1. ALGEBRA
DR . AURELIO BALDOR
F~~dad~~, Di~ec~~~ ~ Jefe de ~a C~-
~ed~a de Ma~e~~~ica~ de~ C~~egi~
Baid~~, Haba~a, C~b~ .
Jefe de ~a C~~ed~a de Ma~e~~~ica~,
STEVENS ACADEMY, H~b~ke~,
Ne~-Je~~e~, U .S .A .
P~~fe~~~ de Ma~e~~~ica~, SAINT
PETER'S COLLEGE . Je~~e~ Ci~~,
Ne~-Je~~e~ .
CULTURAL CENTROAMERICANA, S . A .
CON GR~FICOS Y 6523
EJERCICIOS Y PROBLEMAS
CON RESPUESTAS
Ob~a a~~~bada ~ ~ec~~e~dada c~~~ ~e~~~ ~a~a
~~~ I~~~i~~~~~ de Seg~~da E~~e~a~~a de ~a Re-
~~b~ica ~~~ e~ Mi~i~~e~i~ de Ed~caci~~, ~~e~i~
i~f~~~e fa~~~ab~e de ~a J~~~a T~c~ica de Di-
~ec~~~e~ de I~~~i~~~~~ de Seg~~da E~~e~a~~a .
EDICION 1980
TOTALMENTE REVISADA POR EL AUTOR
De~~~i~~ Lega~ : M . 9 .747-1980
I . S . B . N . : 84-357-0062-3
EDICIONES Y DISTRIBUCIONES CODICE, S . A . MADRID
32. 32 40
~a di~i~i~~, ~a ~~~e~ciaci~~, ~a ~~ga~i~~aci~~ ~ ~a ~adicaci~~ . C~~~ie~e i~
ada~~a~d~ ~a ~e~~a~idad de~ ~~i~ci~ia~~e a~ ca~~c~e~ f~~~a~ (ab~~~ac~~) de e~~a~
~e~e~, ~~e~ e~~~ c~~~~ib~i~~ a ~a c~~~~e~~i~~ de ~~~ ~~~b~e~a~ ~~e ~~~e~i~~~e~~e
~e ~~a~~ea~~~ ~a~ ~a~e~~~ica~ ~~~e~i~~e~ . P~~ ~~~a ~a~~e, e~ c~~~~~~~ de e~~a~
~e~e~ f~~~a~e~ c~~~~i~~i~~ ~~a defi~ici~~ i~di~ec~a de ~~~ ~~~e~~~ ~ea~e~ ~ de
~a~ ~~e~aci~~e~ f~~da~e~~a~e~ . E~~a~ ~e~e~ ~~e ~~ ~e~~ie~e~ de~~~~~aci~~, ~~e~
~~~ de a~~ehe~~i~~ i~~edia~a, ~e ~~a~a~ a~i~~a~ .
IGUALDAD
I . A~i~~a de ide~~idad : a = a .
II . A~i~~a de ~eci~~~cidad : ~i a = b, ~e~e~~~ ~~e b = a .
III . A~i~~a de ~~a~~i~i~idad : ~i a = b ~ b = c, ~e~e~~~ ~~e a = c .
SUMA O ADICION
1 . A~i~~a de ~~if~~~idad : ~a ~~~a de d~~ ~~~e~~~ e~ ~ie~~~e ig~a~,
e~ deci~, ~~ica ; a~~, ~i a = b ~ c = d, ~e~e~~~ ~~e a + c = b + d .
II . A~i~~a de c~~~~~a~i~idad : a + b = b + a .
III . A~i~~a de a~~cia~i~idad : (a + b) + c = a + (b + c) .
IV . A~i~~a de ide~~idad, ~ ~~d~~~ de ~a ~~~a: ha~ ~~ ~~~e~~ ~ ~~~~
~~ ~~~e~~, e~ ce~~, de ~~d~ ~~e a + 0 = 0 + a = a, ~a~a c~a~~~ie~ ~a~~~ de a .
De ah~ ~~e e~ ce~~ ~eciba e~ ~~~b~e'de e~e~e~~~ id~~~ic~ ~ ~~d~~~ de ~a ~~~a .
ALGEBRA
MULTIPLICACION
I . A~i~~a de ~~if~~~idad : e~ ~~~d~c~~ de d~~ ~~~e~~~ e~ ~ie~~~e ig~a~,
e~ deci~, ~~ic~, a~~ ~i a = b ~ c = d, ~e~e~~~ ~~e ac = bd .
II . A~i~~a de c~~~~~a~i~idad : ab = ba .
III . A~i~~a de a~~cia~i~idad : (ab) c = a (bc) .
