Jekyll2021-06-16T02:59:31+01:00https://www.flyingcoloursmaths.co.uk/feed.xmlFlying Colours MathsFlying Colours Maths helps make sense of maths at A-level and beyond.Ask Uncle Colin: A Pathological Conic2021-06-16T00:00:00+01:002021-06-16T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/ask-uncle-colin-pathological-conic<blockquote>
<p>Dear Uncle Colin,</p>
<p>The question tells me that $5x^2 + 4x + 4 + 9y^2 +12xy =0$ and that I have to find $xy$. I’ve tried plotting it in Desmos, but nothing shows up! What should I do?</p>
<p>Dramatically Overthinking Things</p>
</blockquote>
<p>Hi, DOT, and thanks for your message!</p>
<p>I think there are at least a couple of methods here, and I’ll give them both so you can take your pick.</p>
<p>The first, and the one that jumps out at me, is to complete the square on $y$; the second is to mess around with a quadratic formula, which I suspect comes out much the same way.</p>
<h3 id="completing-the-square-on-y">Completing the square on $y$</h3>
<p>If we rewrite it at $9y^2 + 12xy + 5x^2 + 4x + 4$ = 0$, we can replace the first pair of terms with $(3y + 2x)^2 - 4x^2$. This makes the whole thing $(3y+2x)^2 + x^2 + 4x + 4 = 0$.</p>
<p>The last three terms are a perfect square, $(x+2)^2$, so the equation can be rewritten as $(3y+2x)^2 + (x+2)^2 = 0$. This is only true when both of the terms are 0, so $3y=-2x$ and $x=-2$. That means $y = \frac{4}{3}$, and $xy = -\frac{8}{3}$.</p>
<p>Note that there is only point on the curve – that’s why Desmos didn’t plot it!</p>
<h3 id="quadratic">Quadratic</h3>
<p>A little less subtly, you can choose to pretend $x$ or $y$ is a constant. This time, I’ll fix $y$ and let $x$ be the variable.</p>
<p>Rewriting the equation as $5x^2 + (4+12y)x + (4 + 9y^2) = 0$ and applying the quadratic formula gives:</p>
<p>$x = \frac{-(4+12y)\pm \sqrt{(4+12y)^2 - 20(4+9y^2)}}{10}$.</p>
<p>Yuk.</p>
<p>Let’s look at the inside of that square root: it’s $(16 + 96y + 144y^2) - (80-180y^2) or $-36y^2 + 96y - 64$. That factorises as $-(6y-8)^2$.</p>
<p>For us to take the square root of it, it can’t be negative – and the only value of $y$ that gives a non-negative value is $y=\frac{4}{3}$ to make the square root 0.</p>
<p>Putting this into the formula gives $x = \frac{-20}{10}= -2$, as we got in the first part.</p>
<p>Hope that helps!</p>
<p>- Uncle Colin</p>Dear Uncle Colin, The question tells me that $5x^2 + 4x + 4 + 9y^2 +12xy =0$ and that I have to find $xy$. I’ve tried plotting it in Desmos, but nothing shows up! What should I do? Dramatically Overthinking ThingsAn iterative wrinkle2021-06-14T01:00:00+01:002021-06-14T01:00:00+01:00https://www.flyingcoloursmaths.co.uk/an-iterative-wrinkle<p>A wrinkle that came my way via my excellent friend @realityminus3 (Elizabeth A. Williams in real life); I paraphrase slightly:</p>
<blockquote>
<p>Show that $\frac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}} = 0$ (for $x>0$) can be rearranged as $x = \br{\frac{4}{3} - \frac{\sqrt{x}}{12}}^{2/3}$.</p>
</blockquote>
<p>Now, Elizabeth is the best godsdamned maths tutor in her geographical area ((Cardiff)), and is perfectly capable of doing that rearrangement – the question is <em>why</em> that’s the chosen rearrangement.</p>
<p>For the sake of form, let’s go through it:</p>
<p>Since $x > 0$, I can multiply both sides by $4x\sqrt{x}$ to get $12x^2 + x - 16\sqrt{x}=0$. Here is where the paths diverge:</p>
<ul>
<li>You can divide by $\sqrt{x}$ to get $12x^{3/2} + x^{1/2} - 16 =0$, so $x^{3/2} = \frac{16 - x^{1/2}}{12}$, or $x = \br{\frac{4}{3}-\frac{\sqrt{x}}{12}}^{2/3}$</li>
<li>Or you can rearrange in place to get $12x^{2} = 16\sqrt{x} - x$, giving $x^2 = \frac{4}{3}\sqrt{x} - \frac{x}{12}$ and $x = \br{\frac{4}{3}\sqrt{x}-\frac{x}{12}}^{1/2}$.</li>
</ul>
