Principia. Introducción al pensamiento matemático, aritmética y geometría. by MATERIALES EDUCATIVOS UACM

ÂŠ

Principia. IntroducciĂłn al pensamiento matemĂĄtico, aritmĂŠtica y geometrĂa, primera ediciĂłn, 2006 primera reimpresiĂłn, 2007 segunda reimpresiĂłn, 2009

ÂŠ

JosĂŠ Alfredo Del Oso Acevedo, IsaĂ Moreno Roque, Rafael Torres SimĂłn y Myrna Velarde SaldaĂąa

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Universidad AutĂłnoma de la Ciudad de MĂŠxico Av. DivisiĂłn del Norte 906, Col. Narvarte Poniente, DelegaciĂłn Benito JuĂĄrez, C.P. 03020, MĂŠxico, D.F. matsedusuacm@gmail.com

Academia de MatemĂĄticas, Programa de IntegraciĂłn, UACM DiseĂąo de la portada: AarĂłn Aguilar â&#x20AC;˘

Diagramas del texto elaborados por integrantes de la UACM.

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/ =& $

( .! / '% $ / &+ +"% .0 . '0+ $ - '+'! 8 0 ' +8

6 , / '% $ - =& - . ('! + J'+/ '+'/ % +! + !D.%'+0- 0.% $ -8 $ / - ..0$$0 / '% $ / &+ +"% .0

. '0+ $ (& / - . / !. - !'(0-@

> B '% $ - / ! @ -0+

=& $$0- &,

( .! / '% $ !' + &+ +"% .0 '+'!0 / ' . -

0. 7 %($08

B '% $ - - , - @ + -!

I(. -')+8 !0/

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+ $0=& - 0 ( .'0/0- / +"% .0-

+ -! -08 $ - . ('! '+/ '+'/ % +! /' +!0+ - =& - $

( .'0/0 , $ +"% .0 / '% $ - (& / - .' '. -F@ B '3& $ % + .

0. 7 %($08

& $

'+/'

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( .')/' 0 - $ 3.&(0 / ' . - =& - . ('! + '+/ '+'/ % +! ?-F

!' + ( .'0/0 8

!' + ( .'0/0 8 !

B '% $ - - , / @ + $

0.%

/ '% $ + 0+!. %0- &+

( .! =& +0 - . ('!

, 0!.

=& - . ('! '+ '+'! - 2 -

0. 7 %($0

+ $

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=& E % 'D+

, (0. "$!'%0

& $ ?$ 0+7&+!0 / ' . - =& +0 - . ('! + +0 - $ - (0+ .. 0/ %0- 0+ $&'. =& $

. (. - +! ')+ / '% $ / $ +"% .0 . '0+ $ / '% $ '+'!

0 / '% $ ( .')/'

+! .0- ,

Y 8 -

& +) 5& B + )&

L -!

=&F - 6

2'-!0 =& !0/0 +"% .0 . '0+ $ (& / I(. - .- 0%0 &+ +"% .0 / '% $ '+'!0 0 / '% $

( .')/' 0 9 $ (.0 -0

'+2 .-

- .: ' .!0;

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&+0

/ $0- !'(0- / +"% .0- / '% $ - ? @ B ? A E 0+-'/ . %0- $ +"% .0 / '% $ '+'!0 , !. ! %0- / - .' '.$0 + 0.%

/ . ')+

%&$!'($' . , /'2'/'. $ +"% .0 / '% $ ,

(0. 8 8 8 ! 8 J/ ( +/' +/0 / &:+!0- +"% .0- -

+ & +!. +

/ . 6

/ $ (&+!0 / '% $ + -! -08 6 , !. - /F3'!0- / -(&D- / $ (&+!0 / '% $8

-F =&

%&$!'($' %0- , /'2'/'%0- +!. @

0. $0 ! +!08

? @ B ? 1 B 1 + -! -08 % ' . %0- &+ / '% $ ( .')/' 0 (&.0

. ')+

0+-'/ . %0- $ / '% $ $ % . %0- $ / '% -!0 -8

0%0 $

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+!0+ - %&$!'($' %0- (0.

