commit: 6c0949af0918ea9626c2cda1a9c5345d7c336660
parent: df03c50ac9de961054842bed71521dc82edf7785
author: Chris Noxz <chris@noxz.tech>
date: Tue, 24 Sep 2019 13:50:32 +0200
Fix some grammar, more theorems and some formatting
3 files changed, 799 insertions(+), 120 deletions(-)
diff --git a/noxz.tech/dotfiles/index.md b/noxz.tech/dotfiles/index.md
@@ -1,6 +1,6 @@
Dotfiles
========
-I've written a little tool to manage my dotfiles, in a way that symlinks
+I've written a little tool to manage my dotfiles, in a way that it symlinks
everything to where it should be. As I use patched versions of `st` and `dwm`
some parts of my dotfiles doesn't make sense if you don't use my versions of
`st` and `dwm`, but they are not prerequisites.
diff --git a/noxz.tech/guides/mathematics/index.md b/noxz.tech/guides/mathematics/index.md
@@ -8,45 +8,51 @@ out there.
{:: class="toc"}
{- class="toc-title"}Contents{--}
-+ 1 [Divisibility rules](#math-div-rules)
- + 1.1 [Divisibility by 1](#math-div-rules-1)
- + 1.2 [Divisibility by 2](#math-div-rules-2)
- + 1.3 [Divisibility by 3](#math-div-rules-3)
- + 1.4 [Divisibility by 4](#math-div-rules-4)
- + 1.5 [Divisibility by 5](#math-div-rules-5)
- + 1.6 [Divisibility by 6](#math-div-rules-6)
- + 1.7 [Divisibility by 8](#math-div-rules-8)
- + 1.8 [Divisibility by 9](#math-div-rules-9)
- + 1.9 [Divisibility by 10](#math-div-rules-10)
++ 1 [Divisibility theorems](#math-div-theorems)
+ + 1.1 [Divisibility by 1](#math-div-theorems-1)
+ + 1.2 [Divisibility by 2](#math-div-theorems-2)
+ + 1.3 [Divisibility by 3](#math-div-theorems-3)
+ + 1.4 [Divisibility by 4](#math-div-theorems-4)
+ + 1.5 [Divisibility by 5](#math-div-theorems-5)
+ + 1.6 [Divisibility by 6](#math-div-theorems-6)
+ + 1.7 [Divisibility by 7](#math-div-theorems-7)
+ + 1.8 [Divisibility by 8](#math-div-theorems-8)
+ + 1.9 [Divisibility by 9](#math-div-theorems-9)
+ + 1.10 [Divisibility by 10](#math-div-theorems-10)
+ + 1.11 [Divisibility by 11](#math-div-theorems-11)
+ 2 [Fractions](#math-fractions)
+ 2.1 [The fraction flip when dividing](#math-fractions-flip)
{::}
-{- id="math-div-rules"}Divisibility rules{--}
----------------------------------------------
-When it comes to divisibility there exists some neat methods to test a certain
-number's different divisibilities, or factors. Following are those methods and
+{- id="math-div-theorems"}Divisibility theorems{--}
+---------------------------------------------------
+When it comes to divisibility there exists some neat theorems to test a certain
+number's different divisibilities, or factors. Following are those theorems and
their proof. Some of the proofs are more trivial than others, such as the proof
for divisibility by 1, 2, 5 and 10.
-### {- id="math-div-rules-1"}Divisibility by 1{--}
-This rule is quite easy to remember. Every integer is divisible by 1.
+What is the smallest number that is divisible by 1 through 10? The answer is
+{- class="spoiler"}2520{--}. You can try the different divisibility theorems
+ below on it.
+
+### {- id="math-div-theorems-1"}Divisibility by 1{--}
+This theorem is quite easy to remember. Every integer is divisible by 1.
{- class="math block theorem"}
..{- class="expression"}
....{-}1{--}
....{- class="operator"}∣{--}
-....{- class="variable"}n{--}
+....{- class="variable"}a{--}
..{--}
..{- class="operator"}⇔{--}
..{- class="expression"}
-....{- class="variable"}n{--}
+....{- class="variable"}a{--}
....{- class="operator"}∈{--}
....{- class="variable"}ℤ{--}
..{--}
{--}
-### {- id="math-div-rules-2"}Divisibility by 2{--}
+### {- id="math-div-theorems-2"}Divisibility by 2{--}
It's common knowledge that every even number (numbers ending with an even
number) is divisible by 2. This is because even numbers are multiples of 2. In
short, if a number ends with either 0, 2, 4, 6 or 8 it is divisible by 2.
