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1500 questions
67
votes
3 answers
Functions that are Riemann integrable but not Lebesgue integrable
I know there are functions which are Riemann integrable but not Lebesgue integrable, for instance, $$\int_{\mathbb{R}} \frac{\sin(x)}{x} \mathrm{d}x$$ Is Riemann integrable and it is easily shown that its value is $\pi $, nevertheless, it is not…
user438666
- 2,260
67
votes
19 answers
What are the names of numbers in the binary system?
The names we use are very much related to the radix we use $0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9$ zero - one - two - three - four - five - six - seven - eight -nine We repeat the names $21$ twenty one , $22$ twenty two .. and so on. This is not…
user37421
- 927
67
votes
2 answers
What is the integral of 1/x?
What is the integral of $\frac{1}{x}$ ? Do you get $\ln(x)$ or $\ln|x|$ ? In general, does integrating $f'(x)/f(x)$ give $\ln(f(x))$ or $\ln|f(x)|$ ? Also, what is the derivative of $|f(x)|$ ? Is it $f'(x)$ or $|f'(x)|$ ?
hollow7
- 2,555
67
votes
9 answers
What's wrong with this reasoning that $\frac{\infty}{\infty}=0$?
$$\frac{n}{\infty} + \frac{n}{\infty} +\dots = \frac{\infty}{\infty}$$ You can always break up $\infty/\infty$ into the left hand side, where n is an arbitrary number. However, on the left hand side $\frac{n}{\infty}$ is always equal to $0$. Thus…
JobHunter69
- 3,623
67
votes
4 answers
Is it normal to treat Math Theorems as "Black Boxes"
It seems to me that as one goes higher up in mathematics, the proof of theorems get more involved and convoluted, that at some point one must postpone (or even give up?) understanding all prior theorems, and take them as "Black Boxes" instead? Is…
yoyostein
- 20,598
67
votes
2 answers
What is the average rational number?
Let $Q=\mathbb Q \cap(0,1)= \{r_1,r_2,\ldots\}$ be the rational numbers in $(0,1)$ listed out so we can count them. Define $x_n=\frac{1}{n}\sum_{k=1}^nr_n$ to be the average of the first $n$ rational numbers from the list. Questions: What is…
jdods
- 6,682
67
votes
2 answers
When is an infinite product of natural numbers regularizable?
I only recently heard about the concept of $\zeta$-regularization , which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$ where $n\#$ is a primorial , and…
Timmy Turner
- 1,049
67
votes
11 answers
Undergraduate algebraic geometry textbook recommendations
What are the best algebraic geometry textbooks for undergraduate students?
tobias
67
votes
17 answers
Good "history of mathematical ideas" book?
All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living subject in itself, which is not enhanced by knowing…
Awesome Academist
- 773
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67
votes
7 answers
Is the sum and difference of two irrationals always irrational?
If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational?
Miguel Mora Luna
- 3,253
67
votes
2 answers
How do different definitions of "degree" coincide?
I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question: How exactly do the following three notions of "degree" coincide? (1)…
Jesse Madnick
- 33,049
67
votes
1 answer
Direct Sum vs. Direct Product vs. Tensor Product
There are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in the most general sense), and how are they different?
user247773
67
votes
14 answers
Express 99 2/3% as a fraction? No calculator
My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is appreciated: The possible answers are: $1…
lara400
- 886
- 1
- 7
- 9
67
votes
4 answers
Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$
I'm interested in integrals of the form $$I(a,b)=\int_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx,\color{#808080}{\text{ for }a>0,\,b>0}\tag1$$ It's…
Vladimir Reshetnikov
- 32,880
67
votes
10 answers
Why is $1/i$ equal to $-i$?
When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to check it out). Is there a fault in my calculator or…
Arulx Z
- 1,061