tag:blogger.com,1999:blog-62334266427498951372017-07-23T02:47:05.677-07:00MathsMy maths notebook. Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6233426642749895137.post-23614172107015287942016-10-09T01:45:00.000-07:002016-10-09T01:55:10.990-07:00Quadratic Equations General Solution Prooff<h1>Quadratic Equations</h1><h2>General form</h2><div style="font-size: 120%;">$$ax^2 + bx + c = 0 .....(1)$$ </div><h2>Find general solution</h2><br /><div style="font-size: 120%;">Divide both sides by a \begin{align} x^2 + \frac{b}{a}x + \frac{c}{a} & = 0\\\\ x^2 + \frac{b}{a}x & = - \frac{c}{a} .....(2)\\ \end{align} Now \begin{align} \left(x + \frac{b}{2a}\right)^2 &= x^2 + \frac{b}{a}x + \left(\frac{b^2}{4a^2}\right) \end{align} Rewrite (2) \begin{align} \left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} & = - \frac{c}{a}\\\\ \left(x + \frac{b}{2a}\right)^2 &= \frac{b^2}{4a^2} - \frac{c}{a}\\\\ x + \frac{b}{2a} &= \pm\sqrt{\frac{b^2}{4a^2} - \frac{c}{a}}\\\\ x + \frac{b}{2a} & = \pm\sqrt{\frac{b^2-4ac}{4a^2}}\\\\ x + \frac{b}{2a} & = \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\ x & = -\frac{b}{2a} \frac{\pm\sqrt{b^2-4ac}}{2a}\\\\ x & = \frac{-b \pm\sqrt{b^2-4ac}}{2a}\\\\ \end{align} </div><br /> <h2>So the general solution is $$ x = \frac{-b \pm\sqrt{b^2-4ac}}{2a}$$ <!--img src="quadratic.png" alt="General Solution" style="width:237px;height:61px; "--></h2>Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.com0tag:blogger.com,1999:blog-6233426642749895137.post-73973437468779675502016-10-02T02:51:00.001-07:002016-10-02T02:51:13.084-07:00Blogger iPhone and LaTeX<p> I have spent several days working out how to write mathematical equations that can be created and read on Mac, iPad and iPhone. I thought I had a working solution which will be subject of a lengthy blog post later once I have fully tested it. This involved HTML editors, Weebly, Blogger, a couple of iPad apps and a couple of scripts. </p><p>I was disappointed this morning to see that although all worked fine on my iPad the LaTeX commands were not translated to readable equations on my iPhone on Blogger posts. Despair soon vanished when I set Blogger for this particular blog to always use <em>desktop </em>mode. Just navigate to the Blogger <em>design</em> page, select <em>template,</em> click on flower button as shown below and select <em>No</em>, <em>use</em> <em>desktop</em> option on pop up window. </p><div class="separator" style="clear: both; text-align: none;"> </div><div class="separator" style="clear: both; text-align: center;"><a href="https://lh3.googleusercontent.com/-tRDBcYxsO2U/V_DYjkE-cxI/AAAAAAAA_E4/FpPYQZJ_6Rw/s2048/Photo%25252020161002105034256.jpg" target="_blank" style="margin-left: 1em; margin-right: 1em;"><img src="https://lh3.googleusercontent.com/-tRDBcYxsO2U/V_DYjkE-cxI/AAAAAAAA_E4/FpPYQZJ_6Rw/s500/Photo%25252020161002105034256.jpg" id="blogsy-1475401870877.6162" class="aligncenter" width="500" height="375" alt=""></a></div><p> </p><div style="text-align: right; font-size: small; clear: both;" id="blogsy_footer"><a href="http://blogsyapp.com" target="_blank"><img src="http://blogsyapp.com/images/blogsy_footer_icon.png" alt="Posted with Blogsy" style="vertical-align: middle; margin-right: 5px;" width="20" height="20" />Posted with Blogsy</a></div>Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.com0tag:blogger.com,1999:blog-6233426642749895137.post-63014589482890183632016-10-01T08:31:00.003-07:002016-10-01T10:08:49.099-07:00Square Root Of 2 Is Irrational<h1>Square Root Of 2 Is Irrational</h1>Let $\sqrt{2} = \frac{p}{q}$ where p and q are two positive integers such that $\frac{p}{q}$ have no common factors. <br />$$ \left( \frac{p}{q}\right)^2=2 \\ \therefore p^2=2q^2 $$ This means that $p^2$ is an even number and so $p$ is an even number and can be written as $2r$. $$ \therefore 4r^2=2q^2 \\ q^2 = 2r^2 \\ $$ By the same argument as above this means $q$ can be written as $2s$ but then $\frac{p}{q}$ would have the common factor of 2, and so contradicting initial statement. <br /><br /> <h4>Therefore $\sqrt{2}$ must be an irrational number.</h4> Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.com0tag:blogger.com,1999:blog-6233426642749895137.post-26496274726630461812016-09-30T14:46:00.001-07:002016-09-30T14:46:36.919-07:00Testing Blogsy with Blogger<p> This post was written on my iPad using <em>Blogsy</em> app.<\p></p><p>I wanted to see if the <em>LaTeX</em> source would translate correctly. I generated the LaTeX code on the very good <em>MathPad</em> app. </p><p> $$y=x^{2}+\sqrt {x+2}$$ </p><p> </p><div style="text-align: right; font-size: small; clear: both;" id="blogsy_footer"><a href="http://blogsyapp.com" target="_blank"><img src="http://blogsyapp.com/images/blogsy_footer_icon.png" alt="Posted with Blogsy" style="vertical-align: middle; margin-right: 5px;" width="20" height="20" />Posted with Blogsy</a></div>Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.com0tag:blogger.com,1999:blog-6233426642749895137.post-35828795920905646192016-09-30T13:18:00.000-07:002016-09-30T13:32:17.954-07:00Introduction<h3>Background</h3>I enjoy learning and maths has always fascinated me once I got over typical school boy dislike. I know I am not a proper mathematician but know enough to understand most of maths if I put my mind to it and to appreciate its beauty, similar to appreciating a good book even though I am not capable of writing one myself.<br /><br />I studied maths to A-level as two subjects, Pure Maths and Applied Maths and of course it featured a lot in my graduate Physics course.<br /><h3>General Relativity</h3>Every autumn I prepare for winter activities and most recent years I have decided that this is the year I am going to get to grips with General Relativity. Special Relativity I can handle but the university steered away from attempting to teach General Theory. This usually makes me realise that I should at first refresh my maths knowledge, and this is what I am starting out on, again, just now. This winter I shall be 70, so if I am ever going to achieve this I had better get a move on.<br /><br /><h3>Note Taking</h3>I prefer to make notes on line and store in the cloud so that I can always access them when the mood takes me. I am these days completely Apple hardware, iMac, MacBook, iPad and iPhone. My wife has iPad and iPhone. My daughter's family also are Apple dominated, with just a few old windows lap-tops that see the light of day occasionally for grandchildren homework, but this is less as they now can use Mum's MacBook (hand down from Dad).<br /><br />Documenting maths has caused me some concern. What is the best way to write equations which are easy to do, can be edited and look good. I believe I now have the solution which will be the subject of another entry.<br /><br /><h3>Reason for Blog</h3>So primary reason for this blog is so I can jot down my maths notes as they arise, which I can then, if desirable, organise into a more formal web site later.<br /><br /><br />$$y =\int_{0}^{\infty}x^3dx$$Rod Thttp://www.blogger.com/profile/07091370586796916723noreply@blogger.com0