<p>The first is what they ask for in the question; the second is what felt obvious to Elizabeth – and, when put through the iteration hand-wringer, converges on the correct root.</p>
<h3 id="so-why-does-the-question-prefer-the-first">So why does the question prefer the first?</h3>
<p>There are a couple of reasons I can point to ((and I stress that I don’t find Elizabeth’s preferred answer at all unreasonable)) here:</p>
<ul>
<li>It makes sense to remove as much mathematical clutter as possible: since there’s a factor of $\sqrt{x}$ throughout, we may as well remove it;</li>
<li>If you <a href="https://www.desmos.com/calculator/wvvoqmbj4k">plot the graphs</a>, the paper’s right hand side (in blue) is much flatter than Elizabeth’s (in red), with the consequence that it converges more quickly.</li>
</ul>
<p>Now, I don’t like fixed-point iteration as a method at all (give me Newton-Raphson any day, much less guesswork and jiggery-pokery), and I’d be hard pushed to explain why it works, let alone why it’s supposed to be interesting. I suppose the interesting thing here is that there are multiple rearrangements that work. I wonder if there are others? ((But not enough to go looking for them.))</p>
<p>* Many thanks to Elizabeth A. Williams, Ben Sparks (@sparksmaths) and Paul Harrison (@singinghedgehog) for their contributions to the conversation about this.</p>
<p>* Edited 2021-06-14 to put some missing powers back in. Thanks, Adam!</p>A wrinkle that came my way via my excellent friend @realityminus3 (Elizabeth A. Williams in real life); I paraphrase slightly:Ask Uncle Colin: The Constant Term2021-06-09T00:00:00+01:002021-06-09T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/ask-uncle-colin-the-constant-term<blockquote>
<p>Dear Uncle Colin,</p>
<p>I’m supposed to find the coefficient of $x^0$ in the expansion of $\left(x^2 + \frac{1}{x}\right)^{15}$ – I couldn’t figure it out. Any advice?</p>
<p>- Combinations Of Numbers Seem Tough And Nasty, Though</p>
</blockquote>
<p>Hi, CONSTANT, and thanks for your message! As per, I can see several approaches.</p>
<h3 id="approach-1-a-sledgehammer">Approach 1: a sledgehammer</h3>
<p>A perfectly sensible approach here is to work out the first few terms: $\nCr{15}{0} x^{30} + \nCr{15}{1}x^{27} + \nCr{15}{2}x^{24} + \dots$ – oh! It looks like the powers are going down by three each time.</p>
<p>To get the zeroth power, I’ll have $\nCr{15}{10}x^0$, and $\nCr{15}{10} = 3,003$.</p>
<h3 id="approach-2-pulling-out-a-factor">Approach 2: pulling out a factor</h3>
<p>$\left(x^2 + \frac{1}{x}\right)^{15} = \left( \frac{1}{x^{15}}\right)\left(x^3 + 1\right)^{15}$.</p>
<p>To get a constant term when the $\frac{1}{x^{15}}$ is multiplied back in, we need to find the $x^15$ term in the second factor – again, that comes from the 10th term, and the coefficient is again $\nCr{15}{10} = 3,003$.</p>
<h3 id="approach-3-apply-the-formula">Approach 3: apply the formula</h3>
<p>Knowing that the $k$th term of $(a+b)^n$ is $\nCr{n}{k}a^{n-k} b^{k}$ can work nicely here.</p>
<p>We have $a = x^2$, $b = x^{-1}$ and $n=15$, so the $k$th term is $\nCr{15}{k}x^{30-2k} x^{-k}$, or $\nCr{15}{k} x^{30-3k}$.</p>
<p>The power is zero when $k=10$, giving - yet again - a coefficient of $\nCr{15}{10} = 3,003$.</p>
<p>Interesting number, 3,003. It shows up in Pascal’s Triangle eight times. <a href="https://aperiodical.com/2013/01/open-season-singmasters-conjecture/">No-one knows whether a number (other than 1) appears infinitely often.</a></p>
<p>Hope that helps!</p>
<p>- Uncle Colin</p>Dear Uncle Colin, I’m supposed to find the coefficient of $x^0$ in the expansion of $\left(x^2 + \frac{1}{x}\right)^{15}$ – I couldn’t figure it out. Any advice? - Combinations Of Numbers Seem Tough And Nasty, ThoughThe Dictionary of Mathematical Eponymy: The Euler Brick2021-06-07T00:00:00+01:002021-06-07T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/dome-euler-brick<p>The <a href="https://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler">list of things named after Leonhard Euler</a> on Wikipedia runs to about 1500 words, and, I would hazard, omits several such things.</p>