@@ -117,8 +123,10 @@ Say we have a four digit number
..{--}
{--}
+We can now see that the number is divisible by 2 if, and only if, the last
+digit is divisible by 2. And so the theorem is proven.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="expression"}
....{-}2{--}
....{- class="operator"}∣{--}
@@ -132,12 +140,9 @@ Say we have a four digit number
..{--}
{--}
-We can now see that the number is divisible by 2 if, and only if, the last
-digit is divisible by 2. And so the theorem is proven.
-
-### {- id="math-div-rules-3"}Divisibility by 3{--}
-The rule goes that if the sum of all digits in a number is divisible by 3, the
-whole number is divisible by 3, i.e.
+### {- id="math-div-theorems-3"}Divisibility by 3{--}
+The theorem goes that if the sum of all digits in a number is divisible by 3,
+the whole number is divisible by 3, i.e.
{- class="math"}
..{- class="expression"}
....{-}3{--}
@@ -260,17 +265,10 @@ Let's use the four digit number
..{--}
{--}
-{- class="math block theorem"}
-..{- class="operator"}⇒{--}
-..{- class="expression"}
-....{-}3{--}
-....{- class="operator"}∣{--}
-....{- class="variable"}abcd{--}
-..{--}
-..{- class="operator"}⇔{--}
+We can now see that the first term is divisible by 3, and the second term is
+divisible by 3 if, and only if, the sum
+ {- class="math"}
..{- class="expression"}
-....{-}3{--}
-....{- class="operator"}∣{--}
....{- class="fenced parenthesis"}
......{-}({--}
......{- class="variable"}a{--}
@@ -283,12 +281,18 @@ Let's use the four digit number
......{-}){--}
....{--}
..{--}
-{--}
+{--} is divisible by 3. And so the theorem is proven.
-We can now see that the first term is divisible by 3, and the second term is
-divisible by 3 if, and only if, the sum
- {- class="math"}
+{- class="math block theorem"}
..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
....{- class="fenced parenthesis"}
......{-}({--}
......{- class="variable"}a{--}
@@ -301,10 +305,10 @@ divisible by 3 if, and only if, the sum
......{-}){--}
....{--}
..{--}
-{--} is divisible by 3. And so the theorem is proven.
+{--}
-### {- id="math-div-rules-4"}Divisibility by 4{--}
-The rule goes that if the last two digits of a number is divisible by 4, the
+### {- id="math-div-theorems-4"}Divisibility by 4{--}
+The theorem goes that if the last two digits of a number is divisible by 4, the
whole number is divisible by 4, i.e.
{- class="math"}
..{- class="expression"}
@@ -382,8 +386,11 @@ Let's use the four digit number
..{--}
{--}
+We can now see that the first term is divisible by 4, so the whole number is
+divisible by 4 if, and only if, the second term is divisible by 4. And so the
+theorem is proven.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="expression"}
....{-}4{--}
....{- class="operator"}∣{--}
@@ -398,12 +405,8 @@ Let's use the four digit number
..{--}
{--}
-We can now see that the first term is divisible by 4, so the whole number is
-divisible by 4 if, and only if, the second term is divisible by 4. And so the
-theorem is proven.
-
-### {- id="math-div-rules-5"}Divisibility by 5{--}
-The rule goes that if the last digits of a number is divisible by 5, the
+### {- id="math-div-theorems-5"}Divisibility by 5{--}
+The theorem goes that if the last digits of a number is divisible by 5, the
whole number is divisible by 4, i.e.
