Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.

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What really is an indeterminate form?

We can apply l’Hôpital’s Rule to the indeterminate quotients $ \dfrac{0}{0} $ and $ \dfrac{\infty}{\infty} $, but why can’t we directly apply it to the indeterminate difference $ \infty - \infty $ or ...
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Intuitive explanation for formula of maximum length of a pipe moving around a corner?

For one of my homework problems, we had to try and find the maximum possible length $L$ of a pipe (indicated in red) such that it can be moved around a corner with corridor lengths $A$ and $B$ (...
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Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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Geometric interpretation of primitive element theorem?

The primitive element theorem is a basic result about field extensions. I was wondering whether there are nice geometric ways to visualize it or think about it. Since field spectra are singletons, it ...
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Why does gradient descent work?

On Wikipedia, this is the following description of gradient descent: Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable ...
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Intuitive Understanding of the constant “$e$”

Potentially related-questions, shown before posting, didn't have anything like this, so I apologize in advance if this is a duplicate. I know there are many ways of calculating (or should I say "...
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Paradox of the trumpet shape

This is a question I had for long time now, when you rotate the function $y=1/x$, $x>0$ (say $x$ and $y$ both measure meters) about the $x$ axes by $2\pi$ you get a shape which has infinite surface ...
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What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
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What is the intuitive meaning if multiplying by fractional 1?

first post ever on stack exchange in years of using it. Can anyone provide a historical or logical deduction of the reasoning behind multiplication by 1 via a fraction? For instance, in finance ...
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Confusion about geometric interpretation of proof that $\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD

I'm working through a proof that $R=\mathbb R[X,Y,Z]/ \left\langle X^2+Y^2+Z^2 -1 \right\rangle $ is a UFD. The idea is to localize at $1-x$ and show the result is a UFD. Since $R$ is atomic as a ...
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Why is the area under a curve the integral?

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself. However when it comes to the area ...
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Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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1answer
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Least amount of repetitions s.t. probability greater than 1/2

Assume that for a formula $F$ over $n$ variables, there are exactly $k$ allocations that satisfy it. How many random samples from the set $\{0,1\}^n$ are necessary to find an allocation satisfying the ...
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101 views

Why doesn't coordinate difference between two points correspond to distance between two points?

I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\...
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Find the first $4$ Hermite polynomials using a recursion relation

Given the Probabilists' Hermite differential equation: $$U''-xU'+\lambda U=0\tag{1}$$ A book question asks me to: Find the first $4$ polynomial solutions (for $...
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Intuition about the second isomorphism theorem

In group theory we have the second isomorphism theorem which can be stated as follows: Let $G$ be a group and let $S$ be a subgroup of $G$ and $N$ a normal subgroup of $G$, then: The ...
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an outline of “intuitive mathematics”?

This question is related to the third answer in this post. There seems to be a difference between the intuitive idea of a thing (such as a function) and "models" of that thing in mathematics (such ...
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Characteristic function of union of two sets formula and intuition

From http://topologicalmusings.wordpress.com/2008/03/20/inclusion-exclusion-principle-counting-all-the-objects-outside-the-oval-regions-2/ Is there an easier proof or way to calculate $1[A \cup B]$?...
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cycloid of a unit-speed circle

In one of the lectures of the MIT OCW Multivariable Calculus course, the professor introduces the parametric equation of a cycloid in the plane, where $a$ is the radius of the circle that creates it, ...
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1answer
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Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in $\mathbb{R}...
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Why is analysis over the complex numbers so useful vs say other fields?

First I'll state a statement that I hope is false, but I do not know if it is: "Complex analysis is used a lot compared to analysis over other fields (as in it gives a lot of results like the prime ...
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Jacobi identity - intuitive explanation

I am really struggling with understanding the Jacobi Identity. I am not struggling with verifying it or calculating commutators.. I just can't see through it. I can't see the motivation behind it (as ...
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Softmax Derivation Help

I've been reading a paper that derives logistic regression from a few assumptions . Here is the link. If you go to page 5 and look at equation 18 the author claims that this essentially says the ...
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How interpret the dual lattice $\Gamma^*$?

In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \...
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In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
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Understanding the tensor-hom adjunction intuitively

I'm currently trying to teach myself some category theory. Recently, I learned that the tensor product is left adjoint to the hom functor in suitable categories, e.g. vector spaces with linear maps, i....
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Motivation behind standard deviation?

Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 And so the average (magnitude of) ...
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Intuition behind the proof of the validity of the Euclidean algorithm

As the question title suggests, could anybody explain to me their intuition behind the proof of the validity of the Euclidean algorithm?
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1answer
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Intuition for basic fact surrounding Gaussian integers.

