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CONTENIDO MATEMÁTICA MATEMÁ TICA I
1
MATEMÁTICA MATEMÁ TICA II II
81
FÍSICA
187
QUÍMICA
255
RAZONAMIENTO MATEMÁTICO
389
RAZONAMIENTO VERBAL
437
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Me es grato presentar el Compendio Académico del Centro Pre Universitario de la Universidad Nacional del Altiplano de Puno, elaborado por un equipo de docentes con alto nivel académico y de experiencia cur r i cular ul ar y profes profesional ional . El ins in st r umento ument o est á or ientado a est i mul ar y preparar a los estudiantes en los distintos conocimientos teóricos, que posibiliten el estudio y aprendizaje sustantivo para el desarrollo académico y de acuerdo a las exigencias de la formación universitaria. En las distintas asignaturas se vienen incidiendo en los contenidos más importantes y relevantes que se desarrollan en el CEPREUNA, en las áreas de l as Cienc Ci encii as B i om édic di cas, Cie Ci en ci as de l a I ngenie ngeni er ía y Cie Ci encias nci as Soc Soci ale al es, de tal forma que se constituya en un eficiente instrumento y de apoyo a vuestra formación. Esperamos, ahora, que han decidido ingresar a la Universidad Nacional del Altiplano de Puno y a través de su Centro Preuniversitario, aprovechen la información que alcanzamos a vuestra disposición. Estamos seguros, que este material le va a ser de importante ayuda y en este sentido, les deseamos éxito en lo que van a emprender.
Dr . Wenceslao Medina Espinoza
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AUTORIDADES UNIVERSITARIAS Rect Rect or Dr . Luc Lu cio Ávi la Rojas Rojas Vicerrector Académico Dr . Ge Ger mán Yabar Yabar Pilc Pil co Vicerrector Administrativo Dr . Edgar Edgar do Pine Pin eda Qui Quisspe
COMI TÉ EDI TOR TOR Dr. Wenceslao Teddy Medina Espinoza M .S .Scc. Edwin Edwi n Fe F eder der i co Orna Or na Rivas M g. Jul io Ce Cesar Vill Vi ll alt a Pac Pacor i M .S .Scc. Raúl Sanga Ca Catt un untt a Lic. Edgar Pacompia Belizario Est. Mariela Diane Lima Quispe Est. Husam Jeiel Flores Dueñas
EDICION Y DIAGRAMACIÓN Lucio Elías Flores Bustinza Elwin Luis Huamán Quispe
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MATEMÁTICA I
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“Si la Matem Matematica ´ hablase, hablase, emplear´ ıa a la l´ ogica para comunicarse”... La logica consti titu tuye ye una una de las part partes es mas ´ importantes importantes del desarrollo desarrollo ´ tradicional o logica ´ Aristot´ elica cons de la matem´atica como ciencia exacta , estudiarla en detalle demandar´ıa ıa mucho tiempo y dedicaci´on, on, as´ı que en esta parte solo tratare trataremos mos los aspectos aspectos b´asicos asicos pero importante importantess que todo estudiant estudiante, e, que aspira seguir seguir una carrera de ciencias o ingenier´ ingenier´ıas, ıas, debe saber. saber.