<p>So how to settle on one? I’ve come down on one of the greatest “low barrier, high ceiling” problems there is: it’s a conjecture so simple, you can grasp it as soon as you’ve got a sense of Pythagoras’s theorem, but so complicated that people have been searching for perfect bricks since… well, since before Euler really got going((Stigler’s Law of Eponymy states that nothing in science is named after its originator, and was first stated by Merton. The mathematical version is that everything is named after the <em>second</em> person to discover it, or else everything would be named after Euler. This example goes the other way.)).</p>
<h3 id="what-is-an-euler-brick">What is an Euler brick?</h3>
<p>An <em>Euler brick</em> is a cuboid that has integer side lengths, such that the diagonal of each face is also an integer.</p>
<p>For example, Paul Halcke discovered the smallest Euler brick in 1719: it has edges of length 44, 117 and 240. A moment’s work with Pythagoras((or several moments’ work if you, fearing the ninja, decline a calculator)), the rectangular face with sides 44 and 117 turns out to have a diagonal of length 125; the 44 and 240 face has a diagonal of length 244; and the 117 and 240 face a diagonal of length 267.</p>
<p>Euler studied these and came up with several ways to parameterise subsets of bricks, but not <em>all</em> such bricks. So the second most interesting question (for me) about such bricks is, is there a parametric formula to generate all of them?</p>
<p>The first most interesting question relates to <em>perfect Euler bricks</em>. A perfect Euler brick, in addition to its face diagonals, also has an integer main diagonal, from one corner to its opposite. The question: does a perfect Euler brick exist, or can it be proved not to? (If it does, its sides are extremely long.)</p>
<h3 id="why-is-it-important">Why is it important?</h3>
<p>I would class this problem as <em>annoying</em> more than important. It’s something very easy to get your teeth into, but very hard to make significant progress with. Perhaps its importance lies in turning interested young geometers into number theorists? Or perhaps there’s just nothing I like more than an unsolved problem.</p>
<h3 id="who-was-leonhard-euler">Who was Leonhard Euler?</h3>
<p>This is a bit like a football blog asking “who is Pelé?”, or a music blog asking “who was Bach?” <a href="https://en.wikipedia.org/wiki/Leonhard_Euler">Leonhard Euler</a> (1707-1783) was one of the… no, scratch that. He was <strong>the</strong> most important mathematician of all time. There’s barely a corner of maths he didn’t touch, and several corners that he built himself.</p>
<p>If I get started on his achievements, I’ll never finish – much like his written works, which are <em>still being published today</em>, nearly 250 years after he died.</p>
<h3 id="references">References</h3>
<ul>
<li><a href="https://mathworld.wolfram.com/EulerBrick.html">MathWorld</a></li>
<li><a href="https://en.wikipedia.org/wiki/Euler_brick">Wikipedia</a></li>
<li><a href="https://aperiodical.com/2013/03/open-season-the-perfect-cuboid/">Aperiodical</a></li>
</ul>The list of things named after Leonhard Euler on Wikipedia runs to about 1500 words, and, I would hazard, omits several such things.What’s the plot, episode 22021-06-04T01:00:00+01:002021-06-04T01:00:00+01:00https://www.flyingcoloursmaths.co.uk/whats-the-plot-2<p>Another plot, another challenge:</p>
