{"id":25807,"date":"2021-03-20T00:00:03","date_gmt":"2021-03-20T00:00:03","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25807"},"modified":"2025-03-02T19:58:01","modified_gmt":"2025-03-02T19:58:01","slug":"operations-with-natural-numbers-and-order-relations","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/operations-with-natural-numbers-and-order-relations\/","title":{"rendered":"Operations with Natural Numbers and Order Relations"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\"><center><\/p>\n<h1>Operations with Natural Numbers and Order Relations<\/h1>\n<p style=\"text-align:center;\"><strong>Summary:<\/strong><br \/>\nIn this class, we will delve into natural numbers and their basic operations, starting with the origin and properties of addition, multiplication, and exponentiation, in relation to Peano&#8217;s Axioms. We will examine key properties such as commutativity, associativity, distributivity, and rules for simplification and inversion. We will use mathematical induction to demonstrate theorems and properties. Additionally, we will analyze the order relation among natural numbers, including the law of trichotomy and the properties of transitivity and monotonicity, with practical exercises to apply these concepts. Finally, we will address inverse operations (subtraction and division) and explore the exponentiation of natural numbers and their properties.\n<\/p>\n<p><\/center><\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, students will be able to:\n<\/p>\n<p style=\"text-align:justify;\">\n<ol>\n<li><strong>Understand<\/strong> the origin and properties of the basic operations of natural numbers.<\/li>\n<li><strong>Apply<\/strong> the properties of operations with natural numbers, such as commutativity, associativity, distributivity, and the rules for simplification and inverse operation.<\/li>\n<li><strong>Apply<\/strong> mathematical induction for demonstrating simple properties and theorems.<\/li>\n<li><strong>Analyze<\/strong> the properties of order in natural numbers, such as the law of trichotomy and the properties of transitivity and monotonicity.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>TABLE OF CONTENTS<\/u>:<\/strong><br \/>\n<a href=\"#1\">The Origin of Basic Operations of Natural Numbers<\/a><br \/>\n<a href=\"#2\">The Order Induced by Operations of Natural Numbers<\/a><br \/>\n<a href=\"#3\">Inverse Operations: Subtraction and Division of Natural Numbers<\/a><br \/>\n<a href=\"#4\">Powers of Natural Numbers<\/a><br \/>\n<a href=\"#5\">Proposed Problems and Solutions<\/a>\n<\/p>\n<p><center><br \/>\n<iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/jKD71TjMC4s\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">Although operations with natural numbers are well-known, it is necessary to synthesize this knowledge using a bit more \u00abmathematical manners.\u00bb For this reason, we will review the operations of addition, multiplication, and power of natural numbers and their properties.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>The Origin of Basic Operations with Natural Numbers<\/h2>\n<h3>Addition Operation<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=49s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>The foundation of the addition operation was reviewed in the class on<\/strong><\/span><\/a> <a href=\"http:\/\/toposuranos.com\/material\/en\/natural-numbers-and-peanos-axioms\/\" rel=\"noopener\" target=\"_blank\"><strong>Natural Numbers and Peano&#8217;s Axioms,<\/strong><\/a> because the successor of a natural number can also be presented as follows:<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">S(n) = n+1<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">As we said that <span class=\"katex-eq\" data-katex-display=\"false\">2=S(1), 3=S(2), 4=S(3), \\cdots <\/span><\/bdi> and so on, then we can interpret addition as the successive application of the succession operation.<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+1 =S(n),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+2 =S(S(n)),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+3 =S(S(S(n))),<\/span><\/bdi><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\vdots<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">And in general:<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n+m = \\underbrace{S(S(\\cdots S(}_{m\\;times} n)\\cdots)) <\/span><\/bdi><\/p>\n<h4>Properties of Addition<\/h4>\n<p style=\"text-align: justify; color: #000000;\">If <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N},<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=131s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">then from this we can derive the well-known properties of addition:<\/span><\/strong><\/a><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Commutativity<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b=b+a<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Associativity<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b+c=(a+b)+c=a+(b+c)<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Simplification<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\leftrightarrow b=c <\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">All these properties can be demonstrated by induction, but we will skip that work. However, I encourage you to try it as a way to practice the technique of induction.<\/p>\n<h3>Multiplication Operation<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=230s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Similarly, the product of natural numbers<\/span><\/strong><\/a> is defined as a successive application of addition. Therefore, we have<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">n\\cdot m = \\underbrace{n+ n+ \\cdots + n}_{m\\;times}<\/span><\/bdi><\/p>\n<h4>Properties of Multiplication<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=251s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">And analogously<\/span><\/strong><\/a> its properties can be obtained<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Commutativity<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">ab=ba<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Associativity<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">abc=(ab)c=a(bc)<\/span><\/bdi><\/td>\n<\/tr>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Simplification<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">ab=ac \\leftrightarrow b=c <\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Furthermore, from the definition of multiplication, the \u00ab1\u00bb in natural numbers acquires the quality that transforms it into <strong>unit:<\/strong><\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Unit<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">1a=a=a1<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Combined Addition and Multiplication<\/h3>\n<p style=\"text-align: justify; color: #000000;\">When the operations of addition and multiplication are combined, we obtain the distributive property of addition with respect to