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0

While enough counterexamples have been covered and the subjectivity of obvious has been noted, there is another subjective aspect of "it's obvious" that has to do with the student's own mathematical education. I would like to add this one little thing to the answers here: The advances in one's education proceed on many fronts, but a step forward occurs ...


0

Most differences are just derived directly from the language. Sine in Portuguese is "seno" so "sen" makes logical sense. In Russian, tangent is тангенс (pronounced as "tangens"). Just use google translate you you should be able to figure out. Some countries use the comma instead of dot to denote decimals: i.e 0,5 instead of 0.5


0

Because some obvious things are false.


4

Just an addition to the many good answers here: Every axiom is obvious, and every rule of inference is obvious. (Otherwise, they wouldn't be very good axioms.) Thus, every single proof in existence is a long string of trivialities. Thus if you agree that two trivialities make a triviality, then everything in math that is provable is also trivial, and a proof ...


3

Because your first example is not even true without suitable conditions. The limit of 1/n as the integer n increases forever depends on your number system. In the standard reals the limit is indeed zero. However in the reals augmented with infinitesimals (as nonstandard reals) the limit does not exist because every positive infinitesimal is strictly less ...


15

I always say that the most difficult exercise in my undergrad studies was the first question in linear algebra. We were taught about the axioms of a field. Then we had to prove the following thing: For every $x$, $x+0=x$. The catch is that the axioms we were given stated $0+x=x$. So we had to use the axiom of commutativity first, then we could conclude ...


23

I think the answer has three parts Part 1: If you ask a random man at Walmart about what $$\lim_{n\to \infty} \frac{1}{n}$$ is, then you might not get much. If you tell them that it is $0$, then they probably don't think that it is obvious. If you go to a high level research talk, you will hear "it is obvious / it is clear that" a lot. And you might not ...


0

Well those examples mostly use as the definition of the term,for example if you're gonna make a definition for continuity you're gonna start with function $f(x)=x$.Also if you're gonna explain somebody how to prove continuity of a function you'll go with $f(x)=x$.And then again also some things we thought obvious are wrong,like for example that we aren't ...


8

There was just an answer I provided a couple of minutes ago that was wrong. The question was: $$\lim_{n\to\infty}(1+2^{-n})^{2^{n+2}}$$ What I thought: $$\left(1+\frac1{2^{\infty}}\right)^{2^{\infty}}=\left(1+0\right)^\infty=1$$ But the answer was actually $e^4$. Even the computer made a mistake (when $n$ got too high). Sometimes, something that seems ...


8

Because obviously a continuous function must be piecewise monotone. And therefore differentiable at all but at most countably many points. Ampère "proved" this result in 1806 and it was considered a theorem for quite a while. Then Riemann came up with an example of a function than when integrated produces a function (i.e. $x \mapsto \int_a^x f(x)\,dx$) ...


1

The obvious is hard to prove and often wrong.


23

The main procedural reason is to show that your axioms correctly capture what you want them to capture: that is to say they are both "correct" and sufficient. If it turned out that under our axioms $\lim\limits_{n\to\infty}\dfrac{1}{n}\neq0$ then we would probably choose a different definition of $\lim$ (or a different name for it), since it would not be ...


0

In my opinion, only axioms should be treated as obvious, above all while a theory is being explained to others. I think it is simply immoral for a mathematician who is writing a proof of a proposition in a book not to give every smallest detail in the chain of logical inferences, skipping the task of making the subject perfectly clear through a lazy abuse of ...


61

Because sometimes, things that should be "obvious" turn out to be completely false. Here are some examples: Switching doors in the Monty Hall problem "obviously" should not affect the outcome. Since Gabriel's horn has finite volume, then it "obviously" has finite surface area. "Obviously" we cannot decompose a 3-dimensional sphere into a finite number of ...


15

Things that are obvious to one person are not necessarily obvious to another. Futhermore they dispel (most) skeptics. Just thinking they are true does not mean they are true. For example, before I entered university, I was under the impression that there were twice the number of elements in $\mathbb{Z}$ than in $\mathbb{N}$. I would've called this obvious, ...


2

Of course, we have the classic trick $$9\times n=(10\times n)-n$$ which works because of distributivity. This can be generalized as follows: $$99n=100n-n$$ so for example $$99\cdot 54=5400-54=5346$$ It also helps simplify generic calculations. E.g., $$17\cdot 8=17\cdot 10-17\cdot 2=170-34=136$$


1

Here is a trick to multiply two numbers between 10 and 19 together, say $10 + x$ and $10 + y$: Compute $10 + x + y$, put a zero at the end (multiply by 10), and add $x\cdot y$. Thus $(10+x)(10+y) = 10\cdot(10 + x + y) + xy$. Easy with algebra. I learned this from my mother who had only an 8th grade education and no algebra.


