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21

Hamilton (and Graves) wanted to generalize $\mathbb C$ - if viewed as $\mathbb R^2$ with a multiplication that turns it into a field with a multiplicative absolute value. They were looking for something similar in $\mathbb{R}^n$ for $n>2$. It turns out that Hamilton spent 13 years in vain with $n=3$ although it was essentially known since Diophantus ...


11

With the complex numbers in hand, it's natural to wonder what other systems of numbers containing the real numbers one might have. Before constructing the quaternions, Hamilton tried in vain to construct a $3$-dimensional system; it turns out that this is impossible, and you can see a reproduction of Hamilton's own proof in this recent entry: Why should I ...


8

A famous one : Irrational number to an irrational power can be rational. Proof : If $\sqrt 2^{\sqrt 2}$ is rational, we are happy. If $\sqrt 2^\sqrt 2$ is irrational, then $(\sqrt 2^{\sqrt 2})^\sqrt 2=2$ is rational.


8

You should know calculus. You should know how mathematical reasoning is done: what theorems and proofs are. It might help to know some things that are often not taught in first-year calculus: epsilons and deltas, things like the difference between pointwise convergence and uniform convergence, suprema and infima, etc. Knowing some things about power series ...


7

Groups are an abstraction of symmetry. A group representation is a way to realize the abstract symmetry encoded in a group by means of linear transformation, i.e., as geometric transformation of a particular nice nature of a linear space, often a finite dimensional one. With fields the situation is different. A field is not something that encodes symmetry ...


7

$\qquad\qquad\qquad\qquad$ $\qquad\qquad\qquad\qquad\quad$ Geometric Explanation of the Binomial Theorem $\qquad\qquad\qquad\qquad\qquad$ $\qquad\qquad\qquad\qquad\qquad\qquad$ Proof that $~\displaystyle\sum_{k=1}^n(2k-1)=n^2$


7

The nice thing about mathematics is that it is precise. One you define something you can ask whether or not it exists as a mathematical object. Whether or not a mathematical object also exists in any real sense is a debate for philosophers and has little if any consequences for mathematics. Now, the concept of infinity you mention is certainly subtle and ...


6

$\qquad\qquad\qquad\qquad\qquad$ What is the motivation for quaternions? Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. However, $[$he$]$ had been stuck on the problem of multiplication and division for a long time. He could not ...


6

It is correct, however, you might see the way similarly as: $$a(ab)b=a^2b^2=e=(ab)^2=a(ba)b.$$


6

I'm fond of Euclid's proof of the infinitude of primes: For any finite set $S=\{p_1, p_2,\dots, p_k\}$ of prime numbers, let $N=p_1\cdot p_2\cdot\cdots\cdot p_k+1$. Then $N$ isn't divisible by any prime in $S$. Hence it is divisible by some other prime. Hence the set $S$ does not include all primes. Thus there must be infinitely many primes.


6

Cosines and Sines Around the Unit Circle Trigonometric Angle Sum and Difference


5

Proof of Euler's Identity: $$e^{\pi{i}}+1=0$$ BTW, your question is more or less a copy of "Simple" beautiful math proof, so you might wanna check it out too. There's some great colorful stuff there, my answer being somewhere in the middle.


5

I tried to find problems from different areas. My five suggestions are. Sophomore' dream. The formula for the problem is: $$\begin{align}\int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n}\end{align}$$ You can find facts about the problem and the proof of it at Sophomore's dream wikipedia article. Bretschneider's formula. This is an expression for the ...


4

Two important applications come to mind: They model rotations of the sphere, much like the complex numbers model rotations of the circle. You can get the four squares theorem, that every integer $n$ can be represented as $n=a^2+b^2+c^2+d^2$ for some integers $a,b,c,d \geq 0$, using unique factorization-like properties of the quaternions. This has to do ...


4

I think that you are missing an important point about differentiation (but don't feel ashamed, it's not an obvious point at all if you don't have enough background). The important thing here is that differentiation is a local concept, i.e. we have the concept of differentiation at a point, and this depends only on a(n arbitrarily small) open neighborhood of ...


3

“Complex Variables and Applications” by Brown and Churchill is pretty nice. It's compact, and the later chapters show applications of the theory to some physics problems.


