noxz-sites

A collection of a builder and various scripts creating the noxz.tech sites
git clone git://git.noxz.tech/noxz-sites
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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

Diffstat:
Mnoxz.tech/dotfiles/index.md | 2+-
Mnoxz.tech/guides/mathematics/index.md | 887++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++-----------
Mnoxz.tech/pub/style.css | 30++++++++++++++++++++++++++----
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"}&#x2223;{--} -....{- class="variable"}n{--} +....{- class="variable"}a{--} ..{--} ..{- class="operator"}&hArr;{--} ..{- class="expression"} -....{- class="variable"}n{--} +....{- class="variable"}a{--} ....{- class="operator"}&isin;{--} ....{- class="variable"}&#x2124;{--} ..{--} {--} -### {- 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"}&rArr;{--} ..{- class="expression"} ....{-}2{--} ....{- class="operator"}&#x2223;{--} @@ -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"}&rArr;{--} -..{- class="expression"} -....{-}3{--} -....{- class="operator"}&#x2223;{--} -....{- class="variable"}abcd{--} -..{--} -..{- class="operator"}&hArr;{--} +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"}&#x2223;{--} ....{- 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"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} ....{- 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"}&rArr;{--} ..{- class="expression"} ....{-}4{--} ....{- class="operator"}&#x2223;{--} @@ -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"}&rArr;{--} ..{- class="expression"} ....{-}5{--} ....{- class="operator"}&#x2223;{--} @@ -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"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}3{--} +....{- class="operator"}&#x2223;{--} +....{- class="fenced parenthesis"} +......{-}({--} +......{- class="variable"}a{--} +......{- class="operator"}+{--} +......{- class="variable"}b{--} +......{- class="operator"}+{--} +......{- class="variable"}c{--} +......{- class="operator"}+{--} +......{- class="variable"}d{--} +......{-}){--} +....{--} +..{--} +..{- class="operator"}&and;{--} +..{- class="expression"} +....{-}2{--} +....{- class="operator"}&#x2223;{--} +....{- 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"}&#x2223;{--} +....{-}3423{--} +....{- class="operator"}&#63;{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}342{--} +....{- class="operator"}-{--} +....{-}3{--} +....{- class="operator"}&times;{--} +....{-}2{--} +....{- class="operator"}={--} +....{-}336{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}33{--} +....{- class="operator"}-{--} +....{-}6{--} +....{- class="operator"}&times;{--} +....{-}2{--} +....{- class="operator"}={--} +....{-}21{--} +..{--} +{--} + +{- class="math block"} +..{- class="expression"} +....{-}7{--} +....{- class="operator"}&#x2223;{--} +....{-}21{--} +..{--} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}7{--} +....{- class="operator"}&#x2223;{--} +....{-}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"}&#x2223;{--} +..{- class="expression"} +....{- class="variable"}a{--} +....{- class="operator"}-{--} +....{-}2{--} +....{- class="variable"}b{--} +..{--} +{--} then (***B***) + {- class="math"} +..{-}7{--} +..{- class="operator"}&#x2223;{--} +..{- 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"}&#x2223;{--} -....{- class="variable"}abcd{--} +....{- class="expression"} +......{- class="variable"}ab{--} +....{--} ..{--} ..{- class="operator"}&hArr;{--} ..{- class="expression"} -....{-}3{--} +....{-}7{--} ....{- class="operator"}&#x2223;{--} -....{- 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"}&and;{--} -..{- class="expression"} -....{-}2{--} -....{- class="operator"}&#x2223;{--} -....{- 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"}&rArr;{--} ..{- class="expression"} ....{-}8{--} ....{- class="operator"}&#x2223;{--} @@ -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"}&rArr;{--} -..{- class="expression"} -....{-}9{--} -....{- class="operator"}&#x2223;{--} -....{- class="variable"}abcd{--} -..{--} -..{- class="operator"}&hArr;{--} +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"}&#x2223;{--} ....{- 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"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} ..{- class="expression"} +....{-}9{--} +....{- class="operator"}&#x2223;{--} ....{- 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"}&rArr;{--} ..{- class="expression"} ....{-}10{--} ....{- class="operator"}&#x2223;{--} @@ -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"}&#x2223;{--} +....{-}190905{--} +....{- class="operator"}&#63;{--} +..{--} +{--} + +{- 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"}&#x2223;{--} +....{-}-22{--} +..{--} +..{- class="operator"}&rArr;{--} +..{- class="expression"} +....{-}11{--} +....{- class="operator"}&#x2223;{--} +....{-}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"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}11{--} +....{- class="operator"}&#x2223;{--} +....{- 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"}&#x2223;{--} +....{- class="variable"}abcd{--} +..{--} +..{- class="operator"}&hArr;{--} +..{- class="expression"} +....{-}11{--} +....{- class="operator"}&#x2223;{--} +....{- 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"}&rArr;{--} ..{- 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;