Most Popular
1500 questions
73
votes
9 answers
Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?
I've always had this doubt.
It's perfectly reasonable to say that, for example, 9 is bigger than 2.
But does it ever make sense to compare a real number and a complex/imaginary one?
For example, could one say that $5+2i> 3$ because the real part of…
Juanma Eloy
- 1,407
73
votes
11 answers
Where is the flaw in this "proof" that 1=2? (Derivative of repeated addition)
Consider the following:
$1 = 1^2$
$2 + 2 = 2^2$
$3 + 3 + 3 = 3^2$
Therefore,
$\underbrace{x + x + x + \ldots + x}_{x \textrm{ times}}= x^2$
Take the derivative of lhs and rhs and we get:
$\underbrace{1 + 1 + 1 + \ldots + 1}_{x \textrm{ times}}…
user116
72
votes
3 answers
What is the limit of $n \sin (2 \pi \cdot e \cdot n!)$ as $n$ goes to infinity?
I tried and got this
$$e=\sum_{k=0}^\infty\frac{1}{k!}=\lim_{n\to\infty}\sum_{k=0}^n\frac{1}{k!}$$
$$n!\sum_{k=0}^n\frac{1}{k!}=\frac{n!}{0!}+\frac{n!}{1!}+\cdots+\frac{n!}{n!}=m$$
where $m$ is an integer.
$$\lim_{n\to\infty}n\sin(2\pi…
M. Amin
- 721
72
votes
18 answers
Why should we prove obvious things?
Obviously, there are obvious things in mathematics. Why we should prove them?
Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$?
Prove that $f(x)=x$ is continuous on $\mathbb{R}$?
$\dotsc$
Just to list few examples.
x.y.z...
- 1,150
72
votes
13 answers
What Is Exponentiation?
Is there an intuitive definition of exponentiation?
In elementary school, we learned that
$$
a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times})
$$
where $b$ is an integer.
Then later on this was expanded to include rational exponents, so…
baum
- 1,541
72
votes
2 answers
Can $18$ consecutive integers be separated into two groups,such that their product is equal?
Can $18$ consecutive positive integers be separated into two groups, such that their product is equal? We cannot leave out any number and neither we can take any number more than once.
My work:
When the smallest number is not $17$ or its…
Hawk
- 6,540
72
votes
11 answers
Why do we use a Least Squares fit?
I've been wondering for a while now if there's any deep mathematical or statistical significance to finding the line that minimizes the square of the errors between the line and the data points.
If we use a less common method like LAD, where we…
tom
- 3,227
72
votes
11 answers
Is Euclid's proof on the infinitude of primes flawed because it yields some composites?
I've expressed Euclid's proof on the infinitude of primes on Mathematica:
f[x_] := Product[Prime[n], {n, 1, x}] + 1
TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]
Which results in:
$\begin{array}{ll}
3 & \text{True} \\
7 & \text{True} \\
31…
Red Banana
- 23,956
- 20
- 91
- 192
72
votes
4 answers
Difference between basis and subbasis in a topology?
I was reading Topology from Munkres and got confused by the definition of a subbasis. What is/are the difference between basis and subbasis in a topology?
Grobber
- 3,248
72
votes
15 answers
Dot Product Intuition
I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).
For example, suppose I have vector $\mathbf{a} =…
nerdy
- 3,288
72
votes
7 answers
Proof a graph is bipartite if and only if it contains no odd cycles
How can we prove that a graph is bipartite if and only if all of its cycles have even order? Also, does this theorem have a common name? I found it in a maths Olympiad toolbox.
Asinomás
- 105,651
72
votes
12 answers
Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?
Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?
I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases?
Andy
- 2,246
72
votes
9 answers
Intuition on group homomorphisms
So I'm studying for finals now, and came across the idea of homomorphisms again. This is not a new idea for me at all, having seen them in groups, rings, fields ect. However, on reevaluating them I realized suddenly that I really don't understand…
user45793
72
votes
5 answers
Under what condition we can interchange order of a limit and a summation?
Suppose f(m,n) is a double sequence in $\mathbb R$. Under what condition do we have $\lim\limits_{n\to\infty}\sum\limits_{m=1}^\infty f(m,n)=\sum\limits_{m=1}^\infty \lim\limits_{n\to\infty} f(m,n)$? Thanks!
zzzhhh
- 839
72
votes
3 answers
Prove: If a sequence converges, then every subsequence converges to the same limit.
I need some help understanding this proof:
Prove: If a sequence converges, then every subsequence converges to the same limit.
Proof:
Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \geq k$ for all $k$. This easy to prove by induction:…
CodeKingPlusPlus
- 7,342