Questions tagged [intuition]

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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Intuition on independence-like condition

Consider a set of $n$ bounded, non-negative random variables $\{W_i\}^n$ and a set of $n$ arbitrary iid random variables $\{X_i\}^n$. Assume that for all nonnegative, measurable $f$ with $\mathbb{E}f(...
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Probability intuition

Which has the higher probability? A) A fair coin is tossed 26 times. Write down an expression for the probability of seeing exactly 13 heads and 13 tails. B) A pack of 52 cards (containing 26 red and ...
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Motivation for the $3\times 3$ matrix inversion formula

When $A$ is a square non-singular matrix, $A^{-1} = \frac{1}{\det(A)} \mathrm{Adj}(A)$. What is the motivation for this formula and how can I get there? I have been told about how the 2x2 formula is ...
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Non-associativeness of composition in deductive systems?

WARNING: The first three and last two paragraphs of this question concern historical/philosophical matters related to a secondary aim of the question. If you are more interested in the properly ...
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An intuitive explanation for the emergence of the $\pm$ symbol in the expansion of sin, cos or tan of $\frac{\alpha}{2}$ in terms of $\cos\alpha$

For some angle $\theta$, $$\cos(2\theta) = 2\cos^2\theta - 1 \implies \cos(x) = \cos\Big(2\cdot\dfrac{x}{2}\Big) = 2\cos^2\Big(\dfrac{x}{2}\Big)-1$$ $$\implies \cos^2\Big(\dfrac{x}{2}\Big) = \dfrac{1+\...
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Intuition on the special linear group.

For any (unital and commutative) ring $R$ we can define the special linear group as the kernel of the determinant, that is $$ 0 \to \operatorname{SL_n}(R) \to \operatorname{GL_n}(R) \xrightarrow{\det} ...
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Axiom of replacement easy explanation

Can someone explain me the axiom of replacement in easy words? I really dont get it, my book says What i understand from this is that for every set $x$ there is another set $y$ whose members are the ...
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Continuity of a function $f: X\to \mathbb Z$

If you have a set $X$ which is equipped with a topology: what is required for the function $f: X\to \mathbb Z$ to be continuous? The fact that $X$ has a topology should certainly be helpful. But $\...
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Tensor product over non-commutative ring.

I already knew the definition of the tensor product of two modules over a commutative ring : It takes two modules over the ring and spits out a module over a ring. I thought this was nice. I was ...
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Is this a good justification for why integration by substitution works? [duplicate]

It is a well known fact that $$ \int f'(g(x))g'(x)dx=f(g(x))+C $$ This follows directly from the chain rule. However, sometimes it is easier to perform the substitution $u=g(x)$: \begin{align} u&=...
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Does everything in math has to be intuitively explain-able? [closed]

I always try to find an intuitive explanation for everything I study in mathematics. Sometimes, I come across an idea where I wonder if an intuitive explanation is needed for it. Maybe there are some ...
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An (intuitive) explanation for the emergence of $\pm$ in the expansion of $\cos\Big(\frac{x}{2}\Big)$ in terms of $\cos x$?

For some angle $\alpha$, we have : $$\cos(2\alpha) = 2\cos^2\alpha - 1$$ $$\cos\Big(\dfrac{\alpha}{2}\Big) = \pm\sqrt{\dfrac{\cos\alpha + 1}{2}}$$ I want to gain a deeper and more intuitive ...
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Intuition issue with independent events.

Take for example 100 students. Each student studies either French, German, French and German, or none. Let's say 40 students study French and 25 study German. If 10 study both then they are ...
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Conceptually, what is the meaning / the idea of conjugacy? [duplicate]

My background is in geometry, and I have a basic level of understanding of topological conjugacy: It defines an equivalence relation in the space of all continuous surjections of a topological space ...
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Cardinality of set of $a_r$?

Question So I conjectured a formula which was proven: Let $b_r = \sum_{d \mid r} a_d\mu(\frac{m}{d})$. We prove that if the $b_r$'s are small enough, the result is true. Claim: If $\lim_{n \to \infty}...
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Find the points closest to two lines using least squares method

Given are two lines $g(t)=a+bt$ and $h(s)=c+ds$ with $a,b,c,d \in \mathbb R^3$. I need to find the points where the two lines are closest using the least squares method. However I am unable to find a ...
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How does the sum (collection) of zeros become non-zero?