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En general general,, un enunciado es toda toda fras frase e u orac oraci´ i´on o n que que se utili utiliza za en el leng lengua uaje je com comun, ´ por consig consiguie uiente nte,, toda expresi´on on que decimos u o´ımos ımos (con sentido o carente de el) e ´ l) son ejemplos de enunciados. Luego, podemos definir a una proposici´ on como aquel enunciado que expresa un pensamiento o idea cabal y que en muchos casos es suceptible a ser demostrado. Entre las proposiciones destacan las llamadas cuya caracter´ caracter´ıstica ıstica fundamental es la se ser verdadera: “V” o falsa: “F” pero no proposiciones proposiciones logicas ´ ambas a la vez. Por ejemplo: 1. Juan borrador borrador cielo do re cuarzo tres. tres. 2. Vamos Vamos a jugar un partido de futbol. ´ 3. Jupiter ´ es un planeta m´as as grande que Marte. 4. Dos tercios es mayor que la unidad. Claramente, el ejemplo 1 es un enunciado sin sentido, 2 es una proposici´on, on, 3 es una proposici´on on verdadera y 4 es una proposici´on on falsa. Luego, solo 3 y 4 son proposiciones l´ogicas. ogicas. La veracidad o falsedad de una proposici´on l´ogica ogica se obtiene de la disciplina o suceso de donde proviene en base a raciocinios coherentes y sistem´aticamente aticamente organizados. Convensionalmente, a las proposiciones l´ogicas ogicas se les representa mediante las letras min´usculas: p, q , r, s, t, etc. cada una de estas letras constituye una variable proposicional . Dada una proposici´on o n l´ogica ogica p, siempre es posible saber si es verdadera (V) o falsa (F), es decir, siempre se puede averiguar su valor de verdad, esto se dispone en una tabla llamada tabla de verdad: Valor Valor de verdad Verdadero Verdadero
p V
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”; “ p La conjunci´ on: p q , se lee: “ p y q ”; adem´as q ”; ”; “ p tambi´en en q ”; ”; “ p al mismo tiempo que ”; “ p as´ı como q ”; ”; etc. q ”;
”; “ p implica implicaci´ on, se lee: “si p entonces q ”; ”; “q puesto que p”; “q , si p”; “ p es suficiente suficiente para q ”; que q ”; ”; etc. A p se le llama antece antecedente dente y a q consecuente.
∧
en p q , llamada tambi´en La disyun disyunci cion: ´ ”; “ p saldisyunci´ on d´ ebil o inclusiva, se lee: “ p o q ”; vo q ”; ”; “ p a menos que q ”; ”; etc.
∨
La condicional: p
→ q
La bicondicional: bicondicional: p q , se lee: “ p si y solo si q ” o “ p es una condici´on on necesaria y suficiente para q ”. ”.
↔
, llamada tambi´en en
Tablas de verdad de los principales principales conectivo conectivoss logicos: ´
p V V F F
q V F V F
p
∧q
V F F F
p
∨q
V V V F
p
→q V F V V
p
↔ q V F F V
on “ ” es verdadera solo cuando las dos proposiciones son verdaderas, en Observe que: La conjunci´on cualquier otro caso es falsa. La disyunci´on “ ” es falsa solo cuando las dos proposiciones son falsas, en cualquier cualquier otro caso caso es verdadera. verdadera. La condicion condicional al “ ” es falsa solo cuando el antecedente antecedente es verdadero y el consecuente consecuente falso, falso, en cualquier cualquier otro caso caso es verdadera. verdadera. La bicondicional bicondicional “ ” es verdadera verdadera cuando las dos proposiciones son iguales y es falsa cuando son diferentes. Estas observaciones son muy pr´acticas acticas cuando se quiere evaluar la tabla de verdad de una proposici´on on compuesta.
∧
∨
→
↔
convensionalmente mente se lee: “ no p” cuando la proposici´on on es simple, por ejemplo, La negaci negacion ´ 1 : p convensional si p: Juan es m´edico. edico. edico. p: Juan no es m´edico.
∼
⇒ ∼
Y cuando la proposici on ´ es compuesta, compuesta, se suelen suelen usar las las expresione expresiones: s: “no es cierto cierto que p” o “es falso que on compuesta: p” o “no es el caso que p”, etc. Por ejemplo la proposici´on
No es el caso que Juan sea m´ edico y Miguel abogado. abogado. Tiene como proposiciones proposiciones simples a p: Juan es m´ edico. edico. y a q : Miguel es abogado. Luego, su enunciado inicial se representar´ıa ı a como como:: ( p q )
∼ ∧
La negaci´
“ ” cambia el valor de verdad donde se haga presente, es decir:
( V )
F ; tambi´
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on de la conju conjunci´ nci´on, on, se lee lee “no p o no q ”. ”. La negaci negacion ´ alternat alternativa: iva: p q es la negaci´on
|
La negaci negacion ´ conju conjunta nta:: p
on de la disyunci´on, on, se lee “ni p ni q ”. ”. ↓ q es la negaci´on
Sus tablas de verdad son:
p V V F F
q V F V F
p∆q F V V F
p
←q
p q F V V V
|
V V F V
p
↓ q
F F F V
Observe Observe que con 2 proposicione proposicioness simples simples ( p total 4 combinaciones combinaciones posibles. posibles. En general, general, p y q ) existen en total si en una proposici´on on compuesta existen n proposiciones simples, entonces: Numero ´ de combinaciones = 2
n
on compuesta, compuesta, las operaciones operaciones se realizan realizan Jerarqu´ ıa de los conectivos l´ ogicos: En una proposici´on usandos los delimitadores: par´entesis “( )”, corchetes “[ ]” o llaves “ ”; se efect´uan las proposiciones que se encuentran encuentran dentro dentro de estos delimitadores delimitadores y luego siguen sus negaciones negaciones de izquierda izquierda a derecha. derecha.