<blockquote class="twitter-tweet"><p lang="en" dir="ltr">I plotted something… it looked interesting… that’s right: it’s time for another round of What’s My Plot?<br /><br />Tell me what you think the logic behind this diagram is, and YOU could win a coveted postcard! <a href="https://t.co/eXUExwSAM3">pic.twitter.com/eXUExwSAM3</a></p>— Colin Beveridge (@icecolbeveridge) <a href="https://twitter.com/icecolbeveridge/status/1398742848346931200?ref_src=twsrc%5Etfw">May 29, 2021</a></blockquote>
<script async="" src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>
<p>A bit more of a traditional plot, with several good answers all along the same lines:</p>
<ul>
<li>@OneDavidAtATime on Twitter was (I think) first off the mark: “I see the same function twice, which could be… a ratio? Something like $y=\frac{kx}{(a-x)(a+x)}$, $a$ and $k$ parameters. The red area would be the desired interval for the input $x$ so the output ratio is $y_1 \lt y \lt y_2$. $x$ could be some type of spatial shift around point $a$.”</li>
<li>@pretentious7 on Mastodon came in at about the same time with: “Hmmm looks sort of like a sideways logistic with a hyperbolic split.”</li>
<li>And my good friend Matthew on LinkedIn gave the fullest answer:</li>
</ul>
<blockquote>
<p>“Hello. Your mysetery plot - I was going to step thorugh the logic:</p>
<ul>
<li>Area between two curves : $f(x) < y < g(x)$ (although this isn’t consistently true, in places it’s $g(x)< y < f(x)$ - just spotted that)</li>
<li>two vertical asymptotes each (which I’m pretending are -2, 2 and $-\frac{1}{2}$, $\frac{1}{2}$: $f(1/2)=0$, $f(-1/2)=0$, $g(2)=0$, $g(-2)=0$</li>
<li>put the zeros as factors in the denominator
-> crosses the origin with positive slope => numerator is of the order of $x$</li>
</ul>
<p>so $f(x) = -\frac{x}{(x-1/2)(x+1/2)}$ and
$g(x) = -\frac{x}{(x-2)(x+2)}$
but then it isn’t the area between these lines in the usual manner….</p>
</blockquote>
<p>(As an aside, I point at a couple of Matthew’s mathematical problem-solving techniques: one is to make a note of possible problems, like the difference in order of the functions; another is to pick easier numbers than the numbers in the actual problem and look at the <em>structure</em> of the graph rather than the details. He’s a clever chap.)</p>
<p>Postcards for all three!</p>
<hr />
<p>For the record, the actual function I plotted was $\left( x^2 + \frac{x}{y} - 3\right)^2 \le 8$.</p>
<p>Stay tuned for the next unmissable episode of What’s The Plot, probably in a few weeks’ time!</p>Another plot, another challenge:Ask Uncle Colin: Simplify *this*!2021-06-02T01:00:00+01:002021-06-02T01:00:00+01:00https://www.flyingcoloursmaths.co.uk/ask-uncle-colin-simplify-this<blockquote>
<p>Dear Uncle Colin,</p>
<p>I’ve differentiated using the quotient rule and come up with $\frac{\frac{14x}{7x^2+6}x^7 -7x^6\ln\br{7x^2+6}}{\br{x^7}^2}$. That’s… not as simple as I’d like it to be. Any hints?</p>
<p>Quite Ugly Outcome That I’d Expect Neater, Thanks</p>
</blockquote>
<p>Hi, QUOTIENT, and thanks for your message!</p>
<p>You’re right, that’s Not Very Nice. Let’s break it down a bit at a time.</p>
<p>The first question I would ask is, “what’s ugly?”. The second question is “no, what <em>specifically</em>?”, because the first answer is always “… the whole thing.”</p>
<p>For me, the ugliest thing is the stacked fraction, so I’d start by multiplying top and bottom by $7x^2 + 6$ to get:</p>
<p>$\frac{14x^8 - 7x^6\br{7x^2 + 6}\ln\br{7x^2 + 6}}{x^{14}\br{7x^2+6}}$</p>
<p>You’ll notice I’ve tidied up a few powers there as well. There’s also a factor of $x^6$ everywhere I can take out on top and on the bottom (assuming $x\ne0$):</p>
<p>$\frac{14x^2 - 7\br{7x^2 + 6}\ln\br{7x^2+6}}{x^8\br{7x^2 + 6}}$</p>
<p>There’s not a whole lot more to do, other than take out the factor of 7 on top to get:</p>
<p>$\frac{7\br{x^2 - \br{7x^2 + 6}}\ln\br{7x^2+6}}{x^8\br{7x^2 + 6}}$</p>
