But there was a problem. There should be 3 solutions for the above equation, but they could only find 1 solution. <\/p>\n
Why do we expect 3 solutions? Let's look at 3 cases where we know something about the roots.<\/p>\n
Tip:<\/b> A \"root\" of an equation is a value that makes the equation \"true\".<\/p>\n
(a) A linear equation<\/strong> has one solution (eg x<\/em> + 2 = 7<\/span> has solution x<\/em> = 5<\/span>). The highest power of x<\/em><\/span> in a linear equation is 1.<\/p>\n In this image, we see the graph of a linear equation and it cuts the x-axis in one place only (marked with a red dot). <\/p>\n Tip:<\/b> Wherever the graph cuts the x-axis will give you the root of an equation.<\/p>\n [Exceptions: If the linear equation has slope 0, it will be parallel to the x<\/i>-axis and will not pass through it, so will have no solutions, like the following image.<\/p>\n The other exception is if the line is parallel to the y-axis.]<\/p>\n Tip:<\/b> Always draw the graph! The Chinese are right - a picture is worth a 1000 words. When you draw graphs you get a much better idea of what is going on.<\/p>\n (b) A quadratic equation<\/strong> has 2 solutions (eg x<\/em>2<\/sup> − 4x<\/em> + 3 = 0<\/span> can be factored giving solutions x<\/em> = 1<\/span> or x<\/em> = 3<\/span>.) The highest power of x<\/em><\/span> in a quadratic equation is 2. <\/p>\n Here we see a typical quadratic equation with 2 roots (marked with red dots).<\/p>\n We can also have the case where there is only one solution. This might occur if our quadratic is of the form (x − 3)2<\/sup> = 0.<\/p>\n But sometimes when you use the quadratic formula you will get a negative number under the square root. <\/p>\n [Tip:<\/b> The Quadratic Formula is x<\/em> = [−b<\/em> ±√(b<\/em>2<\/sup> − 4ac<\/em>)] \/ 2<\/span>. For more information, see Quadratic Formula<\/a>.]<\/p>\n For example, when we solve x<\/em>2<\/sup> + x<\/em> + 1 = 0<\/span> using the quadratic formula, we get:<\/p>\n x<\/em> = [−1 ±√(1 − 4)] \/ 2<\/span><\/p>\n = [−1 ±√(−3)] \/ 2<\/span><\/p>\n At this point we usually give up and say there are \"no real solutions\". But what does this really mean? We obtained 2 solutions, as expected, but they are \"illegal\".<\/p>\n The graph of this situation is as follows. The curve does not cut through the x-axis at all so there appear to be no solutions.<\/p>\n (c) We expect a cubic equation<\/strong> to have 3 solutions since the highest power of x is 3. In the following image, we see that the curve cuts the x-axis in 3 places, giving us 3 roots.<\/p>\n But sometimes, there is a repeated root, so we only have 2 places where the curve cuts the x-axis. <\/p>\n An example here could be (x − 1)(x − 3)2<\/sup> = 0.<\/p>\n Back to our first cubic equation example, x<\/em>3<\/sup> + x<\/em>2<\/sup> − x<\/em> + 2 = 0<\/span>. The mathematicians could find one solution OK (x<\/em> = −2<\/span>), but where were the other 2?<\/p>\n The graph of this situation is as follows. There is one point where the curve cuts through the x-axis.<\/p>\n By the early 1600s, the famous mathematician and philosopher Rene Descartes called the square root of a negative number \"imaginary\". No-one was really sure if they were legitimate numbers, but because such numbers kept cropping up in different fields of mathematics — and they were found to \"work\" — they were studied more closely.<\/p>\n Around this time, the letter i<\/em> (possibly for \"imaginary\") was used to denote the square root of −1. So we write:<\/p>\n i<\/em> = √−1<\/p>\n A complex number<\/strong> contains a real part and an imaginary part. <\/p>\n Using this, we can solve our cubic equation from before. We cannot see the other 2 solutions but we can apply various methods to find the complex roots<\/a> and we get:<\/p>\n x = (1 ± i√3) \/ 2<\/p>\n So there are 3 roots after all, x = 2 and x = (1 ± i√3) \/ 2.<\/p>\n In 1734 the brilliant Swiss mathematician Leonhard Euler produced the following formula:<\/p>\n (cos θ + i<\/em> sin θ)n<\/em><\/sup> = cos n<\/em>θ + i<\/em> sin n<\/em>θ<\/span><\/p>\n This formula was enormously powerful for calculating<\/a> values of complex numbers and it integrated trigonometry and complex numbers.<\/p>\n In the years since the 18th century, complex numbers have been regarded as legitimate numbers in the field of mathematics. They have proved to be important in electronics<\/strong>.<\/p>\n You can see more about these special numbers, with applications in electronics, here: Complex Numbers<\/a>.