{}
Si en una proposici´on on compuesta se tiene un valor VERDADREO para cualquier combinaci´on on de valores valores veritativos veritativos de sus sus componente componentess se le llama tautolog´ contrario es siempre FALSO FALSO,, se ıa, si por el contrario le llam llama a contradicci por lo menos menos existe existe una V y por por lo menos menos una F, F, se le llam llama a contingencia . contradiccion ´ , y si por Una implicaci´ condicional p ıa, en tal caso a la condicional on l´ ogica es toda condicional q que sea tautolog´ıa, se le denota por “ p ”. Tambi´en, en, una equivale es toda bicondicional p equivalencia ncia l ogica ´ q ”. q que sea tautolog´ıa, ıa, en tal tal caso a la bicondicion bicondicional al se le denota por “ p ”. q ”.
→
⇒
↔
⇐⇒
Dos proposiciones p y q se llaman equivalentes o logicamente ´ equivalentes equivalentes si sus tablas de verdad son id´ identicas. e´ nticas. Esto se simboliza por: “ p ”, lo significa que el resultado es el mismo si en lugar de p se q ”, considera a q .
≡
Leyes del algebra ´ proposicional:
1.
c)
Idempo Idempoten tencia cia::
a) b)
∧ ≡p p ∨ p ≡ p
d)
p p
5.
→ (q ∧ r) ≡ ( p → q ) ∧ ( p → r) p → (q ∨ r ) ≡ ( p → q ) ∨ ( p → r) p
De D’Mo D’Mo
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8.
De la bicond bicondici iciona onal: l:
a) b) 9.
e) f )
↔ q ≡ ( p → q ) ∧ (q → p) p ↔ q ≡ ( p ∧ q ) ∨ (∼ p∧ ∼ q ) p
g)
Equiva Equivalen lencia ciass adicio adicional nales: es:
a) b) c) d)
∨ ∼ p ≡ V p ∧ V ≡ p p ∧ F≡ F p ∧ ∼ p ≡ F ∼ (∼ p) ≡ p p ∆ q ≡ ( p∧ ∼ q ) ∨ (q ∧ ∼ p) p ∆ q ≡ ( p ∨ q )∧ ∼ ( p ∧ q ) p
h)
∼ V ≡ F ∼ F ≡ V p ∨ V ≡ V p ∨ F≡ p
i) j) k)
verdad1 , es decir:
Si la proposici´on: on:
∼ {∼ [∼ (∼ p∨ ∼ q ) → (r∆q )])] ∧ (∼ s ∧ p)}
∼ [∼ (∼ p∨ ∼ q ) → (r∆q )])] ∧ (∼ s ∧ p ) ≡ V F
F
es una cantradicci´on, on, entonces entonces los valores de verdad de p, q , r y s; respectivament respectivamente; e; son: A) VVVV D) VVVF
B) VVFF
C) VFVF E) FFVV
Se define “ p
∗ q ” mediante la tabla: p q p ∗ q V V F F
V F V F
V V F V
F
V
F
V
V
F
⇒
V
V
revise esta deducci´on on de abajo hacia arriba. Si r y q deben ser iguales. (r∆q ) F
≡ ⇒
Si es una contradicci´on, on, entonces es falsa, por lo tanto la expresi´on on dentro de las llaves debe ser
V
Claramente: p
≡ V
; q
≡ V
; r
≡V
; s
≡F
Rpta: D
molecular propuesto:
p V V F
q V F V
[(
∼ p ∗ F F V
F V V
q ) V F V
∗
F V V
p] V V F
→ V F V
(q p)
∗
V F V
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Simplificar Simplificar la proposici´ proposici´on: on:
E = A) p q D) p q
∼ ∧∼ →
∼ [∼ ( p ∧ q ) →∼ q ] B) p ∧ q
justificaci´ justificacion: o´ n:
≡ ∼ [∼∼ ( p ∧ q )∨ ∼ q ] ≡ ∼ [( p ∧ q )∨ ∼ q ] ≡ [∼ ( p ∧ q )∧ ∼∼ q ] ≡ [(∼ p∨ ∼ q ) ∧ q ] ≡ (∼ p ∧ q ) ∴ E ≡ ∼ p ∧ q
E C) p q E) p q
∨ ∼ ∧
Reducimos Reducimos la proposici´ proposici´on on E empleando quivalencias cias logicas. o´ gicas. Note que en cada paso se da su correspondiente
Def. de la condicional Doble negaci´on on D’Morgan D’Morgan, Doble neg. Absorci´on. on.
