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
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"}÷{--}
+..{- 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"}÷{--}
+..{- 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"}×{--}
+......{- class="fraction hi"}
+........{-}5{--}
+........{-}4{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction"}
+........{-}4{--}
+........{-}5{--}
+......{--}
+......{- class="operator hi"}×{--}
+......{- class="fraction hi"}
+........{-}5{--}
+........{-}4{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}3{--}
+........{-}2{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}5{--}
+........{-}4{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}4{- class="operator"}×{--}5{--}
+........{-}5{- class="operator"}×{--}4{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}3{--}
+........{-}2{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}5{--}
+........{-}4{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}20{--}
+........{-}20{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}3{--}
+........{-}2{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}5{--}
+........{-}4{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}1{--}
+........{- class="hidden"} {--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}3{--}
+....{-}2{--}
+..{--}
+..{- class="operator"}×{--}
+..{- 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"}×{--}
+..{- class="fraction"}
+....{-}5{--}
+....{-}4{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}3{- class="operator"}×{--}5{--}
+....{-}2{- class="operator"}×{--}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"}÷{--}
+..{-}2{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}1{--}
+....{-}1{--}
+..{--}
+..{- class="operator"}÷{--}
+..{- 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"}÷{--}
+..{- 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"}×{--}
+......{- class="fraction hi"}
+........{-}1{--}
+........{-}2{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction"}
+........{-}2{--}
+........{-}1{--}
+......{--}
+......{- class="operator hi"}×{--}
+......{- class="fraction hi"}
+........{-}1{--}
+........{-}2{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}1{--}
+........{-}1{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}1{--}
+........{-}2{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}2{- class="operator"}×{--}1{--}
+........{-}1{- class="operator"}×{--}2{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}1{--}
+........{-}1{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}1{--}
+........{-}2{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}2{--}
+........{-}2{--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}
+......{- class="fraction"}
+........{-}1{--}
+........{-}1{--}
+......{--}
+......{- class="operator"}×{--}
+......{- class="fraction"}
+........{-}1{--}
+........{-}2{--}
+......{--}
+....{--}
+....{-}
+......{- class="fraction hi"}
+........{-}1{--}
+........{- class="hidden"} {--}
+......{--}
+....{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}1{--}
+....{-}1{--}
+..{--}
+..{- class="operator"}×{--}
+..{- class="fraction"}
+....{-}1{--}
+....{-}2{--}
+..{--}
+{--}
+
+{- class="math block"}
+..{- class="fraction"}
+....{-}1{--}
+....{-}1{--}
+..{--}
+..{- class="operator"}×{--}
+..{- class="fraction"}
+....{-}1{--}
+....{-}2{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}1{- class="operator"}×{--}1{--}
+....{-}1{- class="operator"}×{--}2{--}
+..{--}
+..{- class="operator"}={--}
+..{- class="fraction"}
+....{-}1{--}
+....{-}2{--}
+..{--}
+{--}
+
+Q.E.D.