What is the intuition behind the following fact? Among the odd primes: Those that have remainder $3$ upon division by $4$ remain prime in $\mathbb{Z}[i]$. Those that leave remainder $1$ ...
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1answer
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Finding the product of a prime function…

If we take the primes $p_k < n$, and raise them to the highest power possible such that $(p_k)^{r_k} \le n$, what is the lower bounds on $\prod{ (p_k)^{r_k} }$? In other words, what are the ...
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What are the disadvantages of non-standard analysis?

Most students are first taught standard analysis, and the might learn about NSA later on. However, what has kept NSA from becoming the standard? At least from what I've been told, it is much more ...
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start mathematics from scratch

I'm sorry if this question goes against the site. I have completed my engineering in computer science; we were taught to get degrees only in our university :(. I love mathematics but time passed ...
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Why if $a = qb + r$, then $\operatorname{gcd}(a,b) = \operatorname{gcd}(b, r)$ intuitively?

Origin - Elementary Number Theory, Jones, p $5$, Lemma $1.5$ Are there any illustrations? I tried Wikipedia's article and the first picture to the right, but I think this delineates Euclid's ...
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1answer
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What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
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Method of Variation of Parameters - Assigning zero works?

I have yet to find a decent answer on this, and so I don't think this question is inappropriate. Also, this question is mainly meant for people that are very familiar with this method. In the method ...
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Problems that are largely believed to be true, but are unresolved

Are there unsolved problems in math that are large believed to be true, but for reasons other then statistical justification? It seems that Goldbach should be true, but this is based on heuristic ...
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Partial differentiability of $f(x, y) := {x^3 - y^3 \over x^2 + y^2}$ at $(0, 0)$

I thought this task up myself, so I'd be good to know whether my solution is correct or not. :-) Given $$f(x, y) := {x^3 - y^3 \over x^2 + y^2}$$ for $(x, y) \in \Bbb R \setminus {0},$ ...
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Is advanced college math (eg analysis, abstract/linear algebra, topology) supposed to be as intuitive as elementary math? [closed]

So I don't know if I'm not smart enough for math, but lately, it seems to me as if some advanced topics are just too unintuitive in my opinion. For example, I have no idea what eigenvalues, ...
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Deriving the Airy functions from first principles

I have just started reading about the Airy functions and am stuck on a particular step of their derivation. But first here is some background information to give this question some meaning, more ...
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Intuition of Immersed versus Embedded Submanifolds

The definitions I read in Lee's Smooth Manifolds is: Embedded Submanifold: $S\subset M$ is an embedded submanifold if $S \to M$ is an embedding. Immersed Submanifold: $S\subset M$ is an immersed ...
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Intuition about heat equation with Neuman boundary data

On a bounded domain, consider the heat equation $u_t - \Delta u = 0$ with $\partial_\nu u = c$ (constant) and initial data $u_0$ which is non-negative. As usual $\nu$ is the outward normal vector. ...
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1answer
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Elements of bounded distributive lattice belonging to same prime ideals are equal?

I have read in a paper that by an easy application of Zorn's lemma one may show that two elements of a bounded distributive lattice are equal iff they are contained in exactly the same prime ideals of ...
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Graphical explanation of the difference between $C^1$ and $C^2$ function?

We are all aware of the intuitive (graphical) explanation of the concepts of continuous and differentiable function. Whenever these two concepts are formally defined, the following elementary ...
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Intuitive Explanation of Bessel's Correction

When calculating a sample variance a factor of (N-1) appears instead of N (see http://en.wikipedia.org/wiki/Sample_variance#Population_variance_and_sample_variance ). Does anybody have an intuitive ...
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Intuition behind the derivative of are of a square? How to properly use the derivative ?

If I derive the formula $$S=16t^2$$, where S denotes the distance and t denotes time I get $$ds/dt= 32t$$. This in return give me a formula for the speed of the object at any time t. However if we ...
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How to come up with the gamma function?

It always puzzles me, how the Gamma function's inventor came up with its definition $$\Gamma(x+1)=\int_0^1(-\ln t)^x\;\mathrm dt=\int_0^\infty t^xe^{-t}\;\mathrm dt$$ Is there a nice derivation of ...
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Is my intuition of dense sets correct?

I am working with the usual definition of a dense set, which is Let $U$ be any non-empty open subset of $X$. A set $A$ is dense in $X$ iff $A \cap U \neq \emptyset$. My highly informal and ...
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Is this symbolic statement impossible?

Is this statement logically impossible if x is a single real number (i.e. not a set)? $$(x<5) \land(x>7)$$ it seems to me that x cannot both be greater than 7 and less than 5 if ...