We know that for a continuous random variable, the probability at a particular real number is 0. But the probability over a range may be a non-zero value that lies between 0 and 1. I know that ...
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Intuition on spectral theorem

In the last month I studied the spectral theorems and I formally understood them. But I would like some intuition about them. If you didn’t know spectral theorems, how would you come up with the idea ...
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Why is $\lambda$ called the *instantaneous* rate of change in the exponential distribution?

In the following paramterisation of the exponential distribution $${\displaystyle f(t;\lambda )={\begin{cases}\lambda e^{-\lambda t}&t\geq 0,\\0&t<0.\end{cases}}},$$ $\lambda$ is called the ...
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Intuition for $\tan\theta=\frac{\sin\theta}{\cos\theta}$

I understand how to prove that using the definitions of the elementary trig functions: $$\frac{\sin\theta}{\cos\theta} = \frac{\dfrac{opp}{hyp}}{\dfrac{adj}{hyp}} = \frac{opp}{adj} = \tan\theta$$ ...
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Undetermined or indeterminate forms: $\frac{0}{0}, \frac{\infty}{\infty}, 0\cdot\infty, 1^\infty, 0^0, +\infty-\infty$

I wanted to know who has decided that for the calculation of the limits of the following forms, $$\color{orange}{\frac{0}{0},\quad \frac{\infty}{\infty},\quad 0\cdot\infty,\quad 1^\infty,\quad 0^0,\...
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Bayesian Statistics : Motivation and Explanation of Marginal Likelihood

$P(\theta|x)$ is the posterior probability. It describes $\textbf{how certain or confident we are that hypothesis $\theta$ is true, given that}$ we have observed data $x$. Calculating posterior ...
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Generating a continuous OD graph from a dicrete OD graph

This is an extract from Donald Bamber's paper regarding OD graphs and ROC plots. Bamber D. The area above the ordinal dominance graph and the area below the receiver operating characteristicgraph. ...
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Importance of the mathematical differentiating of an ellipse (or of a conic)?

Supposing to have a canonical ellipse (or other conics, parabola, circle, or hyperbola): $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad \text{or} \quad \frac{(x-x_0)^2}{a^{2}}+\frac{(y-y_0)^2}{b^{...
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Is there a geometric intuition for integration by parts? [duplicate]

Is there a geometric intuition for integration by parts? $$\int f(x)g'(x)\,dx = f(x)g(x) - \int g(x)f'(x)\,dx$$ This can, of course, be shown algebraically by product rule, but still where is ...
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Euristic and intuitive idea behind the theory of viscosity solutions

As the title suggests, I am kinda struggling to understand the basic idea behind viscosity solutions theory. The theorems are a lot different from what I saw in classical theory for PDEs (with Sobolev,...
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Why “normal subgroups occur as kernels of homomorphisms” is a big deal?

I already knew that normal subgroups where important because they allow for quotient space to have a group structure. But I was told that normal subgroups are also important in particular because they ...
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Intuition behind the horizontal factor in a sinusoidal function

I came across the following question: And here is an excerpt of its solution: I understand how the horizontal factor is calculated: $period=\frac{2\pi}{horizontal factor}$ $365=\frac{2\pi}{...
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Intuition behind why rotating a point $180^\circ$ about the origin give the same result as reflecting it with respect to the origin

I understand that the outcomes of those two actions could be proven mathematically, and we can see that those results match. I also understand that I can visualize those actions and see how the ...
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Intuition behind equation for volume of a cone without calculus [duplicate]

I have come back to study geometry a bit and I'm kind of stuck at deriving the volume formula for a cone. I have read the calculus-based derivation and it totally makes sense, but calculus has been ...
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Given $n$ slots and $k$ objects to fill the slots, what is the probability of a given slot to be filled.

Problem Given: $n$ slots, numbered from $1\ldots n$ $k$ objects a slot can be filled by one object what is the probability that a slot at some position $i$ to be filled? Some Notation For a visual ...
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Intuitive meaning of null-homotopic map between sphere an torus.

I am given an exercise where I have to show that every continuous map from $S^2$ to $T^2$ is null-homotopic. I don't really understand what is the implication of this result intuitively.
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Why is the closed graph theorem called like that and why is my definition using two functions?

