1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
|
% 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
|
