The time is now - Spring classes are filling up!

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k a AMC 10/12 A&B Coming up Soon!
jlacosta   0
Nov 1, 2024
There is still time to train for the November 6th and November 12th AMC 10A/12A and AMC 10B/12B, respectively! Enroll in our weekend seminars to be held on November 2nd and 3rd (listed below) and you will learn problem strategies, test taking techniques, and be able to take a full practice test! Note that the “B” seminars will have different material from the “A” seminars which were held in October.

[list][*]Special AMC 10 Problem Seminar B
[*]Special AMC 12 Problem Seminar B[/list]
For those who want to take a free practice test before the AMC 10/12 competitions, you can simulate a real competition experience by following this link. As you assess your performance on these exams, be sure to gather data!

[list][*]Which problems did you get right?
[list][*]Was the topic a strength (e.g. number theory, geometry, counting/probability, algebra)?
[*]How did you prepare?
[*]What was your confidence level?[/list]
[*]Which problems did you get wrong?
[list][list][*]Did you make an arithmetic error?
[*]Did you misread the problem?
[*]Did you have the foundational knowledge for the problem?
[*]Which topics require more fluency through practice (e.g. number theory, geometry, counting/probability, algebra)?
[*]Did you run out of time?[/list][/list]
Once you have analyzed the results with the above questions, you will have a plan of attack for future contests! BEST OF LUCK to all competitors at this year’s AMC 10 and AMC 12!

Did you know that the day after both the AMC 10A/12A and AMC 10B/12B you can join a free math jam where our AoPS team will go over the most interesting problems? Find the schedule below under “Mark your calendars”.

Mark your calendars for these upcoming free math jams!
[list][*]November 20th: Amherst College Info Session, 7:30 pm ET: Matt McGann, Dean of Admission and Financial Aid at Amherst College, and Nathan Pflueger, math professor at Amherst College, will host an info session exploring both Amherst College specifically and liberal arts colleges generally. Topics include opportunities in math, the admission process, and financial aid for both US and international students.
[*]November 7th: 2024 AMC 10/12 A Discussion, Thursday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 A, administered November 6. We will discuss some of the most interesting problems from each test!
[*]November 13th: 2024 AMC 10/12 B Discussion, Wednesday, 7:30 pm ET:
[*]AoPS instructors will discuss problems from the AMC 10/12 B, administered November 12. We will discuss some of the most interesting problems from each test![/list]
AoPS Spring classes are open for enrollment. Get a jump on the New Year and enroll in our math, contest prep, coding, and science classes today! Need help finding the right plan for your goals? Check out our recommendations page!

Don’t forget: Highlight your AoPS Education on LinkedIn!
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Whether you've taken our classes at AoPS Online or AoPS Academies or reached the top echelons of our competition training with our Worldwide Online Olympiad Training (WOOT) program, you can now add your AoPS experience to the education section on your LinkedIn profile.

Don't miss this opportunity to stand out and connect with fellow problem-solvers in the professional world and be sure to follow us at: https://www.linkedin.com/school/art-of-problem-solving/mycompany/ Check out our job postings, too, if you are interested in either full-time, part-time, or internship opportunities!

Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Nov 1, 2024
0 replies
amc 10 cutoffs
titanlord   33
N 4 hours ago by ItsAniBunny
other discussion ahs too many messages

what do yall think cutoff is
33 replies
titanlord
Wednesday at 4:55 PM
ItsAniBunny
4 hours ago
FTW tournament
booking   3
N Today at 1:49 AM by SpeedCuber7
Hello! I have created a forum where we will host FTW tournaments. Please come and check out this link. Thank you!
3 replies
booking
Today at 1:16 AM
SpeedCuber7
Today at 1:49 AM
Number Sense Marathon (Mental Math)
brainpopper   535
N Today at 12:57 AM by MC_ADe
Number Sense Marathon
Hi, there isn't a number sense marathon that I know of so i'll just make one.