The closed graph theorem as given in my notes, reads: Let $f_1, f_2:(X, \tau) \mapsto (Y, \tau')$ be continuous functions between topological spaces. If $\tau' $ is Hausdorff,then $\Gamma =\{p \in X | ...
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Intuition behind the concept of a topology

So I have tried to understand the basics of topology, but I have some trouble getting a good intuition for it. I know that the idea is supposed to be that we have various open sets telling us ...
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Intuition for weighted average. Why $\frac{w_1}{w_1 + w_2}x_1 + \frac{w_2}{w_1 + w_2}x_2 = \frac{\sum_i w_ix_i}{\sum_i w_i}$?

I know $\dfrac{w_1}{w_1 + w_2}x_1 + \dfrac{w_2}{w_1 + w_2}x_2 = \dfrac{\sum_i w_ix_i}{\sum_i w_i}$, because $\sum_i w_i$ is common denominator. I'm not asking about this algebra. It's intuitive that $\...
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Explain why the characteristic curves generate the solution surface

I recently came across this problem in one of my differential equation texts: Explain why the characteristic curves generate the solution surface of $P(x,y) \frac{\partial z}{\partial x} + Q(x,y) \...
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Visualization for Euclidean Algorithm [duplicate]

I want to really understand the Euclidean Algorithm. A key component in the algorithm is fact that common divisors of two integers are common divisors of their difference. I can see from the ...
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An alternative approach to proof by induction

In my experience, the 'standard' method for proof by induction can often cause confusion. In this post, I propose a slightly different way of conceptualising proof by induction that does not involve ...
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Paradox of Fake Choice

This is a logic sort of question, so if it is better suited for a different stack exchange please let me know, not sure if this is the optimal location I thought of this interesting contradiction / ...
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Is there an intuitive way of justifying why the square of an infinite cardinal is itself?

By no means I am an expert in this subject, but I do have some knowledge of ZFC. While there are many proofs which are difficult to recollect, I feel like I have enough knowledge that if I am given a ...
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Intuition for the linear equation: $Ax+By+C=0$

Is it possible to gain an intuition for the linear equation in the following situations: When being rearranged into the slope-intercept form. The equation becomes: $y=-\frac{A}{B}x-\frac{C}{B}$. I ...
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Intuitive meaning of Diffeomorphism

Let $U\subset\mathbb{R}^n$, $V\subset\mathbb{R}^m$ and a bijection $f:U\to V$ is a diffeomorphism if $f$ and $f^{-1}$ are differentiable. I would like to know the intuitive meaning of two open sets ...
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inner product over two functions

in class we learned about the inner product of two functions as $\int_{a}^{b} f(x)g(x) dx$ but I failed to get an intuition about what it means, can someone please explain it to me? also what the norm ...
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What constitutes a rigorous proof, and can intuitive explanations be made rigorous?

It seems that there is a marked discrepancy between how one might explain a concept in mathematics, and how one might prove it. Obviously, there are some exceptions, and some particularly elegant ...
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The intuition of the round value of $e^{i \pi}$

I can see the value of $e^{i\pi}$ is $-1$, this value is round without decimals. The $e$ value can be defined as the value (v) that the derivative function of $v^{x}$ is still $v^{x}$. And the $\pi$ ...
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A Question on the Rate of Change of the Arc-Length

Main Question Consider some curve $y(x)$, going from a point $(x_0,y_0)$ to a point $(x_1,y_1)$. Let $L$ be the length of the curve, and the function $F$ be the rate of change of the length of this ...
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What does it mean to integrate a partial derivative in the “other” direction?

Intro: A continuous, differentiable function $F(x,y)$ can be pictured as a surface over the $(x,y)$ plane. At each point in space, we could plot another function $\frac{\partial F}{\partial y}(x,y)$, ...
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Conceptual reason why height of unit tetrahedron is the same as the distance between opposite faces of an octahedron?

One of my favorite mathematics visualizations shows why attaching a tetrahedron to a triangular face of a square pyramid results in a polyhedron with five faces instead of the seven faces one might ...
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How to think of cohomology classes

I guess some of my problems come from the fact that I can hardly visualise cohomology. For homology in dimensions 1–3, I claim to have at least some intuition what homology classes look like, and when ...
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About index (set theory)

Definition: A collection of sets $\mathbb{E} $ is said to be indexed by a set A if and only if there is a funcitom F from A onto $\mathbb{E} $. In this case we call A the index set lf $\mathbb{E} $, ...

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