Rpta: E
Todo el el estudi estudio o de la Matem´ Matem´atica atica se realiza realiza sobre sobre determ determina inados dos objeto objetoss llamado llamadoss conjuntos, es as´ as´ı que los conjuntos tienen un rol muy importante en la Matem´atica. atica. En esta parte expondremos las ideas y conc concep epto toss mas ´ elementales elementales sobre la teor´ teor´ıa ıa de conjunto conjuntoss que todo todo estudia estudiante nte debe debe conocer conocer como como m´ınimo. ınimo.
Un conjunto es una colecci´ on o agrupaci´ on de objetos bien definidos (llamados sus elementos) que pueden ser abstractos y/o concretos. concretos. Todo conjunto se puede determinar por: 1.
elementos uno por uno. Por extensi´ on o en forma tabular: tabular: Cuando se citan a sus elementos
2.
Por comprensi´ on o forma constructiva: Cuando se indica una propiedad, cualidad o caracter´ısti ı stica ca com comun ´ de sus elementos.
Por ejemplo, si A es un conjunto 1 cuyos elementos son a, e, i, o, u , entonces: A = a, e, i, o, u es su representaci´ on por exten extensi´ si´on. on. Y: A = x x es una vocal es su representaci´on on por compre comprensi´ nsi´on. on.
{|
}
{
}
Cualqu Cualquier ier conjun conjunto to se puede puede repres represenentar gr´aficame aficamente nte haciendo haciendo uso de los conoci conoci-dos diagramas de Venn Euler. Un diagrama de Venn Venn Euler es cualquier figura plana cerrada. La figur figura a mues muestr tra a la repre represe sent ntac aci´ i´ongr´ o ngr´afica afica para el conjunto A del ejemplo anterior.
−
−
La relaci´ on de pertenencia se d´a exclusivamente entre un conjunto y sus elementos. Si x es un elemento de un conjunto X , se dice que x “pertenence” al conjunto X y se escribe: “ x ” . Si por el X ”.
∈
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Luego, se define el cardinal de un conjunto finito 1 como el n´umero n de elementos no repetidos que posee. Notaci´on: o n: card card(X ) = n Por definici´on, on, el cardinal del conjunto vac´ıo ıo Φ es cero: card(Φ) = 0 Naturalmente, un conjunto con un solo elemento se llama conjunto unitario, por ejemplo, si en X = 5;5;5;5;5 = 5 lo es. A = x, y, z es un conjunto unitario x = y = z ; tambi´en
{
}
⇒
{
} {}
3.
conjunto se llama llama infinito cuando no es finito. Conjunto Conjunto infinito: infinito: Un conjunto
4.
Conjunto universal: Convensionalmente se asume la existencia de un conjunto que contiene todo, todo, a tal conjunto conjunto se le llama conjunto univesal, en este sentido podemos definirlo como:
U = x x = x
{|
}
Sin embargo, por comodidad es frecuente frecuente emplear emplear un universo relativo como por ejemplo si se tratan intervalos de IR, entonces el universo ser´ıa ıa el propio IR.