3

Michael Hardy's answer is excellent. For a specific text I would recommend Flanigan's "Complex Variables." It is extremely clear with many pictures - aiding the geometric aspects M.H. mentioned. This is a Dover publication, so in contrast to the exorbitant cost of most math texts, this is a bargain. ...


3

I found a guide on writing mathematical papers here (about in the middle of the text) that has following guidlines: (...) In addition, avoid numerals because they slow down the reading. Write numbers out if they can be expressed in one or two words and are used as adjectives, unless they are accompanied by units, a percentage sign, or a monetary ...


3

Proofs should follow the same rules as any other kind of writing. If the text naturally forms paragraphs then that's how it should be written. Now, usually if you have five pages of text forming a single paragraph that would be poor style, but I suppose it's possible that you could have a five-page-long paragraph if, say, one of the sentences of that ...


3

You may be surprised to learn that it can be proved that infinity is necessary even for some proofs about finite objects. As a consequence, there can be no self-contained comprehensive discipline of finite mathematics. For an introduction to these ideas see Stephen G. Simpson's writeup of his expository talk Unprovable Theorems and Fast-Growing Functions, ...


3

Stewart Calculus. Don't do Spivak, that is just too much in my opinion. Just start from the beginning of any edition of Stewart Calculus. Section by section work a good amount of problems, to ensure you are learning. Trust me you want a good base to start from in order to get into more theoretical/rigorous calculus. This is coming from someone who took ...


3

Calculus The proof that $\frac{22}{7} > \pi$. $$ \begin {align*} 0 &< \displaystyle\int_0^1 \frac {x^4 \left( 1 - x \right)^4}{1 + x^2} \, \mathrm{d}x \\&= \displaystyle\int_0^1 \frac {x^4 - 4x^5 + 6x^6 - 4x^7 + x^8}{1 + x^2} \, \mathrm{d}x \\&= \frac {22}{7} - \pi. \end {align*} $$ Geometry The Pythagorean Theorem. Algebra Proof ...


3

Here are a few visual proofs that $$\text{arctan}(1) + \text{arctan}(2) + \text{arctan}(3) = \pi$$ One by user KennyTM: More by user dldarek: I think the lattice nature of the proofs would look nice on a door.


3

Hairy Ball theorem (for $n=2$): There are no non-vanishing continuous tangent vector field for $S^2$ Proof:: If such a vector field did exist, let $v_x$ be the vector at $x$. The function $H:S^2 \times [0,1] \to S^2$ mapping $(x,t)$ to the point $t\pi$ radians away from $x$ along the great circle defined by $v_x$ is a homotopy between the identity and the ...


3

The rudimentary differential equation proof of Euler's formula in the complex plane $e^{i \pi}=-1$, where $i=\sqrt{-1}$. First, via $\frac{d}{d\theta}$, $$e^{i\theta}=f(\theta)+ig(\theta) \implies ie^{i\theta}=f^{\prime}(\theta)+ig^{\prime}(\theta)=if(\theta)-g(\theta).$$ Comparing real and imaginary parts, $f(\theta)=g^{\prime}(\theta)$ and ...


2

The proof for the irrationality of $\sqrt{2}$ is pretty simple and satisfying, I think. It's a very easy result to achieve, but the proof is very elegant and has some nice symmetry. Assume $\sqrt{2} = \frac{p}{q}$ with p and q relatively prime (totally simplified). $2q^2 = p^2$ $p^2$ is even the square of an odd number is odd, so $p$ must be even. Let ...


2

The classification of finite simple groups -- so there would finally be a single reference that could be given for this important result. ;)


2

Some suggestions: $1$. The proof for the Gaussian integral $$\int_{-\infty}^{\infty}e^{-x^2} \mathrm dx=\sqrt{\pi}$$ $2$. The proof for Euler's solution to the Basel problem $$\frac {\ \ \pi^2}6=\sum_{n=1}^{\infty}\frac 1{n^2}=\frac 1{1^2}+\frac 1{2^2}+\frac 1{3^2}+\frac 1{4^2}+\cdots+\frac 1{n^2}+\cdots$$ $3$. The proof for Wallis' product $$\frac \pi ...



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