d34, alpha_2, eagleup, J55406, RubixMaster21, Fermat1665, vishwathganesan, Coco7, mathisawesome2169, Sotowa, duskstream, JacobJB, MaverickMan, Practice40hrs, penpencil79, OronSH, Sedro, justJen, detkitten, Anchovy, PowerOfPi_09, YBSuburbanTea, NTfish, DouDragon, ryanbear, akliu, nathanj, GeometryJake, binyubin, Gershwin_Fan, evanhliu2009, vader2008, scannose, RedChameleon, NamePending, ehz2701, iamhungry, purplepenguin2, PatTheKing806, lethan3, PlayfulEagle, Supernova283, channing421, williamxiao, CoolCarsOnTheRun, Pi-rate, amazingxin777, michaelwenquan, Rabbit47, aops-g5-gethsemanea2, MinecraftPlayer404, samrocksnature, akpi2, Auburn4, ObjectZ, megahertz2013, ostriches88, sausagebun, resources, DofL, roblmin235, Pathpaver, NightFury101, VDR, Scienceiscool123, a-dumb-math-kid, chiaroscuro, falco_sparverius, LinkLeap, ttt_jmixlu, Inaaya, happyhippos, Siddharth03, elizhang101412, DDCN_2011, EpicBird08, sehgalsh, captainnobody, EthanHL, amaops1123, Everestbaker, ElsaY456, bpan2021, AlienGirl05, HamstPan38825, Pi31, g943, Lioncub212, alpha_zom, Facejo, ksdicecream, Lighting_Calculator, Zandermaan, juicetin.kim, challengeitmath, akpi, Mathguy49, DaGoldenEagle, hexafandra, pieMax2713, bigfuzzybear, mathmax12, UnrgstrdGst, Bumfuzzle, tigershark123, sabkx, ultimate_life_form, NegativeZeroPlusOne, TerrificChameleon, trk08, BlinkySalamander11, alan9535, FourthRootOfX, MathCone, dbnl, yummy-yum10-2021, Cokevending56, pearlhaas, ShockFish, CrystalFlower, ranu540, skronkmonster, feliciaxu, ethancui0529, thumper12, atharvd, Rice_Farmer, dolphinday, lpieleanu, AndrewC10, Orthogonal., ap246, bjc, Mathbee2020, jf123456, CatsMeow12, ChuMath, zhenghua, FaThEr-SqUiRrEl, Shadow_Sniper, WilliamA, Andrew2019, vincentwant, illusion2415, roribaki, SwordAxe, meduh6849, Turtle09, PartricklnZ, dragonborn56, Awesome_Twin1, miguel00, Blue_banana4, tennisbeast14, Solocraftsolo, ujulee, xytan0585, ahxun2006, WiseTigerJ1, Spiritpalm, stardog157, sharpdragon15, ceilingfan404, TeamFoster-Keefe4Ever, ATGY, rty, piLover314159, anduran, valenbb, c_double_sharp, DearPrince, At777, MathematicalGymnast573, Aaronjudgeisgoat, eggymath, Exponent11, Superstriker, patrickzheng, wkt_1, NaturalSelection, SpeedCuber7, K124659, MajesticCheese, DuoDuoling0, lionlife, Phy6, RuinGuard, bhargchi, awu53, mithu542, Kawhi2, greenpenguin2, tohill, Spacepandamath13, Dustin_Links, 1218865, googolplexinfiinity, zz1227j, giratina3, DreamineYT
Greetings.
I have seen many posts talking about commonly asked questions, such as finding the value of
, 
,
, 
, 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.
2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.
INDETERMINATE VS UNDEFINED
What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like
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.
But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of
, 
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
![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](//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.
WHEN THE WATERS GET MUDDIED
So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example 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.
One example of making something undefined into something defined is the extended real number line, which we define as
![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](//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*}](//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,
![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](//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
![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](//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
![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](//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
![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](//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
![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](//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).
I have seen many posts talking about commonly asked questions, such as finding the value of
, 
,
, 
, 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.
- Firstly, the case of . 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.
- What about ? 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.
- What about ? 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.
- What about if
? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
- What about ? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.
INDETERMINATE VS UNDEFINED
What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like
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.But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of
, 
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.WHEN THE WATERS GET MUDDIED
So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example
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.One example of making something undefined into something defined is the extended real number line, which we define as
![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](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).This post has been edited 13 times. Last edited by Arr0w, Sep 17, 2022, 11:43 PM
and move on with it, really no reason not to (the rationale being, "Because it is").
or 
.
etc. for free, because... we literally have not defined 
as a perfectly valid number, because it starts becoming useful. Particularly, we do define 
in contexts such as measure theory. This differs from the "limit of a function" interpretation, in which 
would be indeterminate.
,
.
.
where 
?
,
.
.
is increasing (i.e. 
everywhere in 
,
converges pointwise to a function 
and moreover
Notice how I allow my functions to take values of 
shows up
What is 
.
.
to make this work, so that 
still.
thing:
,
and 
.
.
?
for any positive integer 
.
for any positive integer 
such that 
then I will provide a sufficient 
for the number of nines needed such that 
We can use this to form a proof by contradiction. More specifically, for a positive value 
if 
then 
is sufficient. Therefore, there will always be a delta such that 
if for every 
which implies 
so 
for a positive value of 
there exists a 