Relacio Relaciones nes entre conjuntos conjuntos : 1.
La incl inclus usiion ´ : “A
conten enid ido o en B ” o “A es subconjun subconjunto to de B ”. Se define como: ⊂ B ”, se lee “A est´a cont A ⊂ B ⇐⇒ ( todo elemento de A est´a en B ) on A ⊂ B tambi´en en se escribe como B ⊃ A y se lee “B contie contiene ne a A”. Nota: La expresi´on Si X es un conjunto conjunto arbitrario arbitrario,, entonces entonces se cumple: cumple: Φ
2.
⊂ X
y tambi´en: en: X
La igualdad : “A = B ”, se lee “ A es igual a B ” y se define como:
A
B
(A
B
∧B
A)
⊂ X
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Conjunto potencia: LLamado tambi´en partes de un conjunto, si X es un conjunto entonces el conjunto potencia de X es la familia P (X ) formada por todos los subconjuntos de X . Por consiguiente, para cualquier cualquier conjunto conjunto X se cumple:
Φ
∈ P (X )
∧
X
∈ P (X )
de modo que la familia P (X ) nunca es vac´ıa. ıa. Por ejemplo, si X = a; b; c P (X ) = Φ; a ; b ; c ; a; b ; a; c ; b; c ; a; b; c
{
}
⇒
{ {}{}{}{ }{ }{ }{
}}
Numero ´ de elementos de P (X ): Si X es un conjunto finito de n elementos, entonces el conjunto elementos.. Es decir: decir: card[P (X )] P (X ) tiene 2n elementos )] = 2card(X ) Operacio Operaciones nes entre conjunto conjuntos s:
1. Uni Union ´ de conjuntos:
A
∪ B = {x|x ∈ A ∨ x ∈ B}
2. Intersecci Intersecci´ on ´ de conjuntos:
A
∩ B = {x|x ∈ A ∧ x ∈ B}
3. Diferencia de conjuntos:
A
− B = {x|x ∈ A ∧ x ∈ B }
4. Diferencia sim´ etrica:
A∆B = xx
{ | ∈ A ∪ B ∧ x ∈ A ∩ B}
5. Complemento: Complemento:
AC = x x
{ | ∈ A} ¯, C (A) Otras notaciones notaciones : A , A 6. Producto cartesiano:
A
× B = {(a, b)|a ∈ A ∧ b ∈ B}
donde (a, b) se llam llama a par ordenado e s la ordenado. “a” es primera primera compon component ente e y “b” es la segu segund nda a comcomponente.
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Cuantificadores: en “para cualquier”. Si P (x) es una funci´ on proposicional1 , 1. Universal: “ ” se lee “para todo” o tambi´en entonces entonces la proposici´ proposici´on: on:
∀
∀ x ∈ A ; P (x) [esto se lee “para todo x en A tal que P (x)”], ser´a verdadera si y solo si P (x) se cumple para todo x de entonces la proposici´ proposici´on on ser´a falsa. A, y si alg´un (por lo menos uno) x de A no cumple P (x) entonces en “existe por lo menos uno”. Si P (x) es una funci´on on proposi2. Existencial: “ ” se lee “existe” o tambi´en cional, entonces la proposici´on: on: x A ; P (x)
∃
∃ ∈
[lo que que se lee lee “exist “existe e un x (por lo menos uno) en A tal que P (x)”], ser´a verdade verdadera ra si en A existe existe un x (por lo menos uno) tal que la proposici´on on P (x) se cumple, y si no existe x en A que verifique P (x) entonces ser´a falsa. El hecho de que “exista por lo menos uno” significa que si existe m´as as de uno, la proposici´on on sigue siendo verdadera, basta que exista uno. on “ !” se lee lee “exi “exist ste e un unico ”. La proposici´on: on: Nota: La expresi´on ´
verdadera ra si y solo P (x) ser´a verdade si en A existe un y solo un elemento x que cumple P (x), en cualqu cualquier ier otro otro caso ser´a falsa.
∃
Negaci´ on de los cuantificadores: x A; P (x) [ x A ; P (x)]
∼ ∀ ∈
≡ ∃ ∈
∼
∼[∃x∈A;
∃! ∈ A ;
P (x)]
≡ ∀ x ∈ A ; ∼ P (x)
Algunas propiedades propiedades del algebra ´ de conjuntos: 1. 2.