<p>I think that’s in its most useful form, although you <em>could</em> split it into two fractions as</p>
<p>$\frac{14}{x^6\br{7x^2+6}} - \frac{7\ln{7x^2+6}}{x^8}$.</p>
<p>Hope that helps!</p>
<p>- Uncle Colin</p>Dear Uncle Colin, I’ve differentiated using the quotient rule and come up with $\frac{\frac{14x}{7x^2+6}x^7 -7x^6\ln\br{7x^2+6}}{\br{x^7}^2}$. That’s… not as simple as I’d like it to be. Any hints? Quite Ugly Outcome That I’d Expect Neater, ThanksBooks for… Real Analysis2021-05-31T00:00:00+01:002021-05-31T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/real-analysis-books<p>My excellent friend @realityminus3 asked:</p>
<blockquote class="twitter-tweet"><p lang="en" dir="ltr">Hey mathematics twitter--do any of you know of a good undergraduate textbook for Real Analysis?</p>— RealityMinus3 (@RealityMinus3) <a href="https://twitter.com/RealityMinus3/status/1340997787043901443?ref_src=twsrc%5Etfw">December 21, 2020</a></blockquote>
<script async="" src="https://platform.twitter.com/widgets.js" charset="utf-8"></script>
<h3 id="tldr">tl;dr</h3>
<p>What would I buy, having read all of the comments and thoughts?</p>
<p>I’d probably pick up <a href="https://amzn.to/2Km7wql">Alcock</a> and <a href="https://amzn.to/2Kp9ZAm">Cummings</a> to build up my intuition, and then <a href="(https://amzn.to/3pkhZRK)">Rudin</a> as a reference. I’d be tempted to download <a href="(http://classicalrealanalysis.info/)">Thomson et al.</a> as well, seeing as it’s free.</p>
<h3 id="book-list">Book list</h3>
<p>What follows is all of the books mentioned (in alphabetical order by [first] author), and then a lightly edited collation of the replies, including some shared elsewhere.</p>
<ul>
<li><a href="https://amzn.to/34DMaLL">Abbott: Understanding Analysis</a></li>
<li><a href="https://amzn.to/2Km7wql">Alcock: How to Think About Analysis</a></li>
<li><a href="https://amzn.to/37IYdti">Apostol: Calculus</a> (very expensive)</li>
<li><a href="https://amzn.to/2WI6KpX">Bear: Introduction to Mathematical Analysis</a></li>
<li><a href="https://amzn.to/3phxAkZ">Bear: A Primer of Lebesgue Integration</a> (advanced)</li>
<li><a href="https://amzn.to/3nJLvjr">Binmore: Mathematical Analysis</a></li>
<li><a href="https://amzn.to/3rpAZjO">Brannan: A First Course in Mathematical Analysis</a></li>
<li><a href="https://amzn.to/34DNwpP">Bressoud: A Radical Approach to Real Analysis</a></li>
<li><a href="https://amzn.to/3mJLWcn">Bryant: Yet Another Introduction To Analysis</a> - also <a href="https://www.cambridge.org/core/books/yet-another-introduction-to-analysis/F9A821905A4F83728DF51BED73989B14#">available on Cambridge core</a></li>
<li><a href="https://amzn.to/2M2Zy5T">Burkill: A First Course in Mathematical Analysis</a></li>
<li><a href="https://amzn.to/3haTx2y">Burn: Numbers and Functions</a></li>
<li><a href="https://amzn.to/34BJ4rW">Clapham: Introduction to Mathematical Analysis</a></li>
<li><a href="https://amzn.to/2Kp9ZAm">Cummings: Real Analysis</a></li>
<li><a href="https://amzn.to/34VAGDR">Garling: A Course in Mathematical Analysis, Vol I</a></li>
<li><a href="https://amzn.to/37G5UjT">Green: Sequences and Series</a></li>
<li><a href="https://amzn.to/3pbtTxo">Haggarty: Fundamentals of Mathematical Analysis</a></li>
<li><a href="https://amzn.to/3rnpTvr">Hart: A Guide to Analysis</a></li>
<li><a href="https://amzn.to/38pYYGQ">Marsden: Elementary Classical Analysis</a></li>
<li><a href="https://amzn.to/3nQIgXO">Reade: Introduction to Mathematical Analysis</a></li>
<li><a href="https://amzn.to/3nMJgfj">Ross: Elementary analysis</a></li>
<li><a href="https://amzn.to/3pkhZRK">Rudin: Principles of Mathematical Analysis</a> (Baby Rudin)</li>
<li><a href="https://amzn.to/3pjvRMd">Spivak: Calculus</a> (advanced)</li>
<li><a href="https://amzn.to/34Dr5RP">Stephenson: Mathematical Methods for Science Students</a></li>
