examples/math_article.adoc in asciidoctor-latex-1.5.0.3.dev vs examples/math_article.adoc in asciidoctor-latex-1.5.0.4.dev

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@@ -1,12 +1,11 @@ :numbered: :latex: -== A Short Math Article -++++ -\( +[env.texmacro] +-- %% Blackboard bold \def\NN{\mathbb{N}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} @@ -16,123 +15,124 @@ \newcommand{\set}[1]{ \{\,#1\, \} } \newcommand{\sett}[2]{ \{\,#1\, \mid\, #2\, \} } \newcommand{\Set}[1]{ \Big\{\,#1\, \Big\} } \newcommand{\Sett}[2]{ \Big\{\,#1\, \Big\vert\; #2\, \Big\} } \newcommand{\mapright}[1]{\ \smash{ \mathop{\longrightarrow}\limits^{#1}}\ } -\) -++++ +-- -This "article" is a demonstration of how one can write -mathematics in Noteshare. Use arbitrary LaTeX inside + +== A Short Math Article + +This "article" is a demonstration of how one can write +mathematics in Noteshare. Use arbitrary LaTeX inside the traditional dollar sign, bracket delimiters, and a form -of the LaTeX environment. Equation -numbering and cross-referencing is supported, as is the numbered -theorem environment.footnote:[We are working on a prototype -asciidoctor-to-latex converter. At the present moment it is +of the LaTeX environment. Equation +numbering and cross-referencing is supported, as is the numbered +theorem environment.footnote:[We are working on a prototype +asciidoctor-to-latex converter. At the present moment it is adequate to convert this file into LaTeX. Stay tuned!] [quote] -- -_Observe the footnote reference in the previous paragraph. -The footnote is *way*at the bottom. However, you can click on -it it reference it._ +Observe the footnote reference in the previous paragraph. +The footnote is *way*at the bottom. However, you can click on +it it reference it. -- === Pythagorean triples -One of the first real pieces of mathematics we +One of the first real pieces of mathematics we learn is this: -[env.theorem#th-pyth] +[env.theorem#th-pyth] -- -_Let $a$, $b$, and $c$ be the sides of a right triangle, -where $c$ is the hypotenuse. Then $a^2 + b^2 = c^2$._ +Let $a$, $b$, and $c$ be the sides of a right triangle, +where $c$ is the hypotenuse. Then $a^2 + b^2 = c^2$. -- The Pythagorean theorem suggests an equation, -[env.equation#eq-pyth] +[env.equation#eqpyth] -- x^2 + y^2 = z^2 -- -If we demand that the unknowns be integers, -then this is a _Diophantine equation_. We all -know one solution, to equation <<eq-pyth>>, the -3-4-5 triangle. However, there are many -more, in fact, infinitely many. One -way of generating more solutions is to rescale existing ones. -Thus 6-8-10 is a solution. However, +If we demand that the unknowns be integers, +then this is a _Diophantine equation_. We all +know one solution, to equation <<eqpyth>>, the +3-4-5 triangle. However, there are many +more, in fact, infinitely many. One +way of generating more solutions is to rescale existing ones. +Thus 6-8-10 is a solution. However, what is interesting is that there are infinitely - many _dissimilar solutions_. One is 5-12-13. -Quite remarkably, there is a clay tablet (Plympton 322) from -Mesopotamia, dated to about 1800 BC, that -contains a list of 15 such + many _dissimilar solutions_. One is 5-12-13. +Quite remarkably, there is a clay tablet (Plympton 322) from +Mesopotamia, dated to about 1800 BC, that +contains a list of 15 such "Pythagorean triples." See http://en.wikipedia.org/wiki/Plimpton_322[Wikipedia]. It is unlikely -that the Babylonian mathematicians, despite -their sophistication and skill, knew that -equation <<eq-pyth>> -has infinitely many solutions. This fact is -equivalent to the statement that the unit circle -centered at the origin has infinitely many points +that the Babylonian mathematicians, despite +their sophistication and skill, knew that +equation <<eqpyth>> +has infinitely many solutions. This fact is +equivalent to the statement that the unit circle +centered at the origin has infinitely many points with rational coordinates. -For more information on Pythagorean triplets, see +For more information on Pythagorean triplets, see http://en.wikipedia.org/wiki/Pythagorean_triple[Wikipedia]. === Another result from ancient times [env.theorem#th-primes] -- -_There are infinitely many primes._ +There are infinitely many primes. -- .Proof -- Suppose that there are only finitely many primes, say -$p_1, p_2, \ldots, p_N$. Let +$p_1, p_2, \ldots, p_N$. Let -- [env.equation#eq-infprimes] -- Q = p_1p_2 \cdots p_N + 1 -- This number is is greater than the greatest prime, $p_N$. -Therefore it is composite, and therefore it is divisible -by $p_i$ for some $i$. But the remainder of $Q$ upon +Therefore it is composite, and therefore it is divisible +by $p_i$ for some $i$. But the remainder of $Q$ upon division by $p_i$ is 1, a contradiction. *Q.E.D.* === The work of Fermat Let us consider <<th-pyth>> once again. Pierre Fermat asked whether this family of Diophantine equations $x^d + y^d = z^d$, where parameter $d$ is greater than two, have any other than the obvious -solutions, e.g., $x, y, z = 1, 0 ,1$ where one variable is zero. +solutions, e.g., $x, y, z = 1, 0 ,1$ where one variable is zero. Fermat conjectured that the answer was no, and he wrote in the margin of Diophantus' treatise that -he had a marvelous proof of this fact, alas, too long to fit int he +he had a marvelous proof of this fact, alas, too long to fit int he space available. More than 300 years later, Andrew Wiles, using techniques developed only in the twentieth century, gave a proof of the theorem of Fermat. === The work of Georg Cantor Georg Cantor introduced the notion of _set_, e.g., [env.equation#eq-set-integers] -- - \NN = \set{ \text{integers} } + \NN = \set{ \text{integers} } = \set{ 1, 2, 3, \ldots } -- Story to be continued ... However, let's check that our cross-references work. We learned about the Pythagorean formula <<eq-pyth>>, and the -number of solutions it has in <<th-pyth>>. We also +number of solutions it has in <<th-pyth>>. We also learned in <<th-primes>> that prime numbers are quite abundant in Nature: there are infinitely many of them. Our last, incomplete section, featuring the set of integers <<eq-set-integers>>, is devoted to the work of Georg Cantor. -