<\/p>\n As an interesting diversion, complex numbers are also involved in art. Fractals<\/b> are made using complex numbers. Here's an example:<\/p>\n See more on this topic: Fractals<\/a>.<\/p>\n Coming up on March 14th is Pi Day, so named because the first 3 digits of pi and the date 3\/14 coincide. Some people go all out and have a special event at 1:59 PM, to recognize the next 3 digits of pi.<\/p>\n Pi is an interesting number and mathematicians have spent countless hours over the centuries learning more about it. <\/p>\n Quick facts about pi:<\/b> <\/p>\n What is your school doing on Pi Day? If you don't have anything planned yet, there are some good ideas in these resources:<\/p>\n Education World<\/a> A poll during Feb 2009 asked readers if they were allowed to use formula sheets during their tests and exams. I asked this question because in a previous poll many students said that one of the main reasons that math is difficult because learning math formulas is hard. <\/p>\n Poll results:<\/b><\/p>\n 36% Some exams only Tip:<\/b> Even if you know you're going to get a formula sheet in your exam, learn your formulas well enough so that you don't need the sheet.<\/p>\n Why?<\/p>\n Because if you know the formulas well, you are more likely to choose the correct one. I have seen many students rely 100% on the formula sheet and it is obvious that they have no idea which formula to use for a particular situation.<\/p>\n Of course, you can always check that you have the formula exactly right using the sheet you are given.<\/p>\n The latest poll<\/b> asks readers what they do on the night before a math exam. Do you study until late, or go to bed early? You can answer on any page in Interactive Mathematics<\/a>.<\/p>\n 1) Friday math movie - math problem solving with mind maps<\/a> 2) Akamai - visualizing Web traffic data<\/a> 3) Friday Math Movie - Math in Art<\/a> 4) Manga Guides to Statistics and Databases<\/a> 5) Energy reduction with Google's help<\/a> Here's a great quote:<\/p>\n \u201cConcern for man and his fate must always form the chief interest of all technical endeavors. Never forget this in the midst of your diagrams and equations.\u201d I realized some time ago that many math classes have no human<\/b> element to them. That is, they are all algebra with no application and no implications for people.<\/p>\n So my suggestion for you, whether you are a student or a teacher, is to always look for what's behind the math that you are doing. <\/p>\n Math wasn’t invented to torture you — it was developed to solve some human problem. Go find what that problem was — you'll understand the math better and find it more interesting, too. (Most chapters in Interactive Mathematics<\/a> start with some applications and\/or history of the topic.)<\/p>\n Until next time, enjoy whatever you learn.<\/p>\n![\"roots-linear-1\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-linear-1.gif\" "\\"roots-linear-1\\"")
<\/p>\n![\"roots-linear-2\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-linear-2.gif\" "\\"roots-linear-2\\"")
<\/p>\n![\"roots-2\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-1.gif\" "\\"roots-1\\"")
<\/p>\n![\"roots-1\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-2.gif\" "\\"roots-2\\"")
<\/p>\n![\"roots-3\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-3.gif\" "\\"roots-3\\"")
<\/p>\n![\"roots-cubic-3\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-cubic-3.gif\" "\\"roots-cubic-3\\"")
<\/p>\n![\"roots-cubic-2\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-cubic-2.gif\" "\\"roots-cubic-2\\"")
<\/p>\n![\"roots-cubic-1\"](\"\/blog\/wp-content\/images\/2009\/03\/roots-cubic-1.gif\" "\\"roots-cubic-1\\"")
<\/p>\n![\"pearl-fractal\"](\"\/blog\/wp-content\/images\/2009\/03\/pearl-fractal.jpg\" "\\"pearl-fractal\\"")
\nImage source<\/a>.<\/p>\n
\n2. Pi Day<\/h2>\n
\n
\nKathi Mitchell<\/a>
\nWikiHow<\/a><\/p>\n
\n3. Latest IntMath Poll - Math formula sheets<\/h2>\n
\n32% Never
\n31% Always<\/p>\n
\n4. From the Math Blog<\/h2>\n
\nThis week's movie gives you some good tips on how to solve math problems.<\/p>\n
\nHere's some interesting statistics about web traffic, attacks on the system and online music downloads.<\/p>\n
\nThis movie shows some beautiful art that comes from the study of dynamical systems in math.<\/p>\n
\nHere's a series of math books based on Japanese manga (comics).<\/p>\n
\nHow much electricity is your home consuming right now? Not sure? Here\u2019s a neat tool from Google that will allow you to see details of your daily energy consumption.<\/p>\n
\n5. Final Thought - Math is human, after all<\/h2>\n
\n
\nAlbert Einstein<\/p>\n<\/blockquote>\n