2

Makes multiplication of multi-digit numbers easier. The above is the following problem: 14759 x 365421 This is how they teach multiplication in Japan. You may be thinking, you draw this in your mind? No, there's a shortcut for this method. Take for instance: 21 x 32 You can draw it to get the answer. But the drawing is basically giving you a ...


0

Allow me to join the party guys... This is another proof of the Pythagorean theorem by The 20th US President James A. Garfield. A nice explanation about Garfield's proof of the Pythagorean theorem can be found on Khan Academy.


0

One thing is when the answer to the first parts leads into the next parts as you said, and I don't have experienced that, in particular. One different thing is an exercise like the integral in your example, where several techniques must be applied in succession in order to solve. Thinking to this second model, comes to my mind the differential equation ...


2

I was asked this exact question by my wife last night. She was looking for an everyday example of the use of complex numbers to explain to her 8th grade math class (whose knowledge of complex numbers consists of i = SQRT(-1) ). My response was this: Imagine an electronic piano. Each key produces a different tone. A volume control changes the amplitude ...


0

If the probability of rain on Saturday is 50% and the probability of rain on Sunday is 50%, which is the probability of rain on the weekend? It can be very fun during dinner with non-mathematicians friends...


3

A recent example I found which is credited to Martin Gardner and is similar to some of the others posted here but perhaps with a slightly different reason for being wrong, as the diagonal cut really is straight. I found the image at a blog belonging to Greg Ross. Spoilers


0

Since I came across this thread several months ago before I created my geometry app, I thought it'd be fitting to add mine to the list now. Isosceles is an app for iPhone/iPod touch/iPad that lets you draw a huge variety of geometric diagrams using an interactive geometry system. I've used it to answer questions on this site as well as do homework and tutor ...


2

The inequality that I know under the name isodiametric inequality is $$ \frac{\text{vol}(K)}{\text{diam}(K)^d} \le \frac{\text{vol}(B)}{\text{diam}(B)^d} $$ for any convex body $K$ in $\mathbb{R}^d$, where $B$ denotes the unit ball. Proof 1: By Steiner symmetrization (which preserves volume, decreases diameter, and tends to the ball if desired). Proof 2: ...


1

\begin{equation} \log6=\log(1+2+3)=\log 1+\log 2+\log 3 \end{equation}


0

Another example would be the interplay between Möbius strip, Klein bottle and real projective plane. They motivate the study of orientability and are fun to visualize, especially how one transform into another.


3

I think my question is already really long so I add here the other inequalities I find out. In the question relation between mean-width and diameter it is pointed out that that for any convex body $K$ in $\mathbb{R}^d$ with positive diameter we have $$\frac{1}{\sqrt{2\pi d}} \simeq\frac{\textrm{Mean-width}(L)}{\textrm{Diameter}(L)} \leq ...


1

This is my favorite. \begin{align}-20 &= -20\\ 16 - 16 - 20 &= 25 - 25 - 20\\ 16 - 36 &= 25 - 45\\ 16 - 36 + \frac{81}{4} &= 25 - 45 + \frac{81}{4}\\ \left(4 - \frac{9}{2}\right)^2 &= \left(5 - \frac{9}{2}\right)^2\\ 4 - \frac{9}{2} &= 5 - \frac{9}{2}\\ 4 &= 5 \end{align} You can generalize it to get any $a=b$ that you'd like ...


3

Here is a measure theoretic one. By 'Picture', if we take a cover of $A:=[0,1]∩\mathbb{Q}$ by open intervals, we have an interval around every rational and so we also cover $[0,1]$; the Lebesgue measure of [0,1] is 1, so the measure of $A$ is 1. As a sanity check, the complement of this cover in $[0,1]$ can't contain any intervals, so its measure is surely ...


0

Let $k$ be a field, and let $A$ be a finite type $k$-algebra. Consider the isomorphism classes of dualizing complexes over $A$. Given two dualizing complexes $R$ and $S$, define their "product" to be the isomorphism class of the Hochschild cohomology complex of their tensor product over $k$: $R\cdot S := RHom_{A\otimes_k A}(A,R\otimes_{k} S)$. Then it is ...


1

Slightly more advanced (and perhaps only tangentially an existential proof), are applications of Jones's Lemma to demonstrate that a topological space is not normal. Jones's Lemma: Let $X$ is a normal space. If $C \subseteq X$ is a closed discrete subset, and $D \subseteq X$ a dense subset, then $2^{|C|} \leq 2^{|D|}$. (So for spaces in which there are ...


9

A favorite of mine was always the following: \begin{align*} \require{cancel}\frac{64}{16} = \frac{\cancel{6}4}{1\cancel{6}} = 4 \end{align*} I particularly like this one because of how simple it is and how it gets the right answer, though for the wrong reasons of course.