∪A =A ; A∩A= A A∪B =B ∪A; A∩B = B ∩A A A
A
A
11. 12.
∩ (B ∪ C ) = ( A ∩ B) ∪ (A ∩ C ) A ∪ (B ∩ C ) = ( A ∪ B ) ∩ (A ∪ C ) A
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tiene: n(A) = 35
;
n(B ) = 55
De la figura, nos piden x. Si el 50% prefiere solo una una de ell ellas as (par (parte te azu azul) l),, el otro otro 50 % ser sera´ el rest resto o la intersecci´on on debe ser 50 x. Luego, es f´acil acil darse darse cuenta cuenta de x 15 y 5 + x.
⇒
−
−
Porr consig Po consiguie uiente nte,, parte parte azul: azul: (x 15)+(5+ x) = 50 de donde x = 30
−
≡ 30 %
Rpta: B
Los cardinales de los conjuntos A, B y C son tres numeros ´ enteros y consecutivos. Si se verifica:
)] = 448 n[P (A)] + n[P (B )] + n[P (C )]
B) 24
a
a+1
C) 18 E) 27
a+2
⇒ 2 + 2 + 2 = 448 ⇐⇒ 2 (1 + 2 + 2 ) = 448 ⇐⇒ 2 = 64 = 2 a
Determine Determine el valor de: de: E = n(A) + n(B ) + n(C ) A) 21 D) 15
Si: n[P (A)] + n[P (B )] + n[P (C )] )] = 448
1
a
2
6
De donde a = 6.
Luego: E = n(A) + n(B ) + n(C ) = 6 + 7 + 8 = 21 Sean: n(A) = a ; n(B ) = a + 1
∧ n(C ) = a + 2.
Rpta: A
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La adici´ on tal que: a, IR le hace corresponde corresponderr un numero: ´ IR llamado llama do a, b s on es una operaci´on “suma” de a y b, y se escribe: a + b = s A los los n umeros ´ a y b se les llama “sumandos”. La suma de numeros ´ reales cumple las leyes:
∀ ∈
1.
Clau Clausu sura ra:: a + b
2.
Conm Conmut utat ativ iva: a: a + b = b + a
3.
Asoc Asocia iati tiva va:: a + (b + c) = ( a + b) + c = a + b + c
4.
Neutro Neutro aditivo aditivo:: a + 0 = a
∈ IR
5.
− 1) = n
2
n(n + 1) S 5 = 1 + 2 + 3 + . . . + n = 2 3
3
3
3
Invers Inverso o aditiv aditivo: o: a + ( a) = 0 , ( a)
− − ∈ IR Estas leyes se cumplen ∀ a, e´ n a , b, c ∈ IR. Tambi en ∀ a ∈ IR, ∃(−a) ∈ IR El neutro neutro e inverso aditivo aditivo son son unicos. ´
Algunas sumas notables: n(n + 1) S 1 = 1 + 2 + 3 + . . . + n = 2
S 3 = 1 + 3 + 5 + . . . + (2n
∈
S 2 = 2 + 4 + 6 + . . . + (2 n) = n(n + 1) S 4 = 1 2 + 2 2 + 32 + . . . + n2 =
2
n(n + 1)(2n + 1) 6
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Tambi´en: El n´ umero de unidades mayor que la base n del sistema de numeraci´ on forma tantas unidades unidades del orden inmediat inmediato o superior superior como grupos grupos de n unidades hay, hay, y si hay residuo (el (el cual debe ser criterio a cada cifra del numeral. numeral. menor que n ) este es la ultima ´ cifra. Se aplica este criterio Por ejemplo: El n´umero 20 en base 5 es: 20 = 40(5) , pues con 20 se forman 4 grupos de 5 y no hay residuo (residuo= 0). El numer um ´ ero o 19 en base base 5 es: 19 = 34(5) , pues con 19 se forman forman 3 grupos grupos de 5 y queda 4 como residu residuo o (residuo= 4). El numer um ´ ero o 14 en base base 4 es: 14 = 32(4) , pues con 14 se forman forman 3 grupos grupos de 4 y queda 2 como residu residuo o (residuo= 2). El n´umero 23 en base 3 es: 23 = 72(3) , pues con 23 se forman 7 grupos de 3 y queda 2 como residuo (residuo= 2). Pero cada cifra debe ser menor que la base, realizamos el mismo proceso a la primera cifra: 7 = 21(3) , finalmente: 23 = 212(3) El numero ´ 3 + 2 en base 4 es: 3 + 2 = 5 = 11 (4) , pues con 5 se forma 1 grupo de 4 y queda 1 como residuo (residuo= 1 ). Luego: Para sumar n´ umeros de m´ as de una cifra se realizan procesos similares a los ejemplos ante-
