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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Collection of Intuitive / Visual Derivations of Mathematical Concepts and Formulas

I find it difficult to simply memorize mathematical formulas in engineering without understanding what it means and what the result is like, but I realized that many mathematical relationships can be ...
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Definition of a Bounded Operator and Some Intuition on the Definition of the Norm

I am confused about the definition of a bounded operator (which is probably a consequence of my unsatisfactory understanding of bounedeness and local boundedness). The definition is ...
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Intuition - Identities with 2-Cycles and 3-Cycles - Mulholland p. 69, 86 - Fraleigh p. 90

Jamie Mulholland p. 69 Theorem 6.1 or Fraleigh p. 90 Corollary 9.12 Any permutation of a finite set of at least two elements is a product of 2-cycles. $1. (a_1, a_2, ···,a_n)= (a_1, a_n)(a_1, ...
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How to Visualize Diagonally Opposite Vertices

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ...
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What is the relationship between the final concentrations of alcohol in the alcohol jug and water in the water jug?

If the mixture is being poured back into the jug with water, then how can the jug with water also contain ${\frac {V}{V+Q}}$. I would assume that when the diluted alcohol is poured into the jug ...
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Isomorphism of Group with the Image of the Group - Fraleigh p. 82 Lemma 8.15

I found multifarious duplicates that I listed at http://math.stackexchange.com/a/631364/53934. I edged the purple part because my answer proves it more efficiently. I remember that any function ...
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Visual Group Theory's Intuitive Proof - Cayley's Theorem - Nathan Carter pp. 85, Theorem 5.1

Theorem 5.1. Cayley's Theorem: Every group is isomorphic to a collection of permutations. Figure 5.31. A multiplication table for the group $V_4$, with nodes numbered 1 through 4 to facilitate ...
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Connections and Differences between these Cayley Diagrams for $A_4$ and $S_4$ - Carter pp. 80, 82

Reference: Nathan Carter pp. 80, 82, ch. 5, Visual Group Theory Figure 5.24. As you will read in the next section, it is no coincidence that [the Cayley digram for $S_4$] looks cube-like. A Cayley ...
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What does “area” really mean?

My professor had an interesting statement at the beginning of first year integral calculus. What does area really mean? How do we know that the area of a circle is $\pi r^2$? Archimedes used ...
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If two sets have the same cardinality, then so do their power sets. Converse can't be answered?

For infinite sets $A, B$, $|A| = |B| \Longrightarrow \require{cancel} \cancel{\Longleftarrow} |P(A)| = |P(B)|$. I recast http://ph.answers.yahoo.com/question/index?qid=20100907061641AAE2Vfq : ...
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Fundamental group of projective plane is $C_{2}$???

I just recently know that there are topology with finite nontrivial fundamental group (homotopy curve). I just can't wrap my mind around it at all. If you have a curve, and somehow cannot shrunk it ...
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Intuition behind the residue at infinity [duplicate]

The residue at infinity is given by: $$\underset{z_0=\infty}{\operatorname{Res}}f(z)=\frac{1}{2\pi i}\int_{C_0} f(z)dz$$ Where $f$ is an analytic function except at finite number of singular points ...
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Questions on Answer to “The cardinality of the set of all finite subsets of an infinite set”

Would someone please enlarge on Arturo Magidin's original answer ? $1.$ Say the question didn't divulge $|S| = |X|$. Then how can $|S|$ be determined? Any intuition? I recast it below with more ...
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“Asymmetric” results in maths analogous to “Parity violation” of the weak force?

Disclaimer: I'm not a physicist and I don't claim to be one so if I have any mistakes I’ll be glad to be corrected. One feature of the standard model of particle physics is that the weak force is not ...
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Intuition — An integer $n > 1$ is composite $\iff \color{purple}{p \le \sqrt{n}}$ divides it.

Origin — Elementary Number Theory — Jones — p32 — Lemma 2.14 Backward direction — I need to prove there exists a divisor $d$ of $n$ satisfying $1<d<n$. Because $p$ is prime, $1 < p$. ...
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Understanding Integration techniques?

Could someone give me a geometric interpretation of: a) Integration by Parts b) Integration by Substitution Thanks!
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Proof - There're infinitely many primes of the form 3k + 2 — origin of $3q_1..q_n + 2$

Origin — Elementary Number Theory — Jones — p28 — Exercise 2.6 To instigate a contradiction, postulate $q_1,q_2,\dots,q_n$ as all the primes $\neq 2 (=$ the only even prime) of the form $3k+2$. ...
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Intuition behind power rule?

I've been using it for a while but still don't really understand why it works. For integer exponents greater or equal to 2, its easy to intuitively understand it using the geometric interpretation of ...
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Why, conceptually, is it that $\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}$? [duplicate]

Why, conceptually, is it that $$\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}?$$ I know how to prove that this is true, but I don't understand conceptually why it makes sense.
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Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
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Intuition — c|a and c|b if and only if $c| \gcd(a,b)$.

