This often comes up in precalculus and calculus, that is sometimes an expression will be said to undefined while at other times undetermined. What is the fundamental difference between the two?  For example, division by zero $x/0$ is said to be undefined, while $0/0$ is said to be undetermined.
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I was on the tube and overheard a dad questioning his kids about maths.  The children were probably about 11 or 12 years old.   After several more mundane questions he asked his daughter what $1/0$ evaluated to. She stated that it had no answer. He asked who told her that and she said her teacher. He then stated that her teacher had "taught it wrong" and it was actually $∞$.  I thought the Dad's statement was a little irresponsible.  Does that seem like reasonable attitude? I suppose this question is partly about morality.
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My question is whether $x/x$ is always equal to $1$. I am mostly intersted in real numbers and particularly wonder whether $x/x$ is defined at $x=0$.  On one hand the division should simplify to $1$, on the other hand you should not be allowed to divide by zero.   I have been trying to find whether the simplification 'goes first' or whether the division causes trouble first, but it has proven impossible for me to find usefull search terms.     Note that this question arose after reading  and its first comment.
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I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It cannot be a stationary value because if so then a smaller value on real number line exist, so it must be a moving value. Moving value towards $0$ so in most places we use its magnitude equal to zero but at the same time we also know that infinitesimal is not equal so in all those places were we use value of infinitesimal equal to $0$ we are making an infinitesimal error and are not $100\%$ accurate, maybe $99.9999\dots\%$ accurate, but no $100\%$! So please explain infinitesimal and its applications and methodology in context to the above paragraph or elsewise intuitively please.
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I can perform operations on polynomials. I can add, multiply, and find their roots. Despite this, I cannot define a polynomial. I wasn't in the advanced mathematics class in 8th grade, then in 9th grade I skipped the class and joined the more advanced class. This question isn't about something I don't understand; it's something I missed. My classes have not covered what really a polynomial is. I can generate one, but not define one. The internet has yielded incomplete definitions: &quot;Consisting of multiple terms&quot; or &quot;A mathematical expression containing 2 or more terms and variables.&quot; Take the following expressions for example: $2x^2-x+12-2x^2+x-12$. Consists of multiple terms, but can also be expressed as $0$. Is zero a polynomial? What about $x^{-1}$? , but I don't understand why. Is $x^2+x+1-x^{-1}-x^{-2}-x^{-3}$? a polynomial? It contains both positive and negative exponents?  tl;dr: What actually is the mathematical definition of a polynomial? Is $0$ a polynomial, and why isn't $x^{-1}$ a polynomial under this definition?
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I have a question that has been bothering me for quite some time. In second order logic sometimes there is an indication that a variable can be both bound and free. The simplest example I can give is using the variable $v_0$ and the constants $c_0, c_1$ who are not equal to one another. How does one interpret the following formula. $$(v_0=c_0)\wedge \left[ \exists v_0 (v_0 = c_1)\right]$$  I see a few possible interpretations:   The statement is not actually valid because variables cannot be both bound and free. The formula will always be false because $(v_0=c_0)$ restricts $v_0$ such that there cannot exist a $v_0=c_1$ The formula is true when $v_0=c_0$ because one essentially treats the free version of a variable differently from the bound version such that even though $v_0=c_0$ in this particular instance, it still theoretically could be that a variable $v_0$ equals the constant $c_1$ and thus the existence clause is satisfied as well.   I also recognize that it could be something completely different from any of these that I have not thought of yet, I was just curious if there was an accepted way to handle variables which are both free and bound because I have not found any satisfying answers as of yet.
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This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero?  Let me give an example:  $$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$  Multiplying both sides by $x(x+1)$ we have:  $$3x+2=A(x+1)+Bx$$  when $x \neq -1$ and $x \neq 0$.  What is traditionally done here is $x$ is set to $-1$ and $0$ to reveal: $$-3+2=-B \implies 1=B$$ and  $$2=A$$  so we find that   $$\frac{3x+2}{x(x+1)}=\frac{2}{x}+\frac{1}{x+1}$$  Why can $x$ be set equal to the roots of the denominator (in this case, $0$ and $-1$) without creating a division by zero problem?
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The definition of a derivative is the slope of a function tangent to a point. It is also defined as $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ If we apply this to $f(x)= |x|$, we get that it is $\lim\limits_{h\to 0} \dfrac{|h|}{h}$, which is undefined. However, if we look at the graph of $|x|$, we see that there can exist a tangent line at x=0, with slope 0. So why is the derivative undefined instead of 0?
