133,847 reputation
15104378
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location Shoulders of Giants
age
visits member for 4 years, 4 months
seen 2 hours ago

The essence of mathematics lies precisely in its freedom. - Georg Cantor

Once you have tasted flight in mathematical skies, you will forever walk the Earth with your eyes turned skyward, for there you have been, and there you will always long to return.


4h
comment Simple doubt about complex numbers
@Ben If not then a denesting does not exist.
4h
comment Solution of $\dfrac{a}{b}=\dfrac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$
@user31782 1. That's what the proof shows. 2. Bt taking fractional parts, e.g $\,\dfrac{5}3 = \dfrac{20}{12}\Rightarrow\dfrac{2}3 = \dfrac{8}{12}$. 3. Yes, as I said, it's equivalent to Euclid's Lemma.
5h
reviewed Leave Open Cantor Pairing Function
5h
reviewed Reopen Combinations. x+y+z=12
5h
reviewed Reopen How to construct a smooth function with compact support satisfying $f(x)+f(x^{-1})=1$
5h
revised For any $p,q\in\mathbb{Z}[i]$, $\mathrm{N}(\gcd(p, q))$ must divide $\gcd(\mathrm{N}(p), \mathrm{N}(q))$
added 211 characters in body
5h
answered For any $p,q\in\mathbb{Z}[i]$, $\mathrm{N}(\gcd(p, q))$ must divide $\gcd(\mathrm{N}(p), \mathrm{N}(q))$
5h
reviewed Leave Open Maximal value, several variables
5h
reviewed Leave Open How can I prove the last digit of $(2^{121985292}-1)$ is $5$
5h
reviewed Leave Open Show that $m^*(E \cup F) \leq m^*(E) + m^*(F)$
5h
reviewed Leave Open boundedness of integral of a bounded function
5h
reviewed Leave Open How to show that a piecewise constant function is integrable, using the upper and lower sums?Please help
5h
reviewed Leave Open Does the boundary of a subset of $\Bbb R^n$ always have measure zero?
6h
comment How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$
@SasQ Norm and trace are met when one studies algebraic extension fields, e.g. in algebraic number theory. If you go on to study Galois theory then that will help you understand better the innate algebraic structure, and why the additive and multiplicative objects are intertwined in the rule.
7h
comment How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$
@SasQ Thanks for elaborating - I didn't realize that was the source of confusion. There $\,w\,$ is used as a common name for the (complex) argument of the norm and trace functions, just like $\,x,y\,$ are used as common names of arguments of bivariate functions (e.g. addition and multiplication).
8h
comment How do I show that $\sqrt{5+\sqrt{24}} = \sqrt{3}+\sqrt{2}$
@SasQ The English summary $\rm \ \, \color{blue}{subtract\ out}\ \sqrt{norm}\:,\ \ {\it then}\ \ \color{#0a0}{divide\ out}\ \sqrt{trace}\ \ $ is meant to be memorable. I do realize that, at first sight, this summary it is a bit ambiguous, but this is clarified once one sees the examples. I see nothing incorrect anywhere.
9h
revised No radical in the denominator — why?
added 1 character in body
13h
reviewed Leave Open Find all positive integers $a,b,c,d$ with given conditions.
13h
reviewed Reopen Decompose $a = a_1\cdots a_k$ and $a_1 + \dots +a_k = 0$
13h
reviewed Reopen Nonhomogeneous equations