Add more missing stuff especially concerning theorems

notes/20241016120254.md

@@ -1,4 +1,4 @@
-# What is a remainder class?
-A remainder class $\overline{a}$ is a set [[20240929155814]] of numbers, that produce the same remainder when divided by a number $m$.
+# What is a remainder system?
+A remainder system $\overline{a}$ is a set [[20240929155814]] of numbers, that produce the same remainder when divided by a number $m$.
 
 #math #adm #numbertheory

notes/20241016120537.md

@@ -1,5 +1,5 @@
-# What rules are there for working with remainder classes? 
-Addition and subtraction work without any restrictions for remainder classes [[20241016120254]]. (Addition working also implies that subtraction works without problems)
+# What rules are there for working with remainder systems? 
+Addition and subtraction work without any restrictions for remainder systems [[20241016120254]]. (Addition working also implies that subtraction works without problems)
 $\overline{a}+\overline{b} = \overline{a+b}$
 $\overline{a} * \overline{b} = \overline{a*b}$
 

notes/20241016120945.md

@@ -1,5 +1,5 @@
-# What is the multiplicative inverse of a remainder class? 
-The multiplicative inverse $\overline{b}$ of a remainder class [[20241016120254]] $\overline{a}$ is defined as the following:
+# What is the multiplicative inverse of a remainder system? 
+The multiplicative inverse $\overline{b}$ of a remainder system [[20241016120254]] $\overline{a}$ is defined as the following:
 $\overline{a}*\overline{b}=\overline{1}$
 
 #math #adm #numbertheory

notes/20241016121406.md

@@ -1,4 +1,4 @@
-# When does a remainder class have a multiplicative inverse ? 
-A remainder class [[20241016120254]] $\overline{a}$ has a multiplicative inverse[[20241016120945]], if $ggT(a,m)=1$ (m is the module).
+# When does a remainder system have a multiplicative inverse ? 
+A remainder system [[20241016120254]] $\overline{a}$ has a multiplicative inverse[[20241016120945]], if $ggT(a,m)=1$ (m is the module).
 
-#math #adm #numbertheory1
+#math #adm #numbertheory

notes/20241016121548.md

@@ -1,7 +1,7 @@
-# How to find the multiplicative inverse of a remainder class? 
+# How to find the multiplicative inverse of a remainder system? 
 
 
-To find the multiplicative inverse [[20241016121548]] of a remainder class [[20241016120254]] $\overline{a} \in \mathbb{Z}_m$
+To find the multiplicative inverse [[20241016121548]] of a remainder system [[20241016120254]] $\overline{a} \in \mathbb{Z}_m$
 1. Check if $ggT(a,m)=1$
 2. If so use the euclidian algorithm to find $p,q \in \mathbb{Z} =pb + qm$
 3. From that you find, that $\overline p$ is the multiplicative inverse of $\overline a$

notes/20241109144025.md

@@ -0,0 +1,5 @@
+# What theorems are there for decimal expansions? 
+
+- The decimal expansion [[20241002204312]] for a rational number [[20240910110436]] is either finite or periodic.
+
+#math #adm

notes/20241109144212.md

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+# What theorems are there for the ggT? 
+- If you run the euclidean algorithm [[20241010161911]] it has to stop at some point because $b>r_0>r_1>r_2>...>=0$. The last remainder is called the greatest common denominator
+- If d is the greatest common denominator of the non-zero numbers a and b, there are numbers e and f, so that $ea + fb = d$. These can easily be calculated using the euclidean algorithm [[20241010161911]]
+
+#math #adm #numbertheory

notes/20241109144704.md

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+# What theorems are there for congruencies/remainder systems? 
+
+- For every positive natural number [[20240910105916]] m there are exactly m remainder systems [[20241016120254]].
+- A remainder system has a multiplicative inverse [[20241016120945]] if and only if ggT(a,m) is 1
+- Let $m=p_1^{e_1}...p_r^{e_r}$ be the prime factorization of m, then $\varphi(m)=m(1-\frac{1}{p_1})...(1-\frac{1}{p_r})$ ($\varphi(m)$ is eulers function [[20241109145707]])
+- For two coprime [[20241010161419]] numbers a,m $a^{\varphi(m)}\equiv 1 \mod m$ is true

notes/20241109145707.md

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+# What is eulers function? 
+Eulers function $\varphi(m)$ returns the number of invertable remainder systems [[20241016120945]] modulo m
+$\varphi(m)=|\{a\in \mathbb{Z} | 1 ≤ a ≤ m, ggT(a,m)=1\}|$. 
+
+#math #adm #numbertheory

notes/20241109150134.md

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+# What is the inclusion-exclusion-principle? 
+The inclusion exclusion principle is concerned with calculating the cardinality of the union of sets with shared elements.
+
+In general the formula is as follows:
+$|\bigcup_{i=1}^n A_i|=\sum_{\emptyset\not=I\subset{1,2,...,n}}^{n} (-1)^{|I|-1} |\bigcap_{i\in I} A_i|$
+
+and 
+
+$|\bigcap_{i=1}^n A'_i|=|(\bigcup_{i=1}^n A_i)'|=\sum_{I\subset \{1,2,...,n\}} (-1)^{|I|} |\bigcap_{i\in I} A_i|$
+
+#math #combinatorics #adm