{- class="math"}
..{- class="expression"}
@@ -481,8 +484,13 @@ Let's use the four digit number
..{--}
{--}
+We can now see that the first term is divisible by 5, so the whole number is
+divisible by 5 if, and only if, the second term is divisible by 5. And so the
+theorem is proven. As the only one digit numbers that are divisible by 5 are 0
+and 5, another way of putting it is -- if last digit is 0 or 5, the number is
+divisible by 5.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="expression"}
....{-}5{--}
....{- class="operator"}∣{--}
@@ -497,50 +505,444 @@ Let's use the four digit number
..{--}
{--}
-We can now see that the first term is divisible by 5, so the whole number is
-divisible by 5 if, and only if, the second term is divisible by 5. And so the
-theorem is proven. As the only one digit numbers that are divisible by 5 are 0
-and 5, another way of putting it is -- if last digit is 0 or 5, the number is
-divisible by 5.
+### {- id="math-div-theorems-6"}Divisibility by 6{--}
+This theorem is a combination of the theorem for [*divisibility by
+2*](#math-div-theorems-2) and [*divisibility by 3*](#math-div-theorems-3).
+
+{- class="math block theorem"}
+..{- class="expression"}
+....{-}6{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}3{--}
+....{- class="operator"}∣{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="variable"}a{--}
+......{- class="operator"}+{--}
+......{- class="variable"}b{--}
+......{- class="operator"}+{--}
+......{- class="variable"}c{--}
+......{- class="operator"}+{--}
+......{- class="variable"}d{--}
+......{-}){--}
+....{--}
+..{--}
+..{- class="operator"}∧{--}
+..{- class="expression"}
+....{-}2{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+### {- id="math-div-theorems-7"}Divisibility by 7{--}
+Probably one of the most useful theorems is the theorem of *divisibility by 7*,
+as it is recursive (just like the theorems of *divisibility by 3 & 9*). The
+theorem states that if the difference between the last digit multiplied by 2
+and the remaining digits in a number is divisible by 7, the whole number is
+divisible by 7. I'll show the procedure with an example below:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}7{--}
+....{- class="operator"}∣{--}
+....{-}3423{--}
+....{- class="operator"}?{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}342{--}
+....{- class="operator"}-{--}
+....{-}3{--}
+....{- class="operator"}×{--}
+....{-}2{--}
+....{- class="operator"}={--}
+....{-}336{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}33{--}
+....{- class="operator"}-{--}
+....{-}6{--}
+....{- class="operator"}×{--}
+....{-}2{--}
+....{- class="operator"}={--}
+....{-}21{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}7{--}
+....{- class="operator"}∣{--}
+....{-}21{--}
+..{--}
+..{- class="operator"}⇒{--}
+..{- class="expression"}
+....{-}7{--}
+....{- class="operator"}∣{--}
+....{-}3423{--}
+..{--}
+{--}
+
+Neat! So how and why does it work? For simplicity's sake we use a two digit
+number
+{- class="math"}
+..{- class="expression"}
+....{- class="variable"}ab{--}
+..{--}
+{--}, represented as
+ {- class="math"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+{--}. The theorem says that if (***A***)
+ {- class="math"}
+..{-}7{--}
+..{- class="operator"}∣{--}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}2{--}
+....{- class="variable"}b{--}
+..{--}
+{--} then (***B***)
+ {- class="math"}
+..{-}7{--}
+..{- class="operator"}∣{--}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+{--}. Let's prove it!