READ THESE RULES:
1. All calculations must be done in the head. No calculators, no paper, no writing.
2. Make problems doable. If you deem a problem too hard, skip it.
3. Please don't fight over a solution or something ridiculous.
4. Use hide tags for your problems and solutions.
5. Make all problems actually number sense. You can find a practice test for number sense here (thanks davidaops)

Don't know how to solve half the problems? Learn some tricks for mental math here.

Please make sure to follow the following format:
To post and hide your solutions, use the S# P# format. It looks like this:
[quote=User1]
S5
P6
[/quote]
[quote=User2]
S6
P7
[/quote]
To use hide tags, simply do
[code]P5[/code]to get
P5.

If you'd like to make your problems and solutions fancy, you can use LaTeX.

Lastly, please don't post if you don't know what you're doing.
535 replies
brainpopper
Jan 6, 2018
MC_ADe
Today at 12:57 AM
Questions about dividing by 0
Arr0w   28
N Today at 12:14 AM by b2025tyx
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
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28 replies
Arr0w
Dec 2, 2020
b2025tyx
Today at 12:14 AM
No more topics!
Questions about dividing by 0
Arr0w   28
N Today at 12:14 AM by b2025tyx
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
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28 replies
Arr0w
Dec 2, 2020
b2025tyx
Today at 12:14 AM
Questions about dividing by 0
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Arr0w
2908 posts
#1 • 2 Y
Y by mobro, DDCN_2011
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
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Math2k06
148 posts
#2 • 3 Y
Y by Mango247, Mango247, Mango247
$1/0$ is not a number.For #3, since anything times $0$ is $0$, the answer should be $0$.

Tldr don't poke into questions including $0$ and division. You will open a parallel univeerse
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aaja3427
1918 posts
#3
Y by
S2

@above that wouldn't be true since you are multiplying by undefined. Anything to do with undefined is undefined.
This post has been edited 1 time. Last edited by aaja3427, Dec 2, 2020, 5:08 PM
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mop
4042 posts
#4
Y by
Undefined numbers are a class of their own. Thus, they cannot be used with real or imaginary numbers without creating an undefined result.
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cubingsoda
19221 posts
#5 • 3 Y
Y by Mango247, Mango247, Mango247
see this video

He divides $a - b$ but since $a=b$ he divides by $0$. He gets $1=2$!

Dividing by $0$ opens the black hole
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correcthorsebatterystaple
620 posts
#6
Y by
Yeah, there are a couple of constructs in higher maths which assign a value to 1/0 (sort of like $i$) but you have to bend the rules of arithmetic a bit and what's going on here (1-3) is definitely not that. They're not that useful, either. We leave 1/0 undefined most of the time, similar to how you would say "no solutions" if you were asked to solve $x^2=-1$ over $\mathbb{R}.$ (to answer 5)

An "undefined value" has no numerical value; I think even calling it a "value" in the first place is a bit of a misnomer.
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ihatemath123
3338 posts
#7
Y by
S4
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cubingsoda
19221 posts
#8
Y by
undefined = no answer so no solution
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SteindorfStrongGirl
116 posts
#9
Y by
i made a presentation on this at school :< its something about 1≠2
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Awesome_Twin1
560 posts
#12
Y by
1. Avoid double posting
2. Don't bump this thread. Just look at A Letter to MSM which has been pinned.
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A_MatheMagician
2251 posts
#13
Y by
please read this post first
sry @above sniped me
This post has been edited 1 time. Last edited by A_MatheMagician, Dec 14, 2023, 1:45 AM
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vrondoS
162 posts
#14
Y by
In some ways, you can think $\frac{1}{0}=\infty$. This isn't rigorous, however, because then $\frac{2}{0}=2\infty=\infty$. Thinking about it this way can help explain some of the questions you had.
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A_MatheMagician
2251 posts
#16
Y by
Post #14 by vrondoS
infinity is not a value
that does not make any sense
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ZekeMath
39 posts
#17
Y by
If it's undefined, eventually somebody will define it :) I say better now than later.
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Yummo
278 posts
#18
Y by
Awesome_Twin1 wrote:
1. Avoid double posting
2. Don't bump this thread. Just look at A Letter to MSM which has been pinned.
A_MatheMagician wrote:
please read this post first
sry @above sniped me

