number-theory

foundations.tex

@@ -310,101 +310,6 @@ \subsection*{\url{https://www.reddit.com/r/explainlikeimfive/comments/61siyw/eli
 not as common as the standard trig functions.
 
 
-\section{Number theory}
-
-
-\begin{definition*}
-
-
-  \defn{prime factorization}
-  \begin{mdframed}
-    \includegraphics[width=400pt]{img/foundations--integers-e251.png}
-  \end{mdframed}
-
- $c | a$, i.e. $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$), i.e. if there exists $d$ such that $cd = a$.
-
- $\gcd(a, b)$, the \defn{greatest common divisor} of $a$ and $b$, is the largest integer $c$ such that $c | a$ and $c | b$.
-
-  $a$ and $b$ are \defn{relatively prime} aka \defn{coprime} if $\gcd(a, b) = 1$.
-
-  $\lcm(c, d)$ is the smallest integer $a$ such that $c|a$ and $d|a$.
-\end{definition*}
-
-  Question: why exactly are we contemplating products here, as opposed to e.g. sums?
-
-
-
-
-\begin{remark*}
-  a \defn{multiple} of a prime is a number whose factorization contains that prime.
-
-  $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$) if $a$'s factorization is a ``multiplicity superset​'' of $c$'s.
-
-  The $\gcd$ is the largest integer whose factorization is a ``subset​'' of both factorizations.
-
-  If $a$ and $b$ are relatively prime then their factorizations have no overlap.
-
-  The $\lcm$ is the smallest integer whose factorization is a ``superset​'' of both factorizations.
-
-  If $a$ is a multiple of $c$ then their $\lcm$ is $a$.
-
-  If $a$ and $b$ are relatively prime then their $\lcm$ is their product.
-\end{remark*}
-
-\begin{example*}
-Consider $2^2 \cdot 3 = 12$ and $2\cdot 3^2 = 18$.
-
-Their factorizations have much in common (they are certainly not coprime), but neither is a
-multiple of the other.
-
-Their $\lcm$ is the first number that is a ``multiplicity superset​'' of the other: i.e. we must add
-a factor of $3$ to $12$'s factorization, or an additional $2$ to $18$'s, either way
-yielding $2^2 \cdot 3^2 = 36$.
-
-Note that their $\lcm$ is not one of them (as it would be if one divided the other), but it is
-smaller than their product.
-
-The $\gcd$ of $2^2\cdot 3$ and $2\cdot 3^2$ is $2\cdot 3$.
-\end{example*}
-
-
-\begin{example*}
-    For example if
-  \begin{align*}
-    a = 12 &= 2^2 \cdot 3 \\
-    b = 40 &= 2^3 \cdot 5 \\
-  \end{align*}
-  then $\gcd(a, b) = 2^2 = 4$ and $\lcm(a, b) = 2^3 \cdot 3 \cdot 5 = 120$.
-
-  This can be written as a general theorem involving mins and maxes in the exponents of a product
-  of primes.
-\end{example*}
-
-
-\begin{theorem*}
-  $\gcd(a,b) \times \lcm(a,b) = ab$
-\end{theorem*}
-
-\begin{example*}
-  The product of  $2^2\cdot 3$ and $2\cdot 3^2$ is their concatenation: $2^2 \cdot 3 \cdot 2 \cdot 3^2 = 12 \cdot 18 = 216$.
-
-  This is the product of their $\gcd$ $2\cdot 3$ and their $\lcm$ $2^2\cdot 3^2$.
-\end{example*}
-
-
-\begin{intuition*}
-  We're performing additive operations on the exponents of the prime factors. The product is the
-  sum of all; the $\gcd$ takes the min from each and is thus missing the maxes; the $\lcm$ takes
-  the max from each and is thus missing the mins; their product contains the full multiplicities of
-  all factors.
-\end{intuition*}
-
-
-\begin{mdframed}
-\includegraphics[width=400pt]{img/foundations--set-theory--number-theory-82fa.png}
-\includegraphics[width=400pt]{img/foundations--set-theory--number-theory-179a.png}
-\end{mdframed}
-
 
