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\u00a0(yes, matrices<\/a> can be considered numbers).<\/li>\n<\/ul>\nSurely you included 1. Almost surely you included 3\/<\/i>5. But what about 0? \u22121? Does your definition include \u221a 2? Do you consider dx\u00a0<\/i>a number? Leibniz<\/a> did. Any of the others? (And, yes, they really are all \u201cnumbers.\u201d)<\/p>\n The lesson in this little demonstration is this: You don\u2019t really have a clear notion of what we mean when we use the word \u201cnumber.\u201d And this is fine. Not knowing is acceptable.<\/p>\n Sometimes it is even encouraged.<\/p>\n A principal goal of this course of study is to rectify this, at least a little bit. When the course is over you may or may not be able to give a better definition of the word \u201cnumber\u201d but you will have a deeper understanding of the real numbers at least. That is enough for now.<\/p>\n Read and understand the following development of the Quadratic Formula<\/a>.<\/p>\n Suppose\u00a0a <\/i> then Now let Were you able to follow the argument? Probably the step from equation (.1) to equation (.2) presented no difficulties. But what about the next step? Do you see where equation (.3) came from? If so, good for you. Most students, in fact most mathematicians, cannot make that step in their heads. But are you sure? Is there, perhaps, some small detail you\u2019ve overlooked?<\/p>\n Check to see.<\/p>\n That is, let\u00a0x\u00a0<\/i>=\u00a0y\u00a0<\/i>\u2212\u00a0b\/ <\/i>2a\u00a0<\/i>in equation (.2) and see if you can get equation (.3). Do it on that handy pad of paper we told you to get out earlier. Do it now. We\u2019ll wait.<\/p>\n If you\u00a0still\u00a0<\/i>haven\u2019t gotten out a pad of paper, give up now. You\u2019re going to fail this course. Seriously. Do you think we would spend so much time on this, that we would repeat it so many times, if it weren\u2019t important. GET OUT A PAD OF PAPER NOW! Last chance. You\u2019ve been warned.<\/p>\n Done? Good.<\/p>\n Perhaps you haven\u2019t been able to fill in the details on your own. That\u2019s ok. Many people can\u2019t. If not then get help: from a classmate, a friend, your instructor, whomever. Unfortunately most people won\u2019t get help in this situation. Instead they will perceive this as \u201cfailure,\u201d hide it and berate themselves or the problem as \u201cstupid.\u201d In short they will let their personal insecurities and demons overwhelm them.\u00a0Don\u2019t do this. Get help.\u00a0<\/i>You are neither dumb nor incapable. There are a thousand reasons that on any given day you might not be able to solve this problem. But don\u2019t let a bad day interfere with the education you are here for. Get someone to help you over this hump. Later you will be able to help your helper in the same way. Really.<\/p>\n At this point we assume that you\u2019ve successfully negotiated the transition from equation (.2) to equation (.5).<\/p>\n See? It really wasn\u2019t that bad after all. Just a lot of elementary algebra<\/a>. Now that you\u2019ve done it (or seen it done), it is easy to see that there really wasn\u2019t much there.<\/p>\n But this is the point! We left those computations out precisely because we knew that they were routine and that you could fill in the details. Moreover, filling in the details yourself gives you a little better insight into the computations. If we\u2019d filled them in for you we would have robbed you of that insight. And we would have made this book longer than it needs to be. We don\u2019t want to do either of those things. If we fill in all of the details of every computation for you, you won\u2019t learn to have confidence in your ability to do them yourself. And this book will easily double in length.<\/p>\n So the lesson here is this: Keep that pad of paper handy whenever you are reading this (or any other) mathematics text. You will need it. Routine computations will often be skipped. But calling them \u201croutine\u201d and skipping them does not mean that they are unimportant. If they were truly unimportant we would leave them out entirely.<\/p>\n Moreover \u201croutine\u201d does not mean \u201cobvious.\u201d Every step we took in the development of the Quadratic Formula was \u201croutine.\u201d But even routine computations need to be understood and the best way to understand them is to do them. This is the way to learn mathematics; it is the only\u00a0<\/i>way that really works. Don\u2019t deprive yourself of your mathematics education by skipping the most important parts. If you didn\u2019t fill in those details you\u2019re being stupid (or at least unduly stubborn). There is a good reason for putting these three lessons first. Stop wasting your time and intellect! Go do it now.<\/p>\n As you saw when you filled in the details of our development of the Quadratic Formula the substitution\u00a0x\u00a0<\/i>=\u00a0y\u00a0<\/i>\u2212\u00a0b<\/i><\/p>\n 2a\u00a0<\/i>was crucial because it turned<\/p>\n where\u00a0k\u00a0<\/i>depends only on\u00a0a<\/i>,\u00a0b<\/i>, and\u00a0c<\/i>. In the sixteenth century a similar technique was used by Ludovico Ferrari<\/a> (1522-1565) to reduce the general cubic equation<\/a><\/p>\n into the so-called \u201cdepressed cubic\u201d<\/p>\n where\u00a0p<\/i>, and\u00a0q\u00a0<\/i>depend only on\u00a0a<\/i>,\u00a0b<\/i>,\u00a0c<\/i>, and\u00a0d<\/i>.<\/p>\n The general depressed cubic had previously been solved by Tartaglia (the Stutterer, 1500-1557) so converting the general cubic into a depressed cubic provided a path for Ferrari to compute the \u201cCubic Formula\u201d \u2014 like the Quadratic Formula but better.<\/p>\n It is not entirely clear why eliminating the quadratic term should be depressing, but there it is.<\/p>\n","protected":false},"excerpt":{"rendered":"2.2 Lesson Two<\/h3>\n
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<\/i>\u00a00. If ![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_640073f0b1e83.png\")
\u00a0(.1)<\/p>\n![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_640073ff639ae.png\")
(.2)<\/span><\/p>\n![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_6400740add153.png\")
\u00a0<\/i>giving<\/p>\n![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_64007419813cd.png\")
<\/p>\n![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_640074911d688.png\")
into![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_6400749a0ceec.png\")
<\/p>\n![\"\"](\"https:\/\/www.booksofall.com\/in\/wp-content\/uploads\/sites\/13\/2023\/03\/img_640074aaf2d5f.png\")
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<\/p>\n