IV . A~i~~a de di~~~ib~~i~idad : c~~ ~e~~ec~~ a ~a ~~~a ~e~e~~~ ~~e
a (b + c) = ab + ac .
V . A~i~~a de ide~~idad, ~ ~~d~~~ de~ ~~~d~c~~ : ha~ ~~ ~~~e~~ ~ ~~~~
~~ ~~~e~~, e~ ~~~ (1), de ~~d~ ~~e a .1 = 1 . a = a, ~a~a c~a~~~ie~ ~a~~~ de a .
VI . A~i~~a de e~i~~e~cia de~ i~~e~~~ : ~a~a ~~d~ ~~~e~~ ~ea~ a 7~= 0
(a di~~i~~~ de ce~~) c~~~e~~~~de ~~ ~~~e~~ ~ea~, ~ ~~~~ ~~~, ~, de ~~d~ ~~e
a~ = 1 . E~~e ~~~e~~ ~ ~e ~~a~a i~~e~~~ ~ ~ec~~~~c~ de a, ~ ~e ~e~~e~e~~a ~~~ 1/a .
AXIOMAS DE ORDEN
I . T~ic~~~~~a : Si ~e~e~~~ d~~ ~~~e~~~ ~ea~e~ a ~ b ~~~~ ~~ede habe~ ~~a
~e~aci~~, ~ ~~~~ ~~a, e~~~e a~b~~, ~~e a > b ; a = b ~ a < b .
M~~~~~~~a de ~a ~~~a : ~i a > b ~e~e~~~ ~~e a + c > b + c .
M~~~~~~~a de ~a ~~~~i~~icaci~~ : ~i a > b ~ c > 0 ~e~e~~~ ~~e ac > bc .
33. AXIOMA DE CONTINUIDAD
1 . Si ~e~e~~~ d~~ c~~~~~~~~ de ~~~e~~~ ~ea~e~ A ~ B, de ~~d~ ~~e ~~d~
~~~e~~ de A e~ ~e~~~ ~~e c~a~~~ie~ ~~~e~~ de B, e~i~~i~~ ~ie~~~e ~~ ~~~e~~
~ea~ c c~~ e~ ~~e ~e ~e~ifi~~e a :5 c :5 b, e~ ~~e a e~ ~~ ~~~e~~ ~~e e~~~
de~~~~ de~ c~~~~~~~ A, ~ b e~ ~~ ~~~e~~ ~~e e~~~ de~~~~ de~ c~~~~~~~ B .
~J!'!-1 ~: : ~. !C . ~, ENTLES CON LOS NUMEROS RELATIVOS
SUMA DE NUMEROS RELATIVOS
E~ ~a ~~~a ~ adici~~ de ~~~e~~~ ~e~a~i~~~ ~~de~~~ c~~~ide~a~ c~a~~~
ca~~~ : ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ; ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ; ~~~a~ ~~
~~~i~i~~ c~~ ~~~~ ~ega~i~~, ~ ~~~a~ e~ ce~~ c~~ ~~ ~~~e~~ ~~~i~i~~ ~ ~ega~i~~ .
I) de (~i~ ~~~ii i~, ~~ i i~~,
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~~~i~i~~~ ~e ~~~cede a ~a ~~~a (+4)+(+2)=+6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~ ~~~e~~~, ~ a~
~e~~~~ad~ ~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ + . A~~ ~e~e~~~ :
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~~~i~i~~~ de~ ~ig~ie~~e ~~d~ :
-4 3
'') S~~a de d~~ ~~~e~~~ ~ega~i~~~
Reg~a
Pa~a ~~~a~ d~~ ~~~e~~~ ~ega~i~~~ ~e ~~~cede a ~a ~~~a (- 4) + (- 2) _ - 6
a~i~~~~ica de ~~~ ~a~~~e~ ab~~~~~~~ de a~b~~, ~ a~ ~e~~~~ad~
~b~e~id~ ~e ~e a~~e~~~e e~ ~ig~~ - . A~~ ~c~~ic~~~ :_ __
P~de~~~ ~e~~e~e~~a~ ~a ~~~a de d~~ ~~~e~~~ ~ega~i~~~ de~ ~ig~ie~~e
~~~~~:
~~~~~~~ ~~~~~ . ~
-1 0 +~
NOTAS SOBRE EL CONCEPTO DE NUMERO
+4-
+Y
FIGURA 2
E- - 2 4
- 7 - 6 - S 4 -3 - 1 0 +1 2 1 3 +4
FIGURA 3
+6 ---- T
+3
A
i
+4
+ 2 -~
+5 i-6 +7
0 33