<li><a href="http://classicalrealanalysis.info/">Thomson, Bruckner and Bruckner: Elementary Real Analysis</a> (free)</li>
<li><a href="https://amzn.to/3nQIUVe">Whitaker and Watson: A Course of Mathematical Analysis</a></li>
</ul>
<h3 id="commentary">Commentary</h3>
<ul>
<li>
<p>@BhaiTeraSakhtHa: Walter <strong>Rudin</strong>.</p>
</li>
<li>
<p>@alephJamesA: <strong>Rudin</strong>.</p>
</li>
<li>
<p>@tasminS: People will say Baby <strong>Rudin</strong>, and it is the classic but I think it’s more fun as a third time round text — for learning from I would say <strong>Abbott</strong>’s Understanding Analysis.</p>
</li>
<li>
<p>@Moon0nASpoon: +1 for <strong>Abbott</strong>, that’s what I taught from this summer and I really liked it. Very readable, and big emphasis on learning how to put proofs together, especially in the early chapters.</p>
</li>
<li>@professorBrenda: Highly recommend “Understanding Analysis” by Stephen <strong>Abbott</strong>. I call it “the book so nice I used it twice” because I learned from the 1st edition while in undergrad, and now I teach my students using the 2nd edition
<ul>
<li>@ilsmythe: <strong>Abbott</strong> is fantastic. I used it to teach an honors section of intro analysis at Rutgers (with the goal that they would be ready for Rudin the next semester) and I thought it worked extremely well.</li>
<li>@benjamindickman: strongly second this choice. FWIW: if the goal ends up being to go further (I don’t think this will happen…) then i think HS <strong>Bear</strong>’s book is an accessible undergrad text on Real Analysis II</li>
</ul>
</li>
<li>@MrMansbridge: Spivak
<ul>
<li>@themathdiva: That would be my choice</li>
</ul>
</li>
<li>
<p>@soupie66 An amazing book to read before you even start is Lara <strong>Alcock</strong>’s fantastic book How to think about Analysis (OUP). I wish I had read it before and during my undergraduate course, for it is BRILL!</p>
</li>
<li>
<p>@Howat_Hazel: have forgotten anything I knew about Analysis but I know Lara <strong>Alcock</strong> writes excellent books</p>
</li>
<li>
<p>@ChrisBMaths: Mary <strong>Hart</strong>’s book, A Guide to Analysis</p>
</li>
<li>@DarrenBrumby: Victor <strong>Bryant</strong>’s Yet Another Introduction to Analysis
<ul>
<li>@MathematicalA: <a href="https://www.cambridge.org/core/books/yet-another-introduction-to-analysis/F9A821905A4F83728DF51BED73989B14#">Available on Cambridge core</a></li>
</ul>
</li>
<li>
<p>@soupie66: <strong>Whittaker and Watson</strong>.</p>
</li>
<li>@Long_tailed_tit: <strong>Brannan</strong> might be too simple. But written by the
@OpenUniversity and so high clarity of explanation.
<ul>
<li>@MathematicalA: <a href="http://www.burhantiryakioglu.com/wp-content/uploads/2015/06/A-First-Course-in-Math.-Analysis.pdf">Seems to be freely available on the internet</a></li>
</ul>
</li>
<li>
<p>@Mathemacricket: My A Level teacher lent me this book before I started my degree:
Fundamentals of Mathematical Analysis by Rod <strong>Haggarty</strong>.</p>
</li>
<li>
<p>@isleofmandan: Fundamentals of Mathematical Analysis by Rod <strong>Haggarty</strong> is quite accessible.</p>
</li>
<li>
<p>@sumsgenius: The maths dept at uni asked us all to work through <strong>Stephenson</strong> before we started the degree. It was a long time ago, though.</p>
</li>
<li>
<p>@JoeHarrisUK: <a href="https://twitter.com/JoeHarrisUK/status/1341000630861783041/photo/1">some options from the Cambridge schedules</a></p>
</li>
<li>
<p>@matthematician: For accessibility and open-educational-resource availability: <strong>Thomson, Bruckner, Bruckner</strong>.</p>
</li>
<li>
<p>@darthkiks: To get a better feel for the motivation behind the theorems, I highly recommend “A radical approach to real analysis” by D. <strong>Bressoud</strong></p>
</li>
<li>@TChihMaths: I think I’ll let @LongFormMath plug his own book!