 \section{$\sqrt{2}$ is irrational}
 \begin{theorem*}

mathematics.tex

@@ -12,6 +12,7 @@
 
 
 \chapter{Foundations}
+\include{number-theory}
 \include{foundations}
 
 \chapter{Discrete Mathematics}

number-theory.tex

@@ -0,0 +1,94 @@
+\section{Number theory}
+
+
+\begin{definition*}
+
+
+  \defn{prime factorization}
+  \begin{mdframed}
+    \includegraphics[width=400pt]{img/foundations--integers-e251.png}
+  \end{mdframed}
+
+ $c | a$, i.e. $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$), i.e. if there exists $d$ such that $cd = a$.
+
+ $\gcd(a, b)$, the \defn{greatest common divisor} of $a$ and $b$, is the largest integer $c$ such that $c | a$ and $c | b$.
+
+  $a$ and $b$ are \defn{relatively prime} aka \defn{coprime} if $\gcd(a, b) = 1$.
+
+  $\lcm(c, d)$ is the smallest integer $a$ such that $c|a$ and $d|a$.
+\end{definition*}
+
+  Question: why exactly are we contemplating products here, as opposed to e.g. sums?
+
+
+
+
+\begin{remark*}
+  a \defn{multiple} of a prime is a number whose factorization contains that prime.
+
+  $c$ \defn{divides} $a$ ($a$ is a \defn{multiple} of $c$) if $a$'s factorization is a ``multiplicity superset​'' of $c$'s.
+
+  The $\gcd$ is the largest integer whose factorization is a ``subset​'' of both factorizations.
+
+  If $a$ and $b$ are relatively prime then their factorizations have no overlap.
+
+  The $\lcm$ is the smallest integer whose factorization is a ``superset​'' of both factorizations.
+
+  If $a$ is a multiple of $c$ then their $\lcm$ is $a$.
+
+  If $a$ and $b$ are relatively prime then their $\lcm$ is their product.
+\end{remark*}
+
+\begin{example*}
+Consider $2^2 \cdot 3 = 12$ and $2\cdot 3^2 = 18$.
+
+Their factorizations have much in common (they are certainly not coprime), but neither is a
+multiple of the other.
+
+Their $\lcm$ is the first number that is a ``multiplicity superset​'' of the other: i.e. we must add
+a factor of $3$ to $12$'s factorization, or an additional $2$ to $18$'s, either way
+yielding $2^2 \cdot 3^2 = 36$.
+
+Note that their $\lcm$ is not one of them (as it would be if one divided the other), but it is
+smaller than their product.
+
+The $\gcd$ of $2^2\cdot 3$ and $2\cdot 3^2$ is $2\cdot 3$.
+\end{example*}
+
+
+\begin{example*}
+    For example if
+  \begin{align*}
+    a = 12 &= 2^2 \cdot 3 \\
+    b = 40 &= 2^3 \cdot 5 \\
+  \end{align*}
+  then $\gcd(a, b) = 2^2 = 4$ and $\lcm(a, b) = 2^3 \cdot 3 \cdot 5 = 120$.
+
+  This can be written as a general theorem involving mins and maxes in the exponents of a product
+  of primes.
+\end{example*}
+
+
+\begin{theorem*}
+  $\gcd(a,b) \times \lcm(a,b) = ab$
+\end{theorem*}
+
+\begin{example*}
+  The product of  $2^2\cdot 3$ and $2\cdot 3^2$ is their concatenation: $2^2 \cdot 3 \cdot 2 \cdot 3^2 = 12 \cdot 18 = 216$.
+
+  This is the product of their $\gcd$ $2\cdot 3$ and their $\lcm$ $2^2\cdot 3^2$.
+\end{example*}
+
+
+\begin{intuition*}
+  We're performing additive operations on the exponents of the prime factors. The product is the
+  sum of all; the $\gcd$ takes the min from each and is thus missing the maxes; the $\lcm$ takes
+  the max from each and is thus missing the mins; their product contains the full multiplicities of
+  all factors.
+\end{intuition*}
+
+
+\begin{mdframed}
+\includegraphics[width=400pt]{img/foundations--set-theory--number-theory-82fa.png}
+\includegraphics[width=400pt]{img/foundations--set-theory--number-theory-179a.png}
+\end{mdframed}