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 d5c3946ad53f5564489f01d67beb4df606079c3e
parent 1b720a34687959e076a75ee7e4cb0383c6626e49
Author: Chris Noxz <chris@noxz.tech>
Date:   Sat, 21 Sep 2019 10:56:29 +0200

Add guides page for mathematics and fix some grammatical stuff

Diffstat:
Mnoxz.tech/guides/groff/index.md | 4++--
Mnoxz.tech/guides/index.md | 9++++++++-
Anoxz.tech/guides/mathematics/index.md | 391+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
3 files changed, 401 insertions(+), 3 deletions(-)

diff --git a/noxz.tech/guides/groff/index.md b/noxz.tech/guides/groff/index.md @@ -4,8 +4,8 @@ As these guides are created for *groff*, I cannot guarantee that they are applicable to *troff*. If you use *troff* and can confirm them to work for it -- please let me know. -Apart from *groff* these guides may include other programs used together with -*groff*, such as *eqn*, *tbl*, and so on. I will try make this clear in the +Apart from *groff*, these guides may include other programs used together with +*groff*, such as *eqn*, *tbl*, and so on. I will try to make this clear in the description of each guide. Crossing out expressions in *eqn* diff --git a/noxz.tech/guides/index.md b/noxz.tech/guides/index.md @@ -3,7 +3,14 @@ Guides The following is a collection of guides, hacks, HOWTOs, tips and tricks -- I have acquired over the years -- for various programs and concepts. I will fill this list over a period of time, so it's far from complete -- and probably -won't be. +never will be. + +Concepts +-------- +{:: class="definition"} +**[Mathematics](/guides/mathematics)** as for many hackers, is an interesting +and fascinating topic. It's literally a language to explain everything. +{::} Programs -------- diff --git a/noxz.tech/guides/mathematics/index.md b/noxz.tech/guides/mathematics/index.md @@ -0,0 +1,391 @@ +Mathematics +=========== +Under this topic I will not just show tricks, as the notion of tricks in +mathematics is probably the most damning to the spread of mathematical +knowledge. I will instead explain how these, so called, tricks work using +logic and reason. So this topic is not for the savvy hackers or mathematicians +out there. + +Fractions +--------- + +### The fraction flip when dividing +Many of you have probably been taught the trick of flipping the right fraction +in a division to instead use simpler multiplication. This is how it works: + +We start of with the division of +{- class="math"} +..{- class="fraction"} +....{-}3{--} +....{-}2{--} +..{--} +{--} + by +{- class="math"} +..{- class="fraction"} +....{-}4{--} +....{-}5{--} +..{--} +{--}. From there we can reconstruct the two fractions as the dividend over +the divisor with a horizontal line, for simplicity's sake, like so: + +{- class="math block"} +..{- class="fraction"} +....{-}3{--} +....{-}2{--} +..{--} +..{- class="operator"}&divide;{--} +..{- class="fraction"} +....{-}4{--} +....{-}5{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}4{--} +........{-}5{--} +......{--} +....{--} +..{--} +{--} + +After that the "trick" can begin. First we multiply both the dividend and the +divisor with the inverse of the divisor. As long as we treat the dividend and +the divisor the same way, this is fine. Be aware of PEMDAS though! If any of +the dividend or the divisor would have been an addition or subtraction you +would have to multiply both terms by *x*, either by *x(a + b)* or *xa + xb*. + +{- class="math block"} +..{- class="fraction"} +....{-}3{--} +....{-}2{--} +..{--} +..{- class="operator"}&divide;{--} +..{- class="fraction"} +....{-}4{--} +....{-}5{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}4{--} +........{-}5{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +......{- class="operator hi"}&times;{--} +......{- class="fraction hi"} +........{-}5{--} +........{-}4{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}4{--} +........{-}5{--} +......{--} +......{- class="operator hi"}&times;{--} +......{- class="fraction hi"} +........{-}5{--} +........{-}4{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}5{--} +........{-}4{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}4{- class="operator"}&times;{--}5{--} +........{-}5{- class="operator"}&times;{--}4{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}5{--} +........{-}4{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}20{--} +........{-}20{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}3{--} +........{-}2{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}5{--} +........{-}4{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}1{--} +........{- class="hidden"}&nbsp;{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}3{--} +....{-}2{--} +..{--} +..{- class="operator"}&times;{--} +..{- class="fraction"} +....{-}5{--} +....{-}4{--} +..{--} +{--} + +As you can see the right fraction has now "flipped", and not by magic, but with +logic and reason. As division by 1 is equal to the dividend, we can then solve +the expression, like so: + +{- class="math block"} +..{- class="fraction"} +....{-}3{--} +....{-}2{--} +..{--} +..{- class="operator"}&times;{--} +..{- class="fraction"} +....{-}5{--} +....{-}4{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}3{- class="operator"}&times;{--}5{--} +....{-}2{- class="operator"}&times;{--}4{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}15{--} +....{-}8{--} +..{--} +{--} + +As the final cherry on top, we can prove the procedure by dividing 1 by 2 as we +know this should result in one half. + +{- class="math block"} +..{-}1{--} +..{- class="operator"}&divide;{--} +..{-}2{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}1{--} +....{-}1{--} +..{--} +..{- class="operator"}&divide;{--} +..{- class="fraction"} +....{-}2{--} +....{-}1{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}2{--} +........{-}1{--} +......{--} +....{--} +..{--} +{--} + +{- class="math block"} +..{- class="fraction"} +....{-}1{--} +....{-}1{--} +..{--} +..{- class="operator"}&divide;{--} +..{- class="fraction"} +....{-}2{--} +....{-}1{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}2{--} +........{-}1{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +......{- class="operator hi"}&times;{--} +......{- class="fraction hi"} +........{-}1{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction"} +........{-}2{--} +........{-}1{--} +......{--} +......{- class="operator hi"}&times;{--} +......{- class="fraction hi"} +........{-}1{--} +........{-}2{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}1{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}2{- class="operator"}&times;{--}1{--} +........{-}1{- class="operator"}&times;{--}2{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}1{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}2{--} +........{-}2{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-} +......{- class="fraction"} +........{-}1{--} +........{-}1{--} +......{--} +......{- class="operator"}&times;{--} +......{- class="fraction"} +........{-}1{--} +........{-}2{--} +......{--} +....{--} +....{-} +......{- class="fraction hi"} +........{-}1{--} +........{- class="hidden"}&nbsp;{--} +......{--} +....{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}1{--} +....{-}1{--} +..{--} +..{- class="operator"}&times;{--} +..{- class="fraction"} +....{-}1{--} +....{-}2{--} +..{--} +{--} + +{- class="math block"} +..{- class="fraction"} +....{-}1{--} +....{-}1{--} +..{--} +..{- class="operator"}&times;{--} +..{- class="fraction"} +....{-}1{--} +....{-}2{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}1{- class="operator"}&times;{--}1{--} +....{-}1{- class="operator"}&times;{--}2{--} +..{--} +..{- class="operator"}={--} +..{- class="fraction"} +....{-}1{--} +....{-}2{--} +..{--} +{--} + +Q.E.D.