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La sustracci´ (llamada tambi´ tambien ´ resta) es una una operaci´ operaci´on tal que que M, corresponder un on (llamada M, S IR le hace corresponder numero ´ D IR llamado “diferencia” de M y S , y se escribe: M S = D al n´umero M se le denomina “minuendo” y a S “sustraendo “sustraendo”. ”. Convensionaleme Convensionalemente nte,, a los numeros ´ en M , S y D se les conoce tambi´en como los “t´ermino erminoss de la sustracci´ sustracci´on” y son tales que cumplen cumplen las propiedades: propiedades:
∈
−
1.
M = D + S
3.
2.
M + D + S = 2M
4.
∀
∈
(M + x)
− (S + x) = D (M − x) − (S − x) = D
Si “ S ” es la suma y “ D” la diferencia de dos n´umeros, entonces: Numero ´ mayor =
S + R 2
;
Numero ´ menor =
S
−R 2
Resta Resta en sistemas sistemas de numeraci´ numeraci´ on diferente al decimal: Para ello observemos primero que al restar 1423 576 en base 10 (como se suele realizar desde la escuela) se procede as´ı: ı:
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La multiplicaci o ntal que que a, hace corres correspon ponder der un numero ´ lla mado o multiplicacion ´ es una operaci´ontal a, b IR le hace p IR llamad “producto” “producto” de a y b, y se escribe: a b = p . En ocasiones ocasiones en lugar de a b, por brevedad, se escribe a.b o tambi´en en ab. Al n´umero “a” se le llama “multiplic “multiplicando” ando” y a “ b” “multiplicad “multiplicador”. or”. El producto producto de numeros ´ reales reales goza de las leyes:
∀ ∈
×
1.
Clau Clausu sura ra:: ab
2.
Conm Conmut utat ativ iva: a: ab = ba
3.
Asoc Asocia iati tiva va:: a(bc) = (ab)c = abc
4.
Neutro Neutro multip multiplic licati ativo: vo: a1 = a
×
∈
± c) = ab ± ac Estas leyes se cumplen ∀ a, e´ n a , b, c ∈ IR. Tambi en − ∀ a ∈ IR − {0}, ∃a ∈ IR .
∈ IR
6.
Distri Distribut butivi ividad: dad: a(b 1
5. Invers Inverso o multip multiplic licati ativo: vo: aa−1 = 1 , a = 0
El neutro neutro e inverso inverso multiplicat multiplicativo ivo (escrito (escrito tam1 bi´ bien e´ n como a ) son unicos. ´
La divisi´ on tal que D, co n D D, d IN con d, le hace corresponder un n´umero q on es una operaci´on llamad llamado o “cocien “cociente” te” y un numero ´ r IN 0 llamado “residuo” tales que verifican:
∀ ∈ ∈ ∪{ }
D = dq + r , 0
≥
∈ IN
≤r
La igualdad igualdad anterior se denomina identidad fundamental o algoritmo de la divisi´ on. Al n´umero “D” se
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2.
Si al divi divide dend ndo o D y divisor d se les multiplica multiplica por x, el cocien cociente te no var´ıa ıa pero el resto queda multiplicado por x: (Dx) = (dx)q + (rx) . Similarmente, Similarmente, si al dividendo D y al divisor divisor d se les divide por x no nulo, el cociente no cambia pero el resto queda dividido por x:
3.