0

Draw any triangle. On each side of the triangle, draw an equilateral triangle such that the new equilateral triangle shares a side with the original triangle. Connect the midpoints of your three new triangles - the result is another equilateral triangle!


3

Geometric constructions are a cool application of field theory. For example, if a number $\alpha$ is constructible as a length, then the field extension $\mathbb{Q}[\alpha]$ is a degree power-of-two field extension over the rationals. Further, asking whether a regular $n$-gon is constructible with a compass and straightedge is equivalent to asking whether ...


2

The impact of Algebra on Number Theory (and vice-versa!) is considerable. Several theorems in Number Theory have nice algebraic proofs. For instance, Fermat's theorem on which numbers can be expressed as sums of two squares has a nice proof in terms of factorisation in the PID $\mathbb Z[i]$. See also Connections between number theory and abstract ...


4

Another topic could be the study of the three famous geometric constructions with compass and straight edge, or better, that they cannot be constructed as such: Squaring a circle, doubling the volume of a cube and trisecting an angle. For that you need rings and fields and all sorts of abstract algebra stuff. Did it a long time ago, forgotten for the most ...


1

I find semigroup theory interesting: it's rich and have applications to computer science (mainly in automata theory). See for instance's Green relations.


2

One topic that I find interesting is finite reflection groups. They have a lot of importance in studying highly symmetric geometric spaces. A related, but distinct, topic would be to discuss the 17 wallpaper groups and illustrate the various patterns that are possible.


6

Here is one I saw on a whiteboard as a kid... \begin{align*} 1=\sqrt{1}=\sqrt{-1\times-1}=\sqrt{-1}\times\sqrt{-1}=\sqrt{-1}^2=-1 \end{align*}


2

To give a contrarian interpretation of the question I will chime in with Goldbach's comet which counts the number of ways an integer can be expressed as the sum of two primes: It is mathematically "wrong" because there is no proof that this function doesn't equal zero infitely often, and it is visually deceptive because it appears to be unbounded with its ...


14

The magnetic pendulum: An iron pendulum is suspended above a flat surface, with three magnets on it. The magnets are colored red, yellow and blue. We hold the pendulum above a random point of the surface and let it go, holding our finger on the starting point. After some swinging this way and that, under the attractions of the magnets and gravity, it ...


1

Check out the "Proofs Without Words" gallery (animated) here: http://usamts.org/Gallery/G_Gallery.php And the related proofs here: http://www.artofproblemsolving.com/Wiki/index.php/Proofs_without_words Many of these are similar to the ones already listed here, but you get a bunch in one place.


12

Steven Wittens presents quite a few math concepts in his talk Making things with math. His slides can be found from his own website. For example bezier curves visually: He has also created MathBox.js which powers his amazing visualisations in the slides.


0

Here's a "proposition" (if I may call it): Let $X$ be a projective scheme over a Noetherian ring $A$ and $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ a ses of sheaves with $\mathcal{F}$ coherent. Then there is some $m$ so that $\Gamma(X,-)$ is exact when applied to the twisted sequence.


1

Differential Operators: $$(D^2-D-2)y=0$$ $$(D-2)(D+1)y=0$$ and the solution follows. Random Substitutions: Let $u=y'-2y$. Note that $$y''-y'-2y=y''-2y'+y'-2y=u'+u=0$$ $$u'=-u$$ $$u=y'-2y=c_1e^{-x}$$ and now use integrating factor to solve the first-order problem. I'll add more as I think of them.


3

Logarithms convert multiplication (difficult) into addition (easy). What follows is not the historical origin of $\log$, just one way of constructing $\log$ and $\exp$ using only tools of elementary calculus. If we define $$ \log x = \int_{1}^{x} \frac{dt}{t},\quad x > 0, $$ then for all $a$, $b > 0$, \begin{align*} \log(ab) = \int_{1}^{ab} ...


18

Construct a rectangle $ABCD$. Now identify a point $E$ such that $CD = CE$ and the angle $\angle DCE$ is a non-zero angle. Take the perpendicular bisector of $AD$, crossing at $F$, and the perpendicular bisector of $AE$, crossing at $G$. Label where the two perpendicular bisectors intersect as $H$ and join this point to $A$, $B$, $C$, $D$, and $E$. Now, ...


5

Not long ago I saw a graph showing the price of bitcoin over the past four years or so (currently in the $\$400\text{--}\$500$ range; it was between $\$2$ and $\$3$ in the fall of 2011). It looked like this: \begin{array}{r} 5000 \\[20pt] 500 \\[20pt] 50 \\[20pt] 5 \\[20pt] 0.50 \\[20pt] 0.05 \\[20pt] & & 2009 & \qquad\qquad & 2010 & ...



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