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1500 questions
96
votes
12 answers
How to convince a math teacher of this simple and obvious fact?
I have in my presence a mathematics teacher, who asserts that $$ \frac{a}{b} = \frac{c}{d} $$ Implies: $$ a = c, \space b=d $$ She has been shown in multiple ways why this is not true: $$ \frac{1}{2} = \frac{4}{8} $$ $$ \frac{0}{5} = \frac{0}{657}…
user86484
- 821
96
votes
0 answers
Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$ , that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$ . Is it true, that for any group word $w$ there…
Chain Markov
- 16,122
96
votes
7 answers
Why can't the Polynomial Ring be a Field?
I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This implies an inverse multiple for every Element in…
IAE
- 1,367
96
votes
21 answers
How do you explain the concept of logarithm to a five year old?
Okay, I understand that it cannot be explained to a 5 year old. But how do you explain the logarithm to primary school students?
Sandbox
- 1,265
95
votes
7 answers
Why is compactness in logic called compactness?
In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I…
vanden
- 1,277
95
votes
15 answers
Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
I'm a second year math student. And I've the following problem. When I prepare myself for an exam, I can distinguish two phases. First I'm mainly interested in whatever is necessary to pass the exam. This means that I do not always read the theory…
Kasper
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95
votes
7 answers
$1=2$ | Continued fraction fallacy
It's easy to check that for any natural $n$ $$\frac{n+1}{n}=\cfrac{1}{2-\cfrac{n+2}{n+1}}.$$ Now, $$1=\frac{1}{2-1}=\frac{1}{2-\cfrac{1}{2-1}}=\frac{1}{2-\cfrac{1}{2-\cfrac{1}{2-1}}}=\cfrac{1}{2-\cfrac{1}{2-\frac{1}{2-\frac{1}{2-1}}}}=\ldots…
Mher
- 5,129
- 1
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95
votes
6 answers
How does Cantor's diagonal argument work?
I'm having trouble understanding Cantor's diagonal argument. Specifically, I do not understand how it proves that something is "uncountable". My understanding of the argument is that it takes the following form (modified slightly from the wikipedia…
johne
- 1,077
95
votes
3 answers
Difference between metric and norm made concrete: The case of Euclid
This is a follow-up question on this one . The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making things more concrete: I am esp. interested in the…
vonjd
- 9,110
95
votes
11 answers
Why not include as a requirement that all functions must be continuous to be differentiable?
Theorem: Suppose that $f : A \to \mathbb{R}$ where $A \subseteq \mathbb{R}$. If $f$ is differentiable at $x \in A$, then $f$ is continuous at $x$. This theorem is equivalent (by the contrapositive) to the result that if $f$ is not continuous at $x…
Perturbative
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- 179
95
votes
11 answers
Results that came out of nowhere.
Most big results in mathematics are built on years and years of groundwork by the author and other mathematicians, such as Wiles' proof of FLT or the classification of finite simple groups. Nevertheless, there is this big romantic idea in pop…
Alexander Gruber
- 28,325
95
votes
30 answers
Best Maths Books for Non-Mathematicians
I'm not a real Mathematician, just an enthusiast. I'm often in the situation where I want to learn some interesting Maths through a good book, but not through an actual Maths textbook. I'm also often trying to give people good Maths books to get…
Edan Maor
- 2,028
95
votes
4 answers
Why does the Mandelbrot set contain (slightly deformed) copies of itself?
The Mandelbrot set is the set of points of the complex plane whos orbits do not diverge. An point $c$'s orbit is defined as the sequence $z_0 = c$, $z_{n+1} = z_n^2 + c$. The shape of this set is well known, why is it that if you zoom into parts of…
anon
95
votes
3 answers
Paul Erdos's Two-Line Functional Analysis Proof
Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes thinking about the original problem, and came up with a…
Emily
- 36,462
95
votes
1 answer
Number of simple edge-disjoint paths needed to cover a planar graph
Let $G=(V,E)$ be a graph with $|E|=m$ of a graph class $\mathcal{G}$. A path-cover $\mathcal{P}=\{P_1,\ldots,P_k\}$ is a partition of $E$ into edge-disjoint simple paths. The size of the cover is $\sigma(\mathcal{P})=k$. I am interested in upper and…
A.Schulz
- 3,828