AoPS Academy
, 
,
, 
, 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.. 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.? 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.? 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.
? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
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.
, 
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. 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.![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](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).
be a parallelogram with 
A point 
is chosen on the extension of ray 
past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
be an integer. We are given a balance and 
weights of weight 
. We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
.
and 
, we have that 
and 
. Find the value of 
.
and 
. He notices that neither nor is a perfect square, but 
is a perfect square. What is the smallest possible value of 
?
whose digits sum to 
?
has 
and 
. The altitude drawn from 
intersects at 
, where 
. What is the area of ? letters in the word for each positive integer is 
. Each letter of the word is chosen randomly and independently from the uppercase English alphabet. The probability of Archyuta’s program outputting “EMCC” can be written as 
for some positive integer 
. What is the greatest nonnegative integer such that 
divides ?
that can be expressed as the sum of seven consecutive whole numbers, as the sum of five consecutive whole numbers, and as the sum of three consecutive whole numbers? with side length 
, square 
is inscribed in such that 
is on side and 
is on side 
such that 
and 
. If square PQRS inscribed in such that 
, find the side length of 
.
seconds to reach the top. Tired, he then walks at one-third of his running speed back down the escalator, taking 30 seconds to reach the bottom. Assuming his running speed and the escalator’s speed are constant, what is the ratio of his running speed to the escalator’s speed? bricks shaped like rectangular prisms each of dimension foot by foot by 
feet which he stores inside a feet by feet by feet hollow box. In how many ways can he fit his bricks into the box? (Rotations and reflections of a configuration are considered distinct.) is a point lying inside rectangle . If 
, 
and 
, find 
in degrees.
be a positive integer. If of ’s divisors are prime and 
of ’s divisors are composite, how many of ’s divisors are perfect squares?
. Let 
be the sum of all distinct possible values of their absolute difference. Find the remainder when is divided by .
. The internal angle bisectors of each of its four vertices are drawn, bounding a square region with area 
. Find the perimeter of the rectangle.
be a right triangle, with a right angle at 
. The perpendicular bisector of hypotenuse 
splits the triangle into a smaller triangle and a quadrilateral. If the triangle has an area of 
and the quadrilateral has an area of 
, find the length of . be the sum of the smallest positive perfect squares minus the sum of the smallest positive odd numbers. What is the largest prime factor of ?
and two congruent equilateral triangles placed inside the square, one in each corner and with one overlapping side. What is the distance between the marked vertices? in each unit square of a two-by-two grid, such that no two isosceles triangles share an edge? One valid construction is shown on the left, followed by an invalid construction on the right., followed by Dr. Drescher writing another on the second turn. Each turn afterwards, they take the two newest numbers on the board and concatenate them, writing the resulting number of the board. For instance, the first few numbers on the board are , , 
, 
, 
, 
How many turns does it take for them to write a number which is divisible by 
?
![$\sqrt[2]{2}$](http://latex.artofproblemsolving.com/1/8/6/1869ac018fb72a26a12ab14499cbf1b1c13b4a10.png)
, ![$\sqrt[3]{3}$](http://latex.artofproblemsolving.com/1/7/a/17a6bb5dc11696b8650b6d3a11e770f861f4c326.png)
, and ![$\sqrt[5]{5}$](http://latex.artofproblemsolving.com/a/b/9/ab90f2d42dc24065e2a64988fe76f22eb9359799.png)
.
Prove that: 
is the sum of 
square numbers
Let 
and is the sum of square numbers. Prove that: 
is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
Let and is the sum of square numbers. Prove that: is the sum of square numbers
be equal to divided by the largest power of that divides . Legendre's three-square theorem says that is expressible as the sum of three squares iff is not mod 
.
is never mod , so even always works. If is odd, if 
is mod , then 
is even and thus 
is even. Thus 
is odd. Observe that if is odd then 
. Thus if is mod then 
is mod . However, any odd number to an odd power is congruent to itself mod , so this means 
, contradiction.
, which would make it trivial.
