5

There is no integer (strictly) between zero and one. Looks pretty stupid, but many irrationality and transcendence proofs rely on it. [and in some sense it is equivalent to the pigeonhole principle] Example: the proof of the transcendence of $e$ given in http://www.math.leidenuniv.nl/~evertse/dio15-4.pdf – note particularly the statement after Corollary 4.3, ...


4

Some geometric construction problems - e.g. doubling the cube - were "posed" in antiquity but only solved in the $19$th century. This gets us a gap of about $\sim 2000$ years at least. Re: the scare quotes above, there is a slight issue here about what exactly constitutes posing a problem. The texts I'm aware of all phrase it as a positive ...


3

Mesopotamians (actually, Sumerians first, then Akkadians of which Assyrians and Babylonians are dialects) used a simple system of bars, stacked if more than 3, with a different mark for 10. So that would be 𒁹 = 1, 𒈫 = 2, 𒐈 = 3, 𒐉 = 4, 𒐊 = 5, 𒐋 = 6, 𒑂 = 7, 𒑄 = 8, 𒑆 = 9 (although there were variants of most of them; keep in mind cuneiform was used for ...


3

Why, according to Hadamard, Pascal could have discovered non-Euclidean geometry? Let start from Euclid's definition: Definition 23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction. The issue is not about substitution of definiens in ...


3

Subtract the first column from the second column and factor out a common term from the second column. $\left|\begin{matrix} 1 & 1 & 1\\ a & b & c\\ a^2 & b^2 & c^2 \end{matrix}\right|=\left|\begin{matrix} 1 & 0 & 1\\ a & b-a & c\\ a^2 & b^2-a^2 & c^2 \end{matrix}\right|=(b-a)\left|\begin{matrix} 1 & 0 &...


2

Poincaré's discovery of homoclinic points grew out of a extremely serious mistake he made in his original submission for a prize essay contest sponsored by Acta Mathematica in 1888. His original 200 page manuscript, on the restricted three-body problem, was evaluated by Weierstrass, Mittag-Leffler, and Phragmén, who had great difficulty following his ...


2

Skolem is working in the context of first-order logic without equality, so any theory with a finite nonempty model has an infinite model. Namely, you can just add infinitely many indistinguishable copies of one of the elements of your model. In any case, though, you seem to be misunderstanding the logic here. Carrying out the first $n$ steps of the ...


2

A typical statistics course will define the standard deviation as "the average of the difference between the data set and the mean ". That is false. I doubt that you've seen that in any textbook on probability or statistics. So if we tried to describe the definition mathematically we should derive this equation (∑(|x-mean|))/n That is NOT the ...


1

The expansion is around $x=0$. Then $$bF(x)=g-\frac{a^2}{c-bx}f\left(\frac a{c-bx}\right)$$ Expanding in Taylor series at $x=0$, the constant term is $$g-\frac{a^2}cf\left(\frac ac\right)$$ The linear term is given by the first derivative: $$\left(g-\frac{a^2}{c-bx}f\left(\frac a{c-bx}\right)\right)'=\frac{a^2}{(c-bx)^2}(-b)f\left(\frac a{c-bx}\right)-\frac{...


1

Since the algebraists were not interested in deductive systems based on syntactic rules, I cannot understand why there would be any reason for them to understand "non-contradictory" as equivalent with "satisfiable" if "contradictory" is meant in the sense of syntactically refutable. Is it true that there is some sort of ...


1

It appeared as Problem 735 in the September 2002 issue of The College Mathematics Journal. The solution in the September 2003 issue referenced the December 2001 issue of FAMOS.


1

It appears that George Boole himself started using these symbols in his book An Investigation of the Laws of Thought He doesn't use the dot notation though. He writes just, for example, $xy$.


1

This is the algebraic notation used for the Boolean semiring ${\Bbb B} = \{0, 1\}$, which is the simplest example of a semiring that is not a ring. It is an idempotent semiring, that is, it satisfies $x + x = x$ for all $x$. Mathematically speaking, it is a very convenient notation, which allows for natural extensions such that Boolean matrices, polynomials ...


1

Note that $$ V=bc(c-b)-ac(c-a)+ab(b-a) = - (a - b)(a - c)(b - c). $$


1

"There is no relation between the two words." feels wrong. The spelling of algorithm was influenced by the Greek árithmos, which is the second compound in logarithm. I quote from Word Origins (2005 2e) by John Ayto. p 16 Left column. algorithm [13] Algorithm comes from the name of a Persian mathematician, in full Abu Ja far Mohammed ibn-Musa al-...


1

One who is not a professional mathematician is Edward Witten, who is a physicist.


1

Diocles was the first prove this in "On burning mirrors". There are no surviving records but fragments were found in Eutocius's notes on Archimedes "Spheres and Cylinders". Conic sections were well-studied in antiquity including parabolas but records were mostly kept by Arabic mathematicians. Ibn Sahl has a surviving proof which likely ...


1

Good explanation from geimetey. My original submission has been done by my friend Zaffar. That's because I am not familiar with formalities if stack exchange. Few points I clarify in this context: (1) The division method can be extended for any n-root. Sat for cube root : Start with a number bB+s where B is the base; a and b are integers. (bB +a) ^3 =(bB)^3 +...


1

For good visualization of the method you can watch the following video at youtube: https://www.youtube.com/watch?v=P7kesuMnqhs


1

When I first read your question, it seemed to me that uniform spaces were much more used than bornological spaces. But as I frequently use uniform spaces in my own research, I thought that my opinion might be biased and I looked for more measurable criteria. (1) It turns out that both topics have their specific entry in the MSC2020-Mathematics Subject ...


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