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.

3,653 questions
Filter by
Sorted by
Tagged with
45 views

Intitutive explanation of why probability of getting an even integer is more when random numbers are multiplied.

If n(>1) random integers are selected, then the probability of their product being odd is $1/2^n$, which is less than that of the product being even. But this doesn't sound intuitive to me. If ...
273 views

Why is $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$ meaningful when $\hat{\imath}$, $\hat{\jmath }$, $\hat{k}$ are not 'alike' quantities?

For the standard form: $a\hat{\imath} + b\hat{\jmath } + c\hat{k}$. Since the $\hat{\imath}$, $\hat{\jmath }$, and $\hat{k}$ directions are different, why are we 'allowed' to write them this way? Isn'...
25 views

Intuition for this definition of "almost orthogonal" in Banach space

I'm looking into an intuitive understanding of a following lemma in functional analysis about "almost orthogonal elements": Let $X$ be a Banach space. $\dim X=\infty$. $U\subseteq X$ is a ...
79 views

Is the Axiom of choice intuitive? How was it first introduced?

I will refer to the Axiom of choice as ($AC$). As far as I know, the widely accepted axioms of set theory is the Zermelo-Fraenkel axioms together with the $AC$. But the last axiom seems to be the most ...
50 views

44 views

Intuition for when a problem may be amenable to the "umbral calculus"?

I've always been interested in situations where we can apply "illegal" operations to objects and still solve problems (as seen here, say), and a common justification for these techniques is ...
35 views

Intuition for orientation of a simplex (in 3 dimensions)

In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it ...
58 views

Intuition for the Cauchy-Schwarz inequality that does not rely on a geometric interpretation of vectors

I know that the Cauchy-Schwarz inequality can be understood by interpreting vectors as arrows in the coordinate plane, but since the inequality should hold for any inner product space, I'm looking for ...
117 views

Duhamel's Principle Intuition

I am trying to understand Duhamel's Principle by applying it to some simple problems. I am thinking of $P(t)$ as expressing a bank account balance at time $t$, to try to gain an intuition for Duhamel'...
36 views

The intuition behind the formula for counting distinct permutations with repetitions

I'm going through a book right now where the author is going over some combinatorics equations. I apologize if the following is an easy question. I'm trying to gain an intuitive understanding of the ...
333 views

What's the intuition for Cross Multiplication?

I already know, I'm NOT asking about, the algebra. It's NOT intuitive why 3 pears x 4 tangelos = 6 quinces x 2 riberries $\iff$ 3 pears/6 quinces = 2 riberries/4 tangelos. I stumbled the picture ...
35 views

What is an intuition for subdirect irreducibility?

I am reading "A Course in Universal Algebra" by Burris and H.P. Sankappanavar and they provide this definition: "An algebra A is subdirectly irreducible if for every subdirect embedding ...
28 views

How to show a sequence of measure converges weakly?

The doubt I have is that in all equivalent definitions of weak convergence of finite measures via the Portmanteau theorems, some knowledge of $\mu$ is required to check those conditions, for example ...
152 views

Expected Geometric Growth Rate (Kelly's Criterion)

The Wikipedia article for Kelly Criterion establishes its main formula using the expected geometric growth rate $r = (1 + fb)^p * (1 - fa)^q$, where $f$ is the fraction of an account (that starts with ...
49 views

Seemingly conflicting notions of a function

Throughout my mathematical education, I have seen a few, seemingly, different and conflicting notions of what a function is: A function is a a type of mathematical object that maps every element of a ...
19 views

Subdifferential, two different definitions.

The subdifferential of the function $W$ at $x$ was defined as $$\partial W(x):=\{\kappa\in\mathbb{R}^d:W(y)−W(x)≥\kappa\cdot(y − x),\text{ for all }y ∈ \mathbb{R}^d\}.$$ I understand what this means ...
36 views

Intuition for derivatives of trigonometric functions

I’m trying to understand the geometric/infinitesimal ‘derivations’ of the formulae for derivatives of trigonometric functions here. What I don’t understand is how the similarity of the triangles gets ...
21 views

What are broad tips for creating pushdown automatas (PDAs) for context free languages (CFLs)?

What are tips for creating PDAs for context free languages? I know there is typically no bit by bit answer for producing these PDAs, however there should be some broad tips to direct one to make them. ...
35 views

Continuity equation with measures / Vizualise the divergence of a measure?

I kind of understand the meaning of the continuity equation when $\rho_t(x)$ is a density $$\partial_t\rho+div_x(\rho_t v)=0$$ Which means that the time variation of the density in a point is the ...
28 views

Intuitively, why does growth proportional to the population size not diverge but growth proportional to pairs diverges to infinity in finite time?

It's interesting that growth which is proportional to the current population doesn't diverge to infinity, while growth that is proportional to higher powers of the current population. That is, the ...
47 views

Why not define boundary to be $\partial S= S-S^o$?

The standard definition of boundary is $\partial S= \bar{S}- S$. Intuitvely in geometric shapes it seems to be the same as $S-S^o$, is there a reason we use the closure definition instead of the ...
43 views

Why is the number of heads and the number of tails independent in $n$ coin flips?

In probability, in different manifestations, I have noticed that the sum of random variables $X$ and $Y$ may be given as $n$, and yet $X$ and $Y$ may still be independent. I find this very counter ...
46 views

Operations with the $\mathcal O$ notation

I'm quite new to the big $\mathcal O$ notation, so my question might sound ridiculous. I was told we wouldn't learn almost anything about $\mathcal O$ and $\mathcal o$ notations during our bachelor ...
29 views

25 views

Equivalent definitions of congruences (in context of universal algebra)

I am familiar with this (informal) definition of congruence relation (or simply congruence): (1) A congruence is an equivalence relation on an algebraic structure that is compatible with the structure ...
I quote Life Insurance Mathematics (Gerber, 1997). Let us consider a person aged $x$ years, denoted by $(x)$. We denote his/her future lifetime by $T$ or, more explicitly, by $T(x)$. Thus $x+T$ will ...