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.

To have intuition about a mathematical truth is to have some insight into why it is true, and to understand the motivation for talking about that truth in the first place. This is usually stated in contrast with merely having a superficial knowledge of a mathematical truth as a fact, or only having skills at applying a mathematical truth to solve a problem without having the conceptual understanding of solution.

For a nice explanation of mathematical intuition with examples, and links to other articles on developing mathematical understanding, see Developing Your Intuition For Math on BetterExplained.com.

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Limit Points of a sequence on the Circle Group

I'm reading ch 7 of Nadkarni's book Spectral Theory of Dynamical Systems and I've come across this statement in a proof I'm currently trying to understand: If $z\in S^1$ (where $S^1$ denotes the Circle Group) and $z^{n_k}$, with $k\in\mathbb{N}$,…
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Maths branch of logics or vice versa?

Is it logics a branch of maths or vice versa? From a the point of view of the definition of a logical system, logics is a 'calculus' which has axioms and rules as any branch of maths. However it seems that we are entitled to use logics rules in any…
Ambesh
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Understanding the Fundamental theorem of Calculus in plain english

I am learning Calculus. I am trying to understand the fundamental theorem of calculus. I am following this wikipedia article: https://en.wikipedia.org/wiki/Integral. I am having a hard time understanding what they refer to as the the Fundamental…
Gilboot
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Intuitive understanding of $\sqrt{a b} \leq \frac{a+b}{2}, \: a,b \ge 0$

How to understand this inequality intuitively? $$\sqrt{a b} \leq \frac{a+b}{2}, \: a,b \ge 0$$ The right part is a number between $a$ and $b$. This is the only thing I have realised about this inequality. Hope you guys some insight to share. Kind…
Xenusi
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Number of onto functions, why does my solution not work?

I have a set $A$ with $4$ elements and a set $B$ with $3$ elements. We need to find all onto functions from $A$ to $B$. My line of thought: Map each element in $A$ to $B$, where only one element in $A$ must be mapped to the same element in…
Max
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Abstracting the general forcing argument from case-specific arguments

I've found myself wondering whether the literature has an abstraction of what the forcing argument shows in general, in order to separate it from the case-specific arguments that result in specific independence results. What I'm thinking would be…
Daniel Schepler
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Why does the curl vector point along the axis of rotation?

Thanks for reading. Say we have a vector-field $F=P(x,y,z)\hat{i}+Q(x,y,z)\hat{j}+R(x,y,z)\hat{z}$, and we compute the curl of $F$ at some point $(x,y,z)$by calculating $\nabla \times F$ at that point. Geometrically, I intuitively understand the…
joshuaronis
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What does $\int \log ( \operatorname{sech}(\log x))dx$ have to do with the area enclosed by $r = \sin(2 \theta)$ and $ r=\cos(2 \theta)$?

Let $f(x) := -\log(\operatorname{sech}(\log(x)))$. Also, let $A$ be the area enclosed by the curves $r=\sin(2\theta)$ and $r=\cos(2\theta)$, overlap not included twice. I have shown that $$\int_0^1f(x)~dx= \frac \pi 2-1 =A.$$ I am wondering if this…
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Intuition behind $\sin(\theta)$ when introducing this to high school students

When first introducing trigonometry to students, the traditional setup is to start with a right-angled triangle with reference angle $\theta$ and we label the sides with "Hypotenuse, Opposite and Adjacent." To keep students engaged with some…
user523384
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Intuitive idea behind orbit stabilizer theorem.

Recently I studied the orbit stabilizer theorem which is as follows: Suppose $G$ is a group acting on $X$ (i.e.$X$ is a $G$-set). Let $x\in X$,then define, $\operatorname {orb}(x):=\{g.x:g\in G\}$ and $\operatorname{stab}(x)=\{g\in G:g.x=x\}$, then…
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Let $H\le G$ be groups such that $x^2 \in H$ for all $x\in G$. Show that $H\unlhd G$.

There is a very standard problem in normal subgroups which is as follows: Let $G$ be a group and $H$ be a subgroup such that $x^2 \in H$ for all $x\in G$. Show that $H$ is a normal subgroup of $G$. In any book you will find it solved in examples…
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Partition of unity & intuition behind it.

Is there some intuition behind this identity? Looks like it should be, but I can't figure it out. $$ (1-\alpha)^{n} + \sum_{i=1}^{n}\alpha(1-\alpha)^{n-i} = 1 $$ $$ \alpha \in [0; 1]$$ Would be grateful for your help!
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Lack of intuition, retention while self studying

I am a first year undergraduate student, currently in second semester. So basically I learnt most of the first year stuff in high school, so I have a lot of free time in this year (currently in second semester), and thus I'm self studying math in…
katana_0
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Is there a geometric interpretation to the Brahmagupta–Fibonacci identity?

The Brahmagupta–Fibonacci identity says that, for integers $a,b,p,q$: $(a^2+b^2)(p^2+q^2)=(ap+bq)^2+(aq-bp)^2$ Is there an intuitive geometric way to interpret this? Thanks!
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How to build intuition for diagonal intersections?

So, I'm currently taking a course in set theory, and we've been using diagonal intersections and diagonal unions. I know the definitions (namely, if $(X_\xi:\xi<\kappa)$ is a sequence of subsets of $\kappa$ for some regular uncountable $\kappa$,…
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