+
+In order to prove the theorem, we must prove both ***A*** and ***B***. So let's
+start with ***A***. If we have
+{- class="math"}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}2{--}
+....{- class="variable"}b{--}
+..{--}
+{--}, and it's divisible by 7, we know that 7 must be a factor of the
+expression. We can now create an equation:
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}2{--}
+....{- class="variable"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}7{--}
+....{- class="variable"}k{--}
+..{--}
+{--}
+
+Multiply the whole equation with 10, and add one extra
+{- class="math"}
+..{- class="expression"}
+....{- class="variable"}b{--}
+..{--}
+{--}:
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="hi"}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{- class="hi"}20{--}
+....{- class="variable"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{- class="hi"}70{--}
+....{- class="variable"}k{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}20{--}
+....{- class="variable"}b{--}
+....{- class="operator hi"}+{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}70{--}
+....{- class="variable"}k{--}
+....{- class="operator hi"}+{--}
+....{- class="variable hi"}b{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{- class="hi"}19{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}70{--}
+....{- class="variable"}k{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+{--}
+
+Now add {- class="math"}
+..{- class="expression"}
+....{-}20{--}
+....{- class="variable"}b{--}
+..{--}
+{--} to each side of the equation, and try to factor out 7:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}19{--}
+....{- class="variable"}b{--}
+....{- class="operator hi"}+{--}
+....{- class="hi"}20{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}70{--}
+....{- class="variable"}k{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+....{- class="operator hi"}+{--}
+....{- class="hi"}20{--}
+....{- class="variable hi"}b{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator hi"}+{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}70{--}
+....{- class="variable"}k{--}
+....{- class="operator hi"}+{--}
+....{- class="hi"}21{--}
+....{- class="variable hi"}b{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression hi"}
+....{-}7{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{-}10{--}
+......{- class="variable"}k{--}
+......{- class="operator"}+{--}
+......{-}3{--}
+......{- class="variable"}b{--}
+......{-}){--}
+....{--}
+..{--}
+{--}
+
+We can now see that the right side of the equation is divisible by 7, and our
+left side says
+{- class="math"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+{--}. Neat, we now know that the two digit number
+ {- class="math"}
+..{- class="expression"}
+....{- class="variable"}ab{--}
+..{--}
+{--} is divisible by 7. Now we must show that ***B*** implies ***A***. That is
+ if
+ {- class="math"}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}2{--}
+....{- class="variable"}b{--}
+..{--} is divisible by 7, then
+ {- class="math"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+{--} is divisible by 7. Let's prove ***B***.
+
+Just as for ***A***, we know that 7 must be a factor of the expression. We can
+now create another equation:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}7{--}
+....{- class="variable"}k{--}
+..{--}
+{--}
+
+Subtract
+{- class="math"}
+..{- class="expression"}
+....{-}21{--}
+....{- class="variable"}b{--}
+..{--}
+{--} from the whole equation, and factorize:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+....{- class="operator hi"}-{--}
+....{- class="hi"}21{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}7{--}
+....{- class="variable"}k{--}
+....{- class="operator hi"}-{--}
+....{- class="hi"}21{--}
+....{- class="variable hi"}b{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}10{--}
+....{- class="variable"}a{--}
+....{- class="operator hi"}-{--}
+....{- class="hi"}20{--}
+....{- class="variable hi"}b{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression"}
+....{-}7{--}
+....{- class="variable"}k{--}
+....{- class="operator"}-{--}
+....{-}21{--}
+....{- class="variable"}b{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression hi"}
+....{-}10{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="variable"}a{--}
+......{- class="operator"}-{--}
+......{-}2{--}
+......{- class="variable"}b{--}
+......{-}){--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="expression hi"}
+....{-}7{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="variable"}k{--}
+......{- class="operator"}-{--}
+......{-}3{--}
+......{- class="variable"}b{--}
+......{-}){--}
+....{--}
+..{--}
+{--}
-### {- id="math-div-rules-6"}Divisibility by 6{--}
-This rule is a combination of the rule for [*divisibility by
-2*](#math-div-rules-2) and [*divisibility by 3*](#math-div-rules-3).
+We can now see that the right side of the equation is divisible by 7, and on
+our left side 10 is not divisible by 7 so the expression inside the
+parenthesis must be. But isn't that expression
+{- class="math"}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{-}2{--}
+....{- class="variable"}b{--}
+..{--}
+{--}. Neat, we now have the proof for the theorem and can conclude that
+ indeed:
{- class="math block theorem"}
..{- class="expression"}
-....{-}6{--}
+....{-}7{--}
....{- class="operator"}∣{--}
-....{- class="variable"}abcd{--}
+....{- class="expression"}
+......{- class="variable"}ab{--}
+....{--}
..{--}
..{- class="operator"}⇔{--}
..{- class="expression"}
-....{-}3{--}
+....{-}7{--}
....{- class="operator"}∣{--}
-....{- class="fenced parenthesis"}
-......{-}({--}
+....{- class="expression"}
......{- class="variable"}a{--}
-......{- class="operator"}+{--}
+......{- class="operator"}-{--}
+......{-}2{--}
......{- class="variable"}b{--}
-......{- class="operator"}+{--}
-......{- class="variable"}c{--}
-......{- class="operator"}+{--}
-......{- class="variable"}d{--}
-......{-}){--}
....{--}
..{--}
-..{- class="operator"}∧{--}
-..{- class="expression"}
-....{-}2{--}
-....{- class="operator"}∣{--}
-....{- class="variable"}d{--}
-..{--}
{--}
-### {- id="math-div-rules-8"}Divisibility by 8{--}
-The rule is quite similar to the rule for *divisibility by 4*. The rule goes
-that if the last three digits of a number is divisible by 8, the whole number
-is divisible by 8, i.e.