multiplication<\/p>\n<table>\n<tbody>\n<tr style=\"text-align: justify; color: #000000;\">\n<td><strong>Distributivity<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a(b+c)=ab+ac<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Order Induced by Operations of Natural Numbers<\/h2>\n<p style=\"text-align: justify; color: #000000;\">From the operations of addition and multiplication we reviewed, an order relation is induced in natural numbers through the following definitions:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is less than <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b := (\\exists k \\in \\mathbb{N}) (a + k = b)<\/span><\/bdi><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is greater than <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b := (\\exists k \\in \\mathbb{N}) (a = b + k)<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Properties of Order in Natural Numbers<\/h3>\n<h4>Law of Trichotomy<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=513s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>From this, it follows that only<\/strong><\/span><\/a> one of the following three situations can occur:<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a = b<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi><\/li>\n<\/ol>\n<p style=\"text-align: justify; color: #000000;\">If it happens that, for example, <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is not less than <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span>, then one of the two must occur: either <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a=b<\/span><\/bdi>, or <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, that is greater than or equal, and it would be written: <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\geq b.<\/span><\/bdi> And analogously, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi> is written when it is less than or equal.<\/p>\n<h4>Transitive Property<\/h4>\n<p style=\"text-align: justify; color: #000000;\">If <span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=625s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>are any natural numbers, then it is fulfilled that:<\/strong><\/span><\/a><\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">[(a\\lt b) \\wedge (b\\lt c)] \\rightarrow (a\\lt c)<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">And analogously:<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">[(a\\gt b) \\wedge (b\\gt c)] \\rightarrow (a\\gt c)<\/span><\/bdi><\/p>\n<h4>Monotonicity Property<\/h4>\n<p style=\"text-align: justify; color: #000000;\">There is a monotonicity property for both addition and multiplication, which is:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>Monotonicity of addition<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\lt b) \\leftrightarrow (a+c \\lt b+c) <\/span><\/bdi><bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\gt b) \\leftrightarrow (a+c \\gt b+c)<\/span><\/bdi><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center; color: #000000;\"><strong>Monotonicity of multiplication<\/strong><br \/>\n<bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\lt b) \\leftrightarrow (a c \\lt b c) <\/span><\/bdi><bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a\\gt b) \\leftrightarrow (a c \\gt b c)<\/span><\/bdi><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Inverse Operations: Subtraction and Division of Natural Numbers<\/h2>\n<h3>Subtraction of Natural Numbers<\/h3>\n<p style=\"text-align: justify; color: #000000;\">If <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N}<\/span><\/bdi>, <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=782s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">we say that the difference between<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (in that order), written as <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b<\/span><\/bdi>, is defined through the relation<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a-b=c \\leftrightarrow a= b+c<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">As we can see, such a relation is true only if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, because there is no <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\in \\mathbb{N}<\/span><\/bdi> with which this relation can be satisfied if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\leq b.<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">Through the definition of subtraction, we have the well-known rule of \u00abwhat is added on one side of the equation can be passed to the other side subtracting, and vice versa\u00bb.<\/p>\n<h3>Division of Natural Numbers<\/h3>\n<p style=\"text-align: justify; color: #000000;\">If <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{N}<\/span><\/bdi>, <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=917s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>we say that the division between<\/strong><\/span><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> and <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> (in that order), written as <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi>, is defined through the relation<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b=c \\leftrightarrow a= bc<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">From the definition of division, we have the rule of \u00abwhat is multiplying on one side of the equation can be passed to the other side dividing, and vice versa\u00bb.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Just as for the subtraction <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a - b<\/span><\/bdi> to exist it must be that <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, for the division <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\/b<\/span><\/bdi> to exist, it is necessary that <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> be \u00abdivisible\u00bb by <span class=\"katex-eq\" data-katex-display=\"false\">b.<\/span> This is represented by writing<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> is divisible by <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> <bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\; :=a|b \\; := \\; (\\exists k \\in \\mathbb{N})(a = kb)<\/span><\/bdi><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Powers of Natural Numbers<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1020s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>With natural numbers, powers can be defined.