(1) ― What's the intuition? Is Bezout Identity supposed to be intuitive? (2) — Why doesn't this try work? Let $c$ be any common divisor of $a, b$. Therefore $c|a, c|b \iff cj_1 = a, cj_2 = b$ for ...
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Connection between even/odd and symmetric/skew symmetric

I read awhile back that the set of continuous real valued functions from $\mathbb{R} \to \mathbb{R} $ has a direct sum decomposition into subspaces of strictly even and odd functions. Any such ...
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Suppose a unique a generates a cyclic subgroup of order. Show ax = xa. - Fraleigh p. 67 6.50

(1.) I don't understand above. How do you magically envisage and envision to let $b = xax^{-1}$? What I did was to start from the answer and see if I can get a chain of equivalences. $ax = xa ...
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Understanding cross ratio and harmonic conjugates

I'm studying projective geometry and I'm really having trouble with ''grokking'' what's it all about. Is there an easy/intuitive/visual way to understand cross ratio? I understand that it's ...
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Intuition and Tricks - Crafty Short Proof - Generators, Order of a Cyclic Group - Fraleigh p. 64 Theorem 6.14

This stronger result and easier proof is based on p. 58. Hence it isn't a duplicate of this. Theorem 206 and 207. Let $G$ be a group, $k \in \mathbb{N}$ and $a \in G$ such that $|a| = n$. Then: 206. ...
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Orthornomal matrices [duplicate]

Is there a more direct reason for the following: If the columns of $n\times n$ square matrix are orthonormal, then its rows are also orthonormal. The standard proof involves showing that left ...
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Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
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Solvable and nilpotent groups, normal series and intuition

I'm reading Hungerford's algebra and I'm on Nilpotent and solvable groups chapter. Hungerford starts with: Consider the following conditions on a finite group G: i) G is the direct product ...
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“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
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Intuition - An inverse of a generator is a generator and powers of generators - Fraleigh p. 58 5.46

(1). How do you envisage or envision a Cyclic Group with only one generator can have at most 2 elements? Solution is based on this. The integers have two generators, 1 and -1. Which should give you ...
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How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
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Intuition, proof, one-sided group definition - Any set with Associativity, Left Identity, Left Inverse is a Group - Fraleigh p.49 4.38

There's a similar post on this question but the third paragraph there is almost impossible to figure out. I explain in my profile why I have to use the big word "prognosticate." Proof that left ...
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What's behind the function $g(x)=\operatorname{inf}\{f(p)+d(x,p):p\in X\}$?

In several books on measure theory, I have seen the following problem: Suppose $(X,d)$ is a metric space, on which $f$ is a nonnegative lower semicontinuous function. Show that $f$ is the ...
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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 ...
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Particles in Free Fields

For the state $\left|\vec{p}\right> = a_{\vec{p}}^{\dagger}\left|0\right>$ we have the energy $H\left|\vec{p}\right>=E_{\vec{p}}\left|\vec{p}\right>$ ...
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Intuition & Proof of rank(AB) $\le$ min{rank(A), rank(B)} (without inverses or maps) [Poole P217 3.6.59, 60]

I'm aware of analogous threads; I hope that mine is specific enough not to be esteemed one. $\mathbf{a^i}$ is a row vector. $A, B$ are matrices. Prove: $1$. $\mathbf{a^i}B$ is a linear ...
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What is the intuition behind the generalized confidence interval?

What is the intuition behind the generalized confidence interval? My best description on GCI that it is the way to derive a formula to calcuate the area of the center region in a asymetry distribution ...
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What is the big picture behind AKS algorithm?

Despite a number of question on AKS algorithm here, there does not seems to anything related to the idea behind it (for those who don't know, AKS primality testing is found in PRIMES is in P). I read ...
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Intuition about the class equation (and flowers).

I apologize in advance for the size of the images I've devoted a lot of time and effort to draw them on the computer and i didn’t manage to re-size. I'd be really thankful if anyone would edit this ...
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Do I have this right Trip integrals

Just finished my proof of the volume of a cone using trip integrals. I think I noticed something. Wonder if I got it right. The first integral defines the line/curve, the second defines the area ...
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Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
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Intuition or Picture for Results on Preimages, Images of Functions, their unions, intersections [Chartrand 3E P235 9.12]

What's the intuition behind the following results? In view of their number, a more helpful question might be how to intuit/naturalise them. Here's a more thorough collectanea. I'm also interested ...
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Clarification on the definition of logical conjunction

First of all, I have never studied Logic seriously before. I am reading this article on Wikipedia. The definition is the following: Logical conjunction is an operation on two logical values, ...
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Intuition for Cayley Table and Cayley Table for identity, inverse but not associativity - Fraleigh p. 47 4.24

$1-2.$ I understand these proofs on pp. 5-6 for Cayley tables but what are the intuitions for Sudoku property : Every element of the group appears only once in each row and each column. Symmetric ...
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Intuition or Motivation behind definition of Homomorphism - Fraleigh p. 29

p.29: A binary algebraic structure is a set $S$ together with a binary operation $*$ on $S$ and is denoted $<S, *>$ p.29: Let $<S,*>$ and $<S',*'>$ be binary algebraic ...
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How to Intuit if these are Linear Transformations or not ? [Strang P380, 7.1.3(c), (d)]

On P376, Strang writes : "You'll get good at recognising which transformations are linear". In his video lectures, he does this; before algebra, he previses whether something's a linear transformation ...
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What are the conditions for a polygon to be tessellated?

Upon one of my mathematical journey's (clicking through wikipedia), I encountered one of the most beautiful geometrical concept that I have ever encountered in my 16 and a half years on this oblate ...
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Intuition/Picture - Theorems on Linear Independence, Span, Basis, Dimension [Poole, Section 6.2]

I'd like to ask about the intuitions for these theorems, absent in David Poole's Linear Algebra (to which the page numbers refer). Also, are there pictures for these theorems?
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How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
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Understanding analytic construction of induced representation

I'd like to get some intuition for analytic construction of induced representations as described on Wikipedia. Algebraic construction also described there is much more intuitive and clear to me, but ...