<ul>
<li>@LongFormMath: (<strong>Cummings</strong>: Real Analysis) - It’s like Abbott’s book on steroids!</li>
</ul>
</li>
<li>
<p>@profgoat: Now this (<strong>Bear</strong>: Lebesgue) isn’t a beginner book, but might be an advanced topic book, or a readable refresher for those of us 30 years out of our comps.</p>
</li>
<li>
<p>@profgoat: <strong>Marsden</strong>’s book is as old as dirt but I love it.</p>
</li>
<li>@mathdocron: I liked “Elementary Analysis: The Theory of Calculus” by Kenneth <strong>Ross</strong>.
<ul>
<li>@_qnlw: Just used it this year. The content and presentation are okay (some minor things I disagree with). But I am not terribly impressed by the choice and phrasing of the exercises.</li>
</ul>
</li>
</ul>
<p>I’ll leave the full run-down to Nicholas Jackson:</p>
<ul>
<li>@njj4: <strong>Hart</strong> is a good introductory book that covers sequences, series, continuity, limits, differentiation. <strong>Alcock</strong> is a very readable introduction that talks about how to think about the subject. <strong>Green</strong> and <strong>Clapham</strong> are nice little books. <strong>Burkill</strong> is a bit old now, and I never found it very readable - it was on the suggested list in my first year (1991) but we used <strong>Binmore</strong> instead, which was much clearer. <strong>Bryant</strong> is quite accessible. <strong>Burn</strong> is readable but strange - the proofs are broken down into guided exercises.
<ul>
<li>@sam_holloway: I remember real analysis was the course I struggled to get a good textbook for (back in 1997/8). <strong>Bryant</strong> I’ve looked at since and it looked like the one that would have helped me!</li>
</ul>
</li>
</ul>My excellent friend @realityminus3 asked:Ask Uncle Colin: Is this an ellipse?2021-05-26T00:00:00+01:002021-05-26T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/ask-uncle-colin-is-this-an-ellipse<blockquote>
<p>Dear Uncle Colin</p>
<p>I know that $\br{\frac{x^2}{4}} + \br{\frac{y^2}{2}}=1$ is an ellipse, but I have $\br{\frac{x^2}{4}} + \br{\frac{y^2}{4}}=1$. Is that also an ellipse?</p>
<p>Explain Loci Like I’m a Precocious Student (Elementary)</p>
</blockquote>
<p>Hi, ELLIPSE, and thanks for your message!</p>
<p>The short answer is yes, that is also an ellipse.</p>
<p>The longer answer is yes, that’s a special case of an ellipse, where the major and minor axis are equal – which makes it a circle.</p>
<p>Now, there are (probably) some who say “that’s not an ellipse! It’s a circle!” However, I am a strong believer in inclusive definitions (a shape can be more than one thing!)</p>
<p>For example, a square is both a rectangle and a rhombus – a rectangle is a quadrilateral with four right angles, and a square satisfies that property; a rhombus is a quadrilateral with four equal sides, and that fits a square too.</p>
<p>In the same way, an ellipse is the intersection of a cone with a plane. That’s also true of a circle, in the special case where the plane is perpendicular to the axis of the cone.</p>
<p>Hope that helps!</p>
<p>- Uncle Colin</p>Dear Uncle Colin I know that $\br{\frac{x^2}{4}} + \br{\frac{y^2}{2}}=1$ is an ellipse, but I have $\br{\frac{x^2}{4}} + \br{\frac{y^2}{4}}=1$. Is that also an ellipse? Explain Loci Like I’m a Precocious Student (Elementary)Curvilinear Asymptotes in GeoGebra2021-05-24T00:00:00+01:002021-05-24T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/curvilinear-asymptotes-in-geogebra<p>GeoGebra has a very useful function called Asymptote: if you have something like $f(x) = \frac{3x^2+4x+3}{x-1}$, typing Asymptote(f) in the input bar gives a list of the linear asymptotes: $\{ y=3x+7; x=1 \}$. Very nice, very useful.</p>
<p>But something like $f(x) = \frac{2x^4 + 3x^3 + 2x + 4}{x^2 +3x + 2}$ is more tricky: GeoGebra only returns the two linear asymptotes, $x=-1$ and $x=-2$. However, there’s also a <em>curvilinear asymptote</em> that GeoGebra doesn’t return. Can we get GeoGebra to find it?</p>