+We have shown that the procedure above will hold for all cases.
+
+### {- id="math-div-theorems-8"}Divisibility by 8{--}
+The theorem is quite similar to the theorem for *divisibility by 4*. The
+theorem goes that if the last three digits of a number is divisible by 8, the
+whole number is divisible by 8, i.e.
{- class="math"}
..{- class="expression"}
....{-}8{--}
@@ -617,8 +1019,11 @@ Let's use the four digit number
..{--}
{--}
+We can now see that the first term is divisible by 8, so the whole number is
+divisible by 8 if, and only if, the second term is divisible by 8. And so the
+theorem is proven.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="expression"}
....{-}8{--}
....{- class="operator"}∣{--}
@@ -633,14 +1038,10 @@ Let's use the four digit number
..{--}
{--}
-We can now see that the first term is divisible by 8, so the whole number is
-divisible by 8 if, and only if, the second term is divisible by 8. And so the
-theorem is proven.
-
-### {- id="math-div-rules-9"}Divisibility by 9{--}
-Much like the rule for *divisibility by 3*, the rule goes that if the sum of
-all digits in a number is divisible by 9, the whole number is divisible by 9,
-i.e.
+### {- id="math-div-theorems-9"}Divisibility by 9{--}
+Much like the theorem for *divisibility by 3*, the theorem goes that if the sum
+of all digits in a number is divisible by 9, the whole number is divisible by
+9, i.e.
{- class="math"}
..{- class="expression"}
....{-}9{--}
@@ -763,17 +1164,10 @@ Let's use the four digit number
..{--}
{--}
-{- class="math block theorem"}
-..{- class="operator"}⇒{--}
-..{- class="expression"}
-....{-}9{--}
-....{- class="operator"}∣{--}
-....{- class="variable"}abcd{--}
-..{--}
-..{- class="operator"}⇔{--}
+We can now see that the first term is divisible by 9, and the second term is
+divisible by 9 if, and only if, the sum
+ {- class="math"}
..{- class="expression"}
-....{-}9{--}
-....{- class="operator"}∣{--}
....{- class="fenced parenthesis"}
......{-}({--}
......{- class="variable"}a{--}
@@ -786,12 +1180,18 @@ Let's use the four digit number
......{-}){--}
....{--}
..{--}
-{--}
+{--} is divisible by 9. And so the theorem is proven.
-We can now see that the first term is divisible by 9, and the second term is
-divisible by 9 if, and only if, the sum
- {- class="math"}
+{- class="math block theorem"}
+..{- class="expression"}
+....{-}9{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
..{- class="expression"}
+....{-}9{--}
+....{- class="operator"}∣{--}
....{- class="fenced parenthesis"}
......{-}({--}
......{- class="variable"}a{--}
@@ -804,10 +1204,10 @@ divisible by 9 if, and only if, the sum
......{-}){--}
....{--}
..{--}
-{--} is divisible by 9. And so the theorem is proven.
+{--}
-### {- id="math-div-rules-10"}Divisibility by 10{--}
-The rule goes that if the last digits of a number is divisible by 10, the
+### {- id="math-div-theorems-10"}Divisibility by 10{--}
+The theorem goes that if the last digits of a number is divisible by 10, the
whole number is divisible by 10, i.e.
