tag:blogger.com,1999:blog-60318898859668087812024-02-19T04:06:37.259-08:00Aprendiendo las matemáticasEste es un trabajo que nos permite conocer y facilitar el aprendisaje de la mateáticaJorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6031889885966808781.post-11914981031288772112010-12-29T11:21:00.003-08:002010-12-29T11:39:11.425-08:00¿Que son los Números Naturales?<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLtamVYgGZLfZooDXjNOGtO0Y8Lhjdb6IVUKpa_IK1ENU4beAJpzVkyY7f_rxdDa6_NSj31swg6d9FxR-AYYWu13_kk_OkY5C9TwKVIcGHtsEIM5cxxDjQle2Toi-XV4Z1Qc53SToyA67l/s1600/numeros+naturales.jpg" imageanchor="1" style="clear: right; cssfloat: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="288" n4="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLtamVYgGZLfZooDXjNOGtO0Y8Lhjdb6IVUKpa_IK1ENU4beAJpzVkyY7f_rxdDa6_NSj31swg6d9FxR-AYYWu13_kk_OkY5C9TwKVIcGHtsEIM5cxxDjQle2Toi-XV4Z1Qc53SToyA67l/s320/numeros+naturales.jpg" width="320" /></a><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Número natural, el que sirve para designar la cantidad de elementos que tiene un cierto conjunto, y se llama cardinal de dicho conjunto.</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los números naturales son infinitos. El conjunto de todos ellos se designa por N: </span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">N = {0, 1, 2, 3, 4,…, 10, 11, 12,…}</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">El cero, a veces, se excluye del conjunto de los números naturales.</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Además de cardinales (para contar), los números naturales son ordinales, pues sirven para ordenar los elementos de un conjunto: </span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">1º (primero), 2º (segundo),…, 16º (decimosexto),…</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><shape alt="http://docente.ucol.mx/grios/Imagenes/numeros_primos.gif" id="Imagen_x0020_1" o:spid="_x0000_s1026" style="height: 106.5pt; left: 0px; margin-left: 3.45pt; margin-top: 16.45pt; position: absolute; text-align: left; visibility: visible; width: 139.5pt; z-index: 251657216;" type="#_x0000_t75"><imagedata o:title="numeros_primos" src="file:///C:\Windows\Temp\msohtmlclip1\01\clip_image003.gif"></imagedata><wrap type="square"></wrap></shape><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los números naturales son los primeros que surgen en las distintas civilizaciones, ya que las tareas de contar y de ordenar son las más elementales que se pueden realizar en el tratamiento de las cantidades.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Entre los números naturales están definidas las operaciones adición y multiplicación. Además, el resultado de sumar o de multiplicar dos números naturales es también un número natural, por lo que se dice que son operaciones internas.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La sustracción, sin embargo, no es una operación interna en N, pues la diferencia de dos números naturales puede no ser un número natural (no lo es cuando el sustraendo es mayor que el minuendo). Por eso se crea el conjunto Z de los números enteros, en el que se puede restar un número de otro, cualesquiera que sean éstos.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La división tampoco es una operación interna en N, pues el cociente de dos números naturales puede no ser un número natural (no lo es cuando el dividendo no es múltiplo del divisor). Por eso se crea el conjunto Q de los números racionales, en el que se puede dividir cualquier número por otro (salvo por el cero). La división entera es un tipo de división peculiar de los números naturales en la que además de un cociente se obtiene un resto.</span></div>Jorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.com0tag:blogger.com,1999:blog-6031889885966808781.post-7568353080893854722010-12-29T11:21:00.001-08:002010-12-29T11:45:31.993-08:00Propiedades de la adición de Números Naturales<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><shapetype coordsize="21600,21600" filled="f" id="_x0000_t75" o:preferrelative="t" o:spt="75" path="m@4@5l@4@11@9@11@9@5xe" stroked="f"><stroke joinstyle="miter"></stroke><formulas><f eqn="if lineDrawn pixelLineWidth 0"></f><f eqn="sum @0 1 0"></f><f eqn="sum 0 0 @1"></f><f eqn="prod @2 1 2"></f><f eqn="prod @3 21600 pixelWidth"></f><f eqn="prod @3 21600 pixelHeight"></f><f eqn="sum @0 0 1"></f><f eqn="prod @6 1 2"></f><f eqn="prod @7 21600 pixelWidth"></f><f eqn="sum @8 21600 0"></f><f eqn="prod @7 21600 pixelHeight"></f><f eqn="sum @10 21600 0"></f></formulas><path gradientshapeok="t" o:connecttype="rect" o:extrusionok="f"></path><lock aspectratio="t" v:ext="edit"></lock></shapetype><shape id="_x0000_s1026" style="height: 125.1pt; left: 0px; margin-left: 285.75pt; margin-top: 3.55pt; position: absolute; text-align: left; width: 138.2pt; z-index: 251655168;" type="#_x0000_t75"><imagedata o:title="numeros naturales" src="file:///C:\Windows\Temp\msohtmlclip1\01\clip_image001.jpg"></imagedata><wrap type="square"></wrap></shape><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La adición de números naturales cumple las propiedades asociativa, conmutativa y elemento neutro.</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><a href="http://wordpress.colegio-alameda.com/matematicas6primaria/files/2010/10/math.jpg" imageanchor="1" style="clear: right; cssfloat: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="264" n4="true" src="http://wordpress.colegio-alameda.com/matematicas6primaria/files/2010/10/math.jpg" width="320" /></a><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">1.- Asociativa:</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si a, b, c son números naturales cualesquiera se cumple que: </span></div></div><div class="MsoNormal" style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(a + b) + c = a + (b + c)</span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Por ejemplo: </span></div></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(7 + 4) + 5 = 11 + 5 = 16</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">7 + (4 + 5) = 7 + 9 = 16</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los resultados coinciden, es decir, </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(7 + 4) + 5 = 7 + ( 4 + 5)</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">2.