D x
=
d x
q +
r x
Si solo solo al divi divide dend ndo o D se le multiplica multiplica por x, entonces:
(Dx ) = d(qx + q ∗ ) + r∗ el nuevo nuevo cociente cociente es: qx + q ∗ y el nuevo residuo es: r∗ , donde q ∗ y r ∗ son el cociente cociente y residuo que que ∗ ∗ resultan al dividir: rx d, esto es: rx = dq + r
÷
(C.A.) Es la cantidad que le falta a un n´umero para ser igual a una Complem Complemento ento algebr algebraico ´ unidad del orden inmediato superior con respecto a su cifra de mayor orden. Ejemplo:
C.A.(80) = 20 C.A.(700) = 300
C.A.(4) = 6 C.A.(970) = 30
C.A.(1) = 9 C.A.(1300) = 8700 , etc
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El divisor y residuo de una divisi´on on son respectivamente 48 y 36. Si se multiplica al dividendo por 25 y se efect´ua nuevamente la divisi´on, o n, el cociente queda multiplicado por 26 y el residuo no se altera. ¿Cu´al al fu´e el dividendo inicial? A) 600 D) 909
B) 900
C) 609 E) 800
cicio cicio se tiene: tiene: = =
D 25D
⇒
48q + 36 48(26q ) + 36
25(48q + 36) = 48(26 q ) + 36
⇒
q = 18
∴ D = 48(18) + 36 = 900
Rpta: B
Sea la divis divisi´ i´on on D = dq + r. Del enunci enunciado ado del ejerejer-
Al dividir A entre B se obtiene un residuo m´aximo. Si el dividendo dividendo disminuye en 170, el cociente disminuye en 3 y el residuo es m´ınimo. ınimo. Siendo
cicio cicio se tiene: tiene:
A
=
Bq + ( B
− 1)
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Sean a on a b es exacta. Del ZZ y b ZZ+ , se dice que “ a es divisible por b” si y solo si la divisi´ algoritmo de la divisi´on, on, se puede concluir que: A es divisible por B si y solo si, existe k ZZ tal que:
∈
∈
÷
∈
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12. Si : n =
◦ a +r ◦ b +r ◦ c +r
⇒
◦ n = M C M (a; b; c) + r
13. Principio de Arqu´ ımedes: Sean a, b ZZ ◦ ´ 0 . Si ab = n y adem´as b n tienen como unico
{}
∧
∈ −
◦
diviso divisorr com´un un a la unidad, entonces: a = n.
diferentes residuos residuos positivos positivos que se obtienen obtienen al analizar analizar las potencias potencias Restos potenciales: Son los diferentes consecutivas de un n´umero entero positivo mayor que la unidad con respecto a cierto m´odulo odulo (divisor). ◦ Esquema: an =m +rn Donde:
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Halle el residuo de la divisi´on: on: 3828 A) 2 D) 3
38
28
B) 4
◦ = 7 +3
28
◦ ◦ ◦ ◦ =7 +(7 +1)4 (4) =7 +(7 +1)(4)
÷7
◦ ◦ =7 +328 =7 +(32 )14
C) 5 E) 6
⇒
◦ 3828 =7 +4
Luego, el residuo es 4.
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A menos que se diga lo contrario, todos los numeros ´ considerados en este cap´ıtulo, ıtulo, ser´an a n n´umeros enteros enteros positivos positivos..
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(PESI): LLaNumeros ume ´ ros primo primos s entres entres s´ ı (PESI): mados tambi´en en primos relativos o coprimos , son aquellos aquellos que tienen tienen como unico ´ divisor com´un a la unidad. unidad.
divisor que es diferente Divisor propio: es un divisor al mismo n´umero.
Divisor elemental: es el menor divisor diferenete de la unidad.
Propiedades: 1.
Dos numeros ´ consecutivos en ZZ+ siempre son PESI.
2.
Si en un grup grupo o de de numeros ´ est´a la unidad,
Propiedades: C D(N ) = C Dsimples + C Dcompuestos
C Dpropios = C D(N )
−1
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Por otro lado tambi´en en se cumple: cumple:
on Teorema: Si : N = pa q b es la descomposici´on
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Propiedades:
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El algoritmo algoritmo de euclides se puede organizar organizar en el
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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The world's largest digital library
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