{- class="math"}
..{- class="expression"}
@@ -884,8 +1284,13 @@ Let's use the four digit number
..{--}
{--}
+We can now see that the first term is divisible by 10, so the whole number is
+divisible by 10 if, and only if, the second term is divisible by 10. And so the
+theorem is proven. As the only one digit number that is divisible by 10 is 0,
+another way of putting it is -- if last digit is 0, the number is
+divisible by 10.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="expression"}
....{-}10{--}
....{- class="operator"}∣{--}
@@ -900,11 +1305,264 @@ Let's use the four digit number
..{--}
{--}
-We can now see that the first term is divisible by 10, so the whole number is
-divisible by 10 if, and only if, the second term is divisible by 10. And so the
-theorem is proven. As the only one digit number that is divisible by 10 is 0,
-another way of putting it is -- if last digit is 0, the number is
-divisible by 10.
+### {- id="math-div-theorems-11"}Divisibility by 11{--}
+The theorem goes that an integer number is divisible by 11 if, and only if, the
+alternate sum of its digits is divisible by 11, like so:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{-}190905{--}
+....{- class="operator"}?{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}1{--}
+....{- class="operator"}-{--}
+....{-}9{--}
+....{- class="operator"}+{--}
+....{-}0{--}
+....{- class="operator"}-{--}
+....{-}9{--}
+....{- class="operator"}+{--}
+....{-}0{--}
+....{- class="operator"}-{--}
+....{-}5{--}
+....{- class="operator"}={--}
+....{-}-22{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{-}-22{--}
+..{--}
+..{- class="operator"}⇒{--}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{-}190905{--}
+..{--}
+{--}
+
+Neat! So how and why does it work? For simplicity's sake we use a four digit
+number
+{- class="math"}
+..{- class="expression"}
+....{- class="variable"}abcd{--}
+..{--}
+{--}, represented as
+ {- class="math"}
+..{- class="expression"}
+....{-}1000{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{-}100{--}
+....{- class="variable"}b{--}
+....{- class="operator"}+{--}
+....{-}10{--}
+....{- class="variable"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}. This expression can also be represented in another way by manipulating
+the terms. We give and take in an alternating fashion, like so:
+
+{- class="math block"}
+..{- class="expression"}
+....{-}1000{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{-}100{--}
+....{- class="variable"}b{--}
+....{- class="operator"}+{--}
+....{-}10{--}
+....{- class="variable"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="variable hi"}a{--}
+....{- class="fenced parenthesis hi"}
+......{-}({--}
+......{-}1000{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable hi"}b{--}
+....{- class="fenced parenthesis hi"}
+......{-}({--}
+......{-}100{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable hi"}c{--}
+....{- class="fenced parenthesis hi"}
+......{-}({--}
+......{-}10{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="variable"}a{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="hi"}1001{--}
+......{- class="operator hi"}-{--}
+......{- class="hi"}1{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="hi"}99{--}
+......{- class="operator hi"}+{--}
+......{- class="hi"}1{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable"}c{--}
+....{- class="fenced parenthesis"}
+......{-}({--}
+......{- class="hi"}11{--}
+......{- class="operator hi"}-{--}
+......{- class="hi"}1{--}
+......{-}){--}
+....{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="hi"}1001{--}
+....{- class="variable hi"}a{--}
+....{- class="operator hi"}-{--}
+....{- class="variable hi"}a{--}
+....{- class="operator"}+{--}
+....{- class="hi"}99{--}
+....{- class="variable hi"}b{--}
+....{- class="operator hi"}+{--}
+....{- class="variable hi"}b{--}
+....{- class="operator"}+{--}
+....{- class="hi"}11{--}
+....{- class="variable hi"}c{--}
+....{- class="operator hi"}-{--}
+....{- class="variable hi"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="expression"}
+....{-}1001{--}
+....{- class="variable"}a{--}
+....{- class="operator hi"}+{--}
+....{- class="hi"}99{--}
+....{- class="variable hi"}b{--}
+....{- class="operator hi"}+{--}
+....{- class="hi"}11{--}
+....{- class="variable hi"}c{--}
+....{- class="operator hi"}-{--}
+....{- class="variable hi"}a{--}
+....{- class="operator hi"}+{--}
+....{- class="variable hi"}b{--}
+....{- class="operator hi"}-{--}
+....{- class="variable hi"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+Now we factorize the expression, like so:
+
+{- class="math block"}
+..{- class="expression"}
+....{- class="hi"}11{--}
+....{- class="fenced parenthesis hi"}
+......{-}({--}
+......{-}91{--}
+......{- class="variable"}a{--}
+......{- class="operator"}+{--}
+......{-}9{--}
+......{- class="variable"}b{--}
+......{- class="operator"}+{--}
+......{-}1{--}
+......{- class="variable"}c{--}
+......{-}){--}
+....{--}
+....{- class="operator"}-{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+....{- class="operator"}-{--}
+....{- class="variable"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+We can see that the first term in the expression is divisible by 11. This means
+that if, and only if, the sum of the other terms is divisible by 11 the whole
+expression is divisible by 11, and so the theorem is proven.