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Can $f(x)=|x|^3$ be differentiated? If yes, how? If no, why?    What I've tried:  lim as $h$ tends to $0$ of $$\frac{f(x+h)-f(x)}{h} = \lim_{h \to 0} (3(x+h)^2|x=h|-3x^2|x|)/h$$  However, I have no idea how to proceed from here. Please help :(  Thanks!
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I'm working with the universal and the existential quantifiers and I was wondering is there an order of operations that applies to them? What is the difference between   $(\forall x)P(x)(\exists x)Q(x)$  And  $(\exists x)Q(x)(\forall x)P(x)$  Lets say we have this relationship  (a) $[(\forall x)P(x) \vee (\exists x)Q(x)] \implies (\forall x)[P(x) \vee Q(x) ]$
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For any $\epsilon$, if $\epsilon&gt;0$ and $|x|&lt;\epsilon$, then $x=0$.  I understand that supposing  $\epsilon=\frac{x}{2}$ will lead to a contradiction, but let’s take a correct case: Let $\epsilon=3$, then $x$ would have a whole set of values. Can you explain what is going on?
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I have a function defined as follows: $f(x,y)= \dfrac{x^2-y^2}{\left(x^2+y^2\right)^2}$, if $(x,y)\neq (0, 0)$ and $f(x,y)=0$ if $(x,y)=(0,0)$.  Now, $$\int_0^1\int_0^1 f(x,y)\,\text{d}x~\text{d}y=-\frac{\pi}{4}$$ and $$\int_0^1\int_0^1 f(x,y)\,\text{d}y~\text{d}x=\frac{\pi}{4}.$$   The question I have is this: why does this not contradict Fubini's theorem?    thanks.
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I have a fairly fundamental question. What is the difference between infinity as shown by the aleph numbers and the infinity we see in algebra and calculus?  Are they interchangeable/transposable in any way? For example, could you describe the infinity of a limit with an aleph number? Does it have cardinality?  Thank you.
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Possible Duplicate:          What will be the answer if zero$(0)$ divided by zero$(0)$ Why It is un answerable?   $$\frac 00=\space ?$$ Thanks in Advance, Vicky
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Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
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I'm reading a  by Andrew Granville on analytic number theory.  On page 22 of the paper, there appears a strange looking symbol, undefined.  I've circled it in red in the screenshot below.    Since it's not defined in the paper, I'm assuming it must be standard notation.   From the context, I'm assuming it means something like "as compared to", or "with reference to", but that's just a guess.  Can anyone identify the symbol, even better explain what it means and/or provide a reference?  Is there a name to speak the symbol?  Thanks in advance.
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From : In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits. Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form.    The indeterminate forms include $0^{0},\frac{0}{0},(\infty - \infty),1^{\infty}, \ \text{etc}\cdots$   My question is can anyone give me a nice explanation of why $1^{\infty}$ is considered to be an indeterminate form? Because, i don't see any justification of this fact. I am still perplexed.
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There are several things that I don't understand. I know that zero is a real number, but I'm confused on the how and why aspects. What defines a real number, compared to a number that's considered non-real? For what reason is zero real, and how is it that a value such as infinity isn't when zero is?  Also, what purpose does zero even have? It's obvious that it's used as a place holder, but what value does it have alone? What makes it any different from other numbers? I realize that it is typically interchanged with nothing, but that would technically mean that zero is nothing—and that's not true. People have told me already so many times that zero does not necessarily mean nothing, and I figured it out a long while ago anyway. So, what I want to know is the actual value that can be associated with zero.  However, my question brings up another topic that has been argued and disputed about and disagreed on for years. Zero divided by zero, or 0/0. There are a lot of varying answers about this, but what's been constant throughout everything is that people claim 0/0 is not a real number. So, if I may ask, why is it that zero is considered a real number, and not 0/0? And why is it considered non-real in the first place? True, there is no defined value, but the expression is anything but meaningless; if it was, then there wouldn't be so many people trying to understand it. I guess what I'm trying to figure out is the reason behind the expression's classification as non-real—especially when the sole number of zero is real, despite not having an evident value.  This covers a few different things, so I was uncertain about what to make the title of these questions. I apologize for that. If anyone can answer, though, I'd be extremely grateful!