Do you realize who posted that? @Arr0w, I thought you left AoPS a while ago.
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mathboy282
2966 posts
#20
Y by
vrondoS wrote:
In some ways, you can think $\frac{1}{0}=\infty$. This isn't rigorous, however, because then $\frac{2}{0}=2\infty=\infty$. Thinking about it this way can help explain some of the questions you had.

I would argue that this isn't true. $1/0 is undefined.$ The limit of it also does not exist, because:
$\lim_{x->0^+}\frac1x = +\infty$
but also:
$\lim_{x->0^-} \frac1x = -\infty$
This post has been edited 2 times. Last edited by mathboy282, Dec 14, 2023, 3:33 AM
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JoyfulSapling
664 posts
#21
Y by
Arr0w wrote:
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
1
2
3
4
5

Solutions:
1
2
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5
This post has been edited 4 times. Last edited by JoyfulSapling, Mar 14, 2024, 2:33 PM
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MathPerson12321
3160 posts
#22
Y by
I would think a variable like b would be defined as $\frac{1}{0}$, similar to $i$.
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the_mathmagician
467 posts
#23
Y by
Note: defining a number like $b=\frac{1}{0}$ serves no purpose. Why did we define $i$? Because we created the quadratic formula and realized that we couldn't describe the full range of solutions. Plus, we found something called casus irreducibilis, in which we found that describing a root of a cubic that was a real number required the use of imaginary numbers. After we accepted that we found a lot of useful other things to do with complex numbers. As for with "$b$", it serves no purpose. What can we do with this? There's nothing we can do with it. Actually, all it does is completely break our current framework of math.

Edit: Also, read this by the OP, written a few years later. It's an announcement but MSM unfortunately isn't known for paying attention to those ;)
This post has been edited 2 times. Last edited by the_mathmagician, Mar 14, 2024, 4:10 AM
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yummy-yum10-2021
94 posts
#24
Y by
Arr0w wrote:
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ 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.
[...]
What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.
[...]
What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[...]
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 $0/0$ 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 $0/0$ 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 $0$, $1$ or $\infty$, 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
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]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 $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.


Full post: A Letter to MSN
This post has been edited 1 time. Last edited by yummy-yum10-2021, Nov 11, 2024, 3:41 AM
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greenplanet2050
1150 posts
#25
Y by
yummy-yum10-2021 wrote:
Arr0w wrote:
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ 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.
[...]
What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.
[...]
What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[...]
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 $0/0$ 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 $0/0$ 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 $0$, $1$ or $\infty$, 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
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]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 $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.


Full post: A Letter to MSN
There’s no need to bump this thread.
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vincentwant
1132 posts
#26
Y by
me when the person who posted this is the same person who posted a letter to msm:
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Turtle09
1779 posts
#27
Y by
vincentwant wrote:
me when the person who posted this is the same person who posted a letter to msm:

this is crazy lol
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sadas123
103 posts
#28
Y by
honestly I think that 1/0 is infinity because if you graph this on a graphing calculator it looks like this and you can see the largest red line is when you divide with 0 and if you do it with a smaller number with 1 which I will also include makes it a very large slope change, which looks like it is going to infinity.
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club52
712 posts
#29
Y by
@above it goes to both positive and negative infinity.
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b2025tyx
1402 posts
#30
Y by
Sol 4
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blimpo
798 posts
#31
Y by
I would like to point out that undefined and infinity are not numbers, but definitions. So you can't say "x is equal to infinity" or "when dividing this by undefined"
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Catcumber
92 posts
#32
Y by
guys stop bumping this thread... its 4 years old
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b2025tyx
1402 posts
#33
Y by
Catcumber wrote:
guys stop bumping this thread... its 4 years old

Whoops, forgot to check that
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