<\/strong><\/span><\/a> Raising a natural number <span class=\"katex-eq\" data-katex-display=\"false\">b,<\/span> called the base, to another natural number <span class=\"katex-eq\" data-katex-display=\"false\">n,<\/span> called the exponent, means multiplying <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> by itself <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> times. Thus<\/p>\n<p style=\"text-align: center; color: #000000;\"><bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^n = \\underbrace{bb\\cdots b}_{n\\;times}<\/span><\/bdi><\/p>\n<p style=\"text-align: justify; color: #000000;\">If <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a,b,n,m\\in\\mathbb{N},<\/span><\/bdi> by (double) induction, the following properties can be demonstrated:<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle b^nb^m=b^{n+m}<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{b^n}{b^m} = b^{n-m},<\/span><\/bdi> provided that <bdi><span class=\"katex-eq\" data-katex-display=\"false\">n\\lt m<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (ab)^n=a^nb^n<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{a}{b}\\right)^n = \\frac{a^n}{b^n}<\/span><\/bdi><\/li>\n<li><bdi><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle (b^n)^m=b^{nm}<\/span><\/bdi><\/li>\n<\/ol>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Proposed and Solved Problems<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>All the properties shown here can be demonstrated using mathematical induction (either simple or double), but I have not developed them because the resulting demonstration is unnecessarily long for these intuitive results. However, those following these classes can try to perform these demonstrations as an exercise. <strong>[Only proposed]<\/strong><\/li>\n<li>Is <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{n^m}<\/span><\/bdi> (which is defined as <bdi><span class=\"katex-eq\" data-katex-display=\"false\">b^{(n^m)})<\/span><\/bdi> the same as <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(b^n)^m<\/span><\/bdi>? <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1298s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[Solution]<\/span><\/strong><\/a><\/li>\n<li>Using the properties seen, verify the equalities:<br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(c+d) = ac+ad+bc+bd<\/span><\/bdi><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1556s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(c-d) = ac-ad+bc-bd,<\/span><\/bdi>; if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1660s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nc)<bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)(c-d) = ac-ad-bc+bd,<\/span><\/bdi>; if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">a\\gt b<\/span><\/bdi>, <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1730s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a>&nbsp;<\/li>\n<li>Demonstrate that <\/br><br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)^2 = a^2 + 2ab + b^2<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1903s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^2 = a^2 - 2ab + b^2<\/span><\/bdi>; if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1953s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)(a-b) = a^2-b^2<\/span>; if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=1978s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nd) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a+b)^3 = a^3 + 3a^2b+3ab^2+b^3<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2008s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\ne) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)^3 = a^3 - 3a^2b+3ab^2-b^3<\/span><\/bdi>; if <bdi><span class=\"katex-eq\" data-katex-display=\"false\">c\\gt d<\/span><\/bdi><a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2124s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"> <strong>[Solution]<\/strong><\/span><\/a><\/p>\n<p>&nbsp;<\/li>\n<li>Prove by complete induction the following properties:<\/br><br \/>\na) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1+2+3+4+\\cdots+n = \\displaystyle \\frac{n(n+1)}{2}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2328s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a> <\/br><br \/>\nb) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1^2+2^2+3^2+4^2+\\cdots+n^2 = \\displaystyle \\frac{n(n+1)(2n+1)}{6}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2505s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a><\/br><br \/>\nc) <bdi><span class=\"katex-eq\" data-katex-display=\"false\">1^3+2^3+3^3+4^3+\\cdots+n^3 = \\displaystyle \\frac{n^2(n+1)^2}{4}<\/span><\/bdi> <a href=\"https:\/\/www.youtube.com\/watch?v=jKD71TjMC4s&amp;t=2972s\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #ff0000;\"><strong>[Solution]<\/strong><\/span><\/a>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Operations with Natural Numbers and Order Relations Summary: In this class, we will delve into natural numbers and their basic operations, starting with the origin and properties of addition, multiplication, and exponentiation, in relation to Peano&#8217;s Axioms. We will examine key properties such as commutativity, associativity, distributivity, and rules for simplification and inversion. We will [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25803,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":19,"footnotes":""},"categories":[583,1031,567],"tags":[],"class_list":["post-25807","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-and-geometry","category-general-algebra","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Operations with Natural Numbers and Order Relations - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Everything about Operations and Order Relations of Natural Numbers, Peano&#039;s Axioms, Induction, Trichotomy, and Monotonicity\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/operations-with-natural-numbers-and-order-relations\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Operations with Natural Numbers and Order Relations\" \/>\n<meta property=\"og:description\" content=\"Everything about Operations and Order Relations of Natural Numbers, Peano&#039;s Axioms, Induction, Trichotomy, and Monotonicity\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/operations-with-natural-numbers-and-order-relations\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-20T00:00:03+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-03-02T19:58:01+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/OPERACIONES-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Operations with Natural Numbers and Order Relations\" \/>\n<meta name=\"twitter:description\" content=\"Everything about Operations and Order Relations of Natural Numbers, Peano&#039;s Axioms, Induction, Trichotomy, and Monotonicity\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/12\/OPERACIONES.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Operations with Natural Numbers and Order Relations\",\"datePublished\":\"2021-03-20T00:00:03+00:00\",\"dateModified\":\"2025-03-02T19:58:01+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/\"},\"wordCount\":1503,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/12\\\/OPERACIONES.jpg\",\"articleSection\":[\"Algebra and Geometry\",\"General Algebra\",\"Mathematics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/operations-with-natural-numbers-and-order-relations\\\/\",\"name\":\"Operations with Natural Numbers and Order Relations - 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