Mathematics Stack Exchange
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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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Serge Lang´s remarks on the superiority of algebra. What it actually means?

I read two comments of Lang that basically places algebra over other math subjects. One of this comments is on his calculus book preface (see Remark 1 below); I am not finding his other comment, but it was an interaction he had with someone at…
Espinho da Flor
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Why does the log-log scale on my Slide Rule work?

For a long time I've eschewed bulky and inelegant calculators for the use of my trusty trig/log-log slide rule. For those unfamiliar, here is a simple slide rule simulator using Javascript. To demonstrate, find the $LL_3$ scale, which is on the…
Justin L.
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Intuition on the sum of first (n-1) numbers is equal to the number of ways of picking 2 items out of n.

While going through an equation today i realized that sum of first (n-1) numbers is [n*(n-1)/2] which is equal to combinations of two items out of n i.e [n!/((n-2)! * 2!)]. I need some intuition on how these two things are related?
Shashank Tomar
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What is a smooth curve in $\mathbb{R}^2$ intuitively?

While studying for my exam, I've run into some problems understanding what a smooth curve in $\mathbb{R}^2$ is. I first thought that, intuitively, I could think of a piece of string on a piece of paper, where the string is the curve and my paper is…
Joshua
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What's the intuitive explanation that the volume of a solid is $\frac{1}{3} A_{base} h$?

I can see why the area of a triangle is $A = \frac{1}{2} bh$ because it's half of a rectangle with sides $b$ and $h$, but I fail to see the intuitive explanation for this general volume formula. (Yes, I am aware it doesn't work for everything). If I…
MT_
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Sketch $x^2 e^{-x}$

In my interview to join a university to study physics, I was asked to sketch $y = x^2 e^{-x}$ at the time I could not do so. The interviewer told me that I need to have mathematical intuition like this to study physics. Now, I am studying physics at…
Wanting to be anAndroidDevelor
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Examples of advanced results and ideas explained in a down-to-earth way

Are there any advanced topics--preferably at the research frontier--which can nevertheless be explained accurately, if not efficiently, using very down-to-earth ideas which would be accessible to most college students, not necessarily even math…
Zach Conn
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Gauge fractions with exponents - No Calculator

How does one (without the use of Calculator) determine that $5/6$ is less than $(35/36)^6$? How is this done mentally?
QRIUS2KNW
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How is it possible that we can solve some equations numerically, but can't solve them analytically?

For example, there is the Abel–Ruffini theorem that states there is no analytical solution to polynomial equations of degree five or higher, mentioning $x^5-x-1=0$ as the simplest example. Now if we take $$x^5-x-a=0$$ we can calculate $x$…
MaxD
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If A $\propto$ B and A $\propto$ C while keeping each other constant, then why is A $\propto$ BC?

Possible Duplicate: Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables I guess some people may find this obvious, but I really don't. My question is: If $A\propto B$ while $C$ is constant and $A\propto C$ while B…
Alraxite
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Is it a general truth that when two equations can be described with the same sentence, only varying intonation they are equivalent?

My question is the effect of something that I have noticed during some of the courses I am taking as a first year maths bachelor, that in tutorials we will be asked to proof that two statements are equal but that when I try to describe the left and…
Poseidaan
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Why is multiplication a commutative operation?

This trivial question is all about reasoning (intuition) and obviously not proving. I know $a\cdot b = b\cdot a$ from very early school years and it's considered intuitive. A simple proof is by taking a rectangle that is $2 \cdot 7$ and calculate…
doc_id
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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 theory, specifically the DuPont formula, we see that we…
someonesomewhere
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What are some "deep" questions to explore in elementary school math?

My first grader is very advanced in math. Rather than doing more and more math and making school math even more boring for him, I recently decided to start going "deeper" rather than "faster." Some of the questions we've explored recently are: Why…
Akdinv
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Intuition Behind Geometric Means

Suppose $w \in L^{1}(G)$ and $w \geq 0$. The geometric mean of $w$ is defined by $$ \Delta(w) = \exp \int_{G} \log w(x) \ dx$$ where $\Delta(w) = 0$ if the integral is $-\infty$. What is the intuition behind this definition? Source: Fourier…
NebulousReveal
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