-Conmutativa </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si a, b son números naturales cualesquiera se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">a + b = b + a</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">En particular, para los números 7 y 4, se verifica que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">7 + 4 = 4 + 7</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Gracias a las propiedades asociativa y conmutativa de la adición se pueden efectuar largas sumas de números naturales sin utilizar paréntesis y sin tener en cuenta el orden.</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">3.- Elemento neutro </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">El 0 es el elemento neutro de la suma de enteros porque, cualquiera que sea el número natural a, se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">a + 0 = a</span></div>Jorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.com0tag:blogger.com,1999:blog-6031889885966808781.post-22654732210039209412010-12-29T11:20:00.002-08:002010-12-29T11:20:55.966-08:00Propiedades de la Multiplicación de Números Naturales<span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La multiplicación de números naturales cumple las propiedades asociativa, conmutativa, elemento neutro y distributiva del producto respecto de la suma.</span> <div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">1.-Asociativa </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si a, b, c son números naturales cualesquiera se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(a · b) · c = a · (b · c)</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Por ejemplo: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(3 · 5) · 2 = 15 · 2 = 30</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">3 · (5 · 2) = 3 · 10 = 30</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los resultados coinciden, es decir, </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">(3 · 5) · 2 = 3 · (5 · 2)</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">2.- Conmutativa </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si a, b son números naturales cualesquiera se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">a · b = b · a</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Por ejemplo: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">5 · 8 = 8 · 5 = 40</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">3.-Elemento neutro </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">El 1 es el elemento neutro de la multiplicación porque, cualquiera que sea el número natural a, se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">a · 1 = a</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;"><br />
4.- Distributiva del producto respecto de la suma </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si a, b, c son números naturales cualesquiera se cumple que: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">a · (b + c) = a · b + a · c</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Por ejemplo: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">5 · (3 + 8) = 5 · 11 = 55</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">5 · 3 + 5 · 8 = 15 + 40 = 55</span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los resultados coinciden, es decir, </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">5 · (3 + 8) = 5 · 3 + 5 · 8</span></div>Jorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.com0tag:blogger.com,1999:blog-6031889885966808781.post-1581253178139571212010-12-29T11:20:00.000-08:002010-12-29T11:20:06.198-08:00Propiedades de la Sustracción de Números Naturales<div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Igual que la suma la resta es una operación que se deriva de la operación de contar. </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si tenemos 6 ovejas y los lobos se comen 2 ovejas ¿cuantas ovejas tenemos?. Una forma de hacerlo sería volver a contar todas las ovejas, pero alguien que hubiese contado varias veces el mismo caso, recordaría el resultado y no necesitaría volver a contar las ovejas. Sabría que 6 - 2 = 4. </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los términos de la resta se llaman minuendo (las ovejas que tenemos) y sustraendo (las ovejas que se comieron los lobos). </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; mso-outline-level: 4; text-align: justify;"><span lang="ES-MX" style="color: #666666; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Propiedades de la resta: </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La resta no tiene la propiedad conmutativa (no es lo mismo a - b que b - a) </span></div>Jorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.com0tag:blogger.com,1999:blog-6031889885966808781.post-62652901316317232332010-12-27T16:16:00.000-08:002010-12-29T11:19:31.685-08:00Propiedades de la División de Números Naturales<div align="center" class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; mso-outline-level: 4; text-align: center;"><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La división es la operación que tenemos que hacer para repartir un numero de cosas entre un número de personas. </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Los términos de la división se llaman dividendo (el número de cosas), divisor (el número de personas), cociente (el numero que le corresponde a cada persona) y resto (lo que sobra). </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Si el resto es cero la división se llama exacta y en caso contrario inexacta. </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 0pt; mso-outline-level: 4; text-align: justify;"><span lang="ES-MX" style="color: #666666; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">Propiedades de la división </span></div><div class="MsoNormal" style="line-height: normal; margin: 0cm 0cm 10pt; mso-margin-bottom-alt: auto; mso-margin-top-alt: auto; text-align: justify;"><span lang="ES-MX" style="color: #333333; font-family: "Times New Roman", "serif"; font-size: 12pt; mso-ansi-language: ES-MX; mso-fareast-font-family: "Times New Roman"; mso-fareast-language: ES-MX;">La división no tiene la propiedad conmutativa. No es lo mismo a/b que b/a. </span></div></div>Jorge Sanchez Reyeshttp://www.blogger.com/profile/05743144496384398664noreply@blogger.com0