% This is a demonstration file distributed with the
% Lecturer package (see lecturer-doc.pdf).
%
% You can recompile the file with a basic TeX implementation,
% using pdfTeX or LuaTeX with the plain format.
%
% The reusable part ends somewhere around line 150.
%
% Author: Paul Isambert.
% Date: July 2010.
\input lecturer
%\showgrid{1cm}
\setparameter job:
fullscreen = true
font = \mainfont
\setparameter slide:
width height = 12cm
top = 0cm
bottom = 3cm
left = 5cm
right = 1.6cm
topskip = 0pt
background = black % For the lines between squares.
foreground = white
vpos = center
hpos = ff
\setarea{area1 area2 matharea}
width = 3cm
\setarea{area1 area2}
height = 4.3cm
background = white
\setarea{area2}
vshift = 4.7cm
\setarea{matharea footnotearea}
height = 2.6cm
vshift* = 0pt
topskip = .1cm
baselineskip = .333cm
vpos = center
\setarea{matharea}
background = blue
foreground = white
hpos = rr
% Below the slide's text.
\setarea{mainarea}
hshift = 3.4cm
hshift* = 0pt
height = 9cm
background = red
\setarea{footnotearea}
hshift = 3.4cm
hshift* = 1.8cm
left right = .3cm
background = white
font = \footnotefont
hpos = ff
\setarea{area6 area7}
width = 1.4cm
hshift* = 0pt
height = 1.1cm
\setarea{area6}
vshift = 9.4cm
background = white
\setarea{area7}
vshift* = 0pt
background = yellow
\setparameter step:
vskip = \baselineskip
\setstep\emptystep
vskip = 0pt
% Used for the maths.
\def\mathstep#1 #2 #3{\emptystep[on=#1,off=#2]\position{matharea}[0pt,0pt]{$#3$}\ignorespaces}
\def\Mathstep#1 #2{\emptystep[off=#1,visible=true]\position{matharea}[0pt,0pt]{$#2$}\ignorespaces}
% The footnote. Since the same name is given
% to each footnote step, there must be at most
% one per slide. Otherwise, a name should be given
% manually (or automatically with a number in the name,
% and a counter to increment it on every footnote call).
\def\footnote#1{%
\showorhide{toggle=footnote}{\super{\usecolor{blue}{*}}}
\emptystep[footnote,on=]
\position{footnotearea}{\quitvmode\llap{* }#1}%
}
% On the last equation.
\newsymbol\cross[1.7em]{%
pen 0.1,
move 0 -.5, .5 .9, stroke,
move 0 .9, .5 -.5, stroke,
}
% Font for text.
\font\mainfont=cmss10
\font\supmainfont=cmss10 at 7.5pt
% Simple superscript (I'm not very good with
% math font family... I'm no mathematician after all).
\def\super#1{\raise.3em\hbox{\supmainfont#1}}
\font\footnotefont=cmss8 at 7pt
\mainfont
\frenchspacing
% Font for maths.
\font\mathfont=cmss12 at 16pt
\font\scriptmathfont=cmss12 at 10pt
\font\scriptscriptmathfont=cmss12 at 8pt
\textfont0=\mathfont
\scriptfont0=\scriptmathfont
\scriptscriptfont0=\scriptscriptmathfont
\font\mathfont=cmss12 at 16pt
\font\scriptmathfont=cmss12 at 12pt
\font\scriptscriptmathfont=cmss12 at 8pt
\textfont1=\mathfont
\scriptfont1=\scriptmathfont
\scriptscriptfont1=\scriptscriptmathfont
\font\Tensy=cmsy10 at 18pt
\font\scriptTensy=cmsy10 at 10pt
\font\scriptscriptTensy=cmsy10 at 8pt
\textfont2=\Tensy
\scriptfont2=\scriptTensy
\scriptscriptfont2=\scriptscriptTensy
\font\Tenex=cmex10 at 18pt
\font\scriptTenex=cmex10 at 12pt
\font\scriptscriptTenex=cmex10 at 8pt
\textfont3=\Tenex
\scriptfont3=\scriptTenex
\scriptscriptfont3=\scriptscriptTenex
% \endinput
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% UNCOMMENT THE PREVIOUS LINE TO USE THIS FILE AS A TEMPLATE, %
% OR REMOVE EVERYTHING BELOW. %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\slide[A simple assumption]
\mathstep A B {{a\over b}=\sqrt2}
\mathstep B C {({a\over b})^2=2}
\mathstep C D {{a^2\over b^2}=2}
\mathstep D {} {a^2=2b^2}
\step Some say the square root of 2 isn't rational.
\step Suppose it were.
\step[A] Then we could write it as this, where a and b are
integers without a common factor.%
\footnote{%
Suppose c/d is a rational number. If c and d have no common
factor, then a=b and b=d. If they have a common factor,
divide both by their greatest common divisor. The result
is a/b, with no common factor.}
\step[B] But then we can also write this.
\step[C] And this.
\step[D] And finally this.
\step Which means that a\super2 is even.
\endslide
\slide[Its consequences]
\Mathstep A {a^2=2b^2}
\mathstep A B {(2k)^2=2b^2}
\mathstep B C {4k^2=2b^2}
\mathstep C D {2k^2=b^2}
\step[visible=true] So what?
\step So a is even. Because only even numbers produce even squares.%
\footnote{%
An even number, by definition, is expressible
in the form 2k, where k is any integer.
On the other hand, an odd number is expressible
by 2k+1. Thus the square of an odd number is
(2k+1)\super2, i.e. 4k\super2+4k+1, i.e.
2x2(k\super2+k)+1, which is of the form 2k+1,
with 2(k\super2+k) as k.
Hence, an odd number produces an odd square,
and thus if a square is even its root is even too.}
\step[A] Being even means being expressible in the form 2k, where k is
any integer.
\step[B] And (2k)\super2 square gives 4k\super2.
\step[C] Let's simplify.
\step Thus b\super2 is even.
\step And b is too.
\endslide
\slide[The problem]
\Mathstep A {{a\over b}=\sqrt2}
\mathstep A {} {\rlap\cross{a\over b}=\sqrt2}
\step[visible=true] And but so we said a and b have no common factor.
\step If both are even they do have a common factor: 2.
\step Which is absurd.
\step Thus, our basic assumption is false.
\step[A] There are no such a and b.
\step The square root of 2 is irrational.
\endslide
\bye