+
+{- class="math block theorem"}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{- class="operator"}-{--}
+....{- class="variable"}a{--}
+....{- class="operator"}+{--}
+....{- class="variable"}b{--}
+....{- class="operator"}-{--}
+....{- class="variable"}c{--}
+....{- class="operator"}+{--}
+....{- class="variable"}d{--}
+..{--}
+..{- class="break"}{--}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}abcd{--}
+..{--}
+..{- class="operator"}⇔{--}
+..{- class="expression"}
+....{-}11{--}
+....{- class="operator"}∣{--}
+....{- class="variable"}a{--}
+....{- class="operator"}-{--}
+....{- class="variable"}b{--}
+....{- class="operator"}+{--}
+....{- class="variable"}c{--}
+....{- class="operator"}-{--}
+....{- class="variable"}d{--}
+..{--}
+{--}
+
+We have shown that the procedure above will hold for all cases, as the integer
+can be extended with infinite digits and still follow the same pattern.
{- id="math-fractions"}Fractions{--}
------------------------------------
@@ -1288,8 +1946,9 @@ know this should result in one half.
..{--}
{--}
+Q.E.D.
+
{- class="math block theorem"}
-..{- class="operator"}⇒{--}
..{- class="fraction"}
....{- class="variable"}a{--}
....{- class="variable"}b{--}
@@ -1310,5 +1969,3 @@ know this should result in one half.
....{- class="variable"}c{--}
..{--}
{--}
-
-Q.E.D.
diff --git a/noxz.tech/pub/style.css b/noxz.tech/pub/style.css
@@ -2,6 +2,8 @@ body {
background-color: #eee;
color : #222;
font-family : monospace, sans-serif;
+ font-size : 1em;
+ line-height : 2em;
padding : 0;
margin : 0;
}
@@ -22,6 +24,7 @@ a:hover {
border-bottom : solid 2px #526587;
background : #333a56;
height : 2.75em;
+ font-size : 1.3em;
line-height : 2.75em;
padding : 0 1.35ex;
}
@@ -70,19 +73,21 @@ h4 {
p, li {
color : #444;
- line-height : 1.5em;
+ /*line-height : 2em;*/
}
code {
font-family : monospace;
background-color: #efefef;
- padding : 0 4px;
+ padding : 0.3em;
}
pre {
font-family : monospace;
+ font-size : 1em;
+ /*line-height : 2em;*/
background-color: #efefef;
- padding : 10px;
+ padding : 1em;
overflow : auto;
}
@@ -104,6 +109,7 @@ pre code {
float : left;
margin : 0 1px 0 0;
padding : 1em 0;
+ line-height : 1.5em;
border-right : 1px dotted #ccc;
width : 200px;
}
@@ -183,6 +189,16 @@ pre code {
text-align : center;
}
+.spoiler {
+ color : #000;
+ background : #000;
+}
+
+.spoiler:hover {
+ color : inherit;
+ background : inherit;
+}
+
.article h1 {
margin-bottom : 0em;
}
@@ -210,6 +226,7 @@ ul.repo-log {
list-style-type : none;
margin : 0;
padding : 0;
+ line-height : 1.5em;
}
ul.repo-log li {
@@ -219,7 +236,7 @@ ul.repo-log li {
ul.repo-log li .log-date {
font-weight : bold;
- padding-right : 10px;
+ padding-right : 1em;
}
/* table of contents */
@@ -262,6 +279,11 @@ ul.repo-log li .log-date {
margin-left : 2em;
}
+.math .break:before {
+ content : '\A';
+ white-space : pre-line;
+}
+
.math.theorem {
padding : 1em 1.5em;
border : 1px double #000;