% 0- $ /0- / $

'3& $/ /@

?60. 8 . -! %0- J / J @

ZZZZZZZZZZZZZ

0. "$!'%08 . -0$2 %0- $

& ')+ B -( 7 +/0 @

& / - 0%(.0 . $ . -&$! /08 !& +/0 $

/'2'-')+ / +!.

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0+-'/ . %0-

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8 +!0+ - %&$!'($' %0- (0.

% 0- $ /0- / $

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. -!

J / J @

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& ')+ B -( 7 +/0 @

& / - 0%(.0 . $

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0+2 .!'. &+ / '% $ ( .')/' 0 %'I!0

. ')+

E0% %0- 0%0 7 %($0 $ +"% .0 / '% $ $ % %0- $ +"% .0 -!0 -@

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=& $ +& 20 +"% .0 -

/ '% $ ( .')/' 0 (&.0 + -! -08

% 0- $ /0- / $

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&+ / '% $ ( .')/' 0 (&.0@

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? 0%0 $ +!'/ / / /F3'!0- =& 0.% + $ ( .'0/0 - /0- J + -! -0 $ ( .'0/0 - 8

%&$!'($'=& %0-

% 0- $ /0- / $

& ')+ J (0. @

J !

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+! .'0.8 $$ 3 %0-

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$ / '% $ ( .')/' 0 %'I!0 $ % %0- $ +"% .0 -!0 -8

+ -! -08

% 0- $ /0- / $

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$ !& . $

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%&$!'($'=& %0-

% 0- $ /0- / $

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J !

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J J

5& 6 )

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/ '% $@

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%'-%

0.% \/ '% $ 0 . ')+\ ,

!& . $

0( . ')+ '+/' /

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/ '% $@

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% &' ) )& B )&6 & ) B) &' ) B

&' ) )& B

L %0- 0% +! /0 + (:3'+ -

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Y - &+ # ! 0 &+ # - , 1 -&$!

=& $

(.0(0-' ')+ '+2 .-

! % 'D+ - 2 ./ / . @ E0/0 +"% .0 " 0 # - &+ !. ! #C

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( .'0/0;

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7 %($0-

+! .'0. - - 0+0 + 0%0

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1 -&$!

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% /'/

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8 %' +!. - -

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% ! %:!'

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$0 -&%0 - $$ 3)

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=& -

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2'/ + '

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3 0% !.F

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J _

8 -!&/' +! / $

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'. &+ . + '

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0 I,! D 01 ! # ! $ " F : ! ! D ! ,! 2 !. : F #" ! -# ! # 3 !

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=& 6 %0- &- /0 6 -!

60. @ v

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$ !.

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+ 3 !'20-8 $ +"% .0 .0 , $0- +"% .0- +! .0- (0-'!'20- ? -! 0+7&+!0 - $ - $$ %

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$ !. B -!

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/ +0! %0- 0+ $

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' +8

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/ $ F. &$0 , $ . /'0

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-0(0! %' 8 %:- 0 % +0- (0. $

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) *+

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J B )&6 & ) B) &' ) B

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Esta obra se termin贸 de imprimir en el mes de agosto de 2009 en el taller de impresi贸n de la Universidad Aut贸noma de la Ciudad de M茅xico con un tiraje de 3500 ejemplares.

Principia es un libro dirigido, en general, a todo aquel que le interese adentrarse en la matemática a un nivel básico universitario, y especialmente a los estudiantes de la Universidad Autónoma de la Ciudad de México que cursan el módulo inicial del Programa de Integración. La propuesta metodológica se basa en la comprensión de los principios matemáticos como elementos esenciales para el entendimiento de un universo tan maravilloso como vasto. Privilegia, antes que el conocimiento, el pensamiento y la imaginación; el uso creativo de la razón. Siguiendo la ﬁlosofía del matemático Pólya, quien habla de la “excitación del descubrimiento”, Principia se estructura bajo la consigna de aprender por medio del razonamiento y la resolución de problemas. Cada sección, tanto la de aritmética como la de geometría, y en particular la de razonamiento inductivo, incluyen ejercicios y notas históricas amenas que permiten que el lector perciba la génesis de algunos tópicos matemáticos. Introducir al lector en este mundo y despojarlo de sus miedos, ayudarlo a comprender lo que ocurre de manera natural y sumergirlo en su goce y disfrute son los propósitos fundamentales de esta obra.