<p>Of course we can. It’s a tiny bit tricky, but it’s not as bad as I first thought.</p>
<p>The first step is to split the function into a numerator and a denominator, a top and a bottom:</p>
<ul>
<li>$N(x) = 2x^4 + 3x^3 + 2x + 4$</li>
<li>$D(x) = x^2 + 3x + 2$</li>
</ul>
<p>The key is then to use the Division command:</p>
<ul>
<li>$L = Division(N, D)$</li>
</ul>
<p>This gives a list containing the quotient and remainder. Here, we need the quotient:</p>
<ul>
<li>$q(x) = Element(L, 1)$ ((Note that GeoGebra lists start counting from 1)).</li>
</ul>
<p>And that’s it! $q(x)$ is the curvilinear asymptote to $f(x)$.</p>GeoGebra has a very useful function called Asymptote: if you have something like $f(x) = \frac{3x^2+4x+3}{x-1}$, typing Asymptote(f) in the input bar gives a list of the linear asymptotes: $\{ y=3x+7; x=1 \}$. Very nice, very useful.What’s the plot2021-05-21T00:00:00+01:002021-05-21T00:00:00+01:00https://www.flyingcoloursmaths.co.uk/whats-the-plot-1<p>Last week, I invited my followers on Mathstodon, LinkedIn and Twitter to ascertain the logic behind this plot:</p>
<blockquote class="twitter-tweet" data-partner="tweetdeck"><p lang="en" dir="ltr">Let’s play a game of “What’s the plot?”!<br /><br />A postcard for the first person to tell me what the logic behind this plot is (or, failing that, the answer I like best). <a href="https://t.co/pVWWvnaSMC">pic.twitter.com/pVWWvnaSMC</a></p>— Colin Beveridge (@icecolbeveridge) <a href="https://twitter.com/icecolbeveridge/status/1393942112664801286?ref_src=twsrc%5Etfw">May 16, 2021</a></blockquote>
<p>Before I reveal the answer and the winners, let me go through some honourable mentions in the responses:</p>
<ul>
<li>@bbarber_ suggested a projection of the positive orthant of the 4D lattice</li>
<li>@springdot noted that doodles of feathers are what you draw when you want to be truthful</li>
<li>@RobAnthony01 said it was a fractal where the central branch is repeated at about 80% of the original, which might be correct but I’m not going to measure</li>
<li>@vatsasir said “bifurcation”. If anything, it’s a trifurcation.</li>
<li>@phil_dubious <a href="https://twitter.com/Phil_Dubious/status/1394225940226576384?s=20">tried to reconstruct the fractal</a>, which is a good effort!</li>
<li>@realityminus3 suggested “Plot: a person starts out telling three versions of the truth. Consequences ensue, disparate storylines entangle, the situation goes dark. Only one character emerges alive.” Love it.</li>
<li>@jgfwstone reckoned it looked like a computer learning to ride a bike.</li>
</ul>
<h3 id="the-winners">The winners</h3>
<ul>
<li><strong>First response</strong> ((In future, I reserve the right to decide what constitutes a response. Going ‘first!!!’ isn’t going to get you a postcard.)): @MrAllanMaths, who suggested <a href="https://twitter.com/mrallanmaths/status/1393984338178490369?s=20">the Chinese Method for powers</a></li>
<li><strong>Most interesting response</strong>: On LinkedIn, Deborah Castle asked it it was a <a href="https://en.wikipedia.org/wiki/Ishikawa_diagram">Fishikawa</a>, which sent me searching. Lovely idea!</li>
<li><strong>Closest response</strong>: @shahlock suggested “some (wacky) visualization of (x+y+z)^n?”, which I think is morally quite close to the true answer.</li>
</ul>
<p>All three will be getting postcards :-)</p>
<h3 id="the-answer">The answer</h3>
<p>In fact, the coordinates of the nodes of the tree represent the legs of primitive Pythagorean triples (plotted on a log scale). The edges are the links in this <a href="https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples#Alternative_methods_of_generating_the_tree">tree of triples</a> – every primitive triple can be seen as the parent of three others. (There are several possible ways to do this; I picked the one that was simplest to code up.)</p>
<hr />
<p>Wasn’t that fun? Let me know if you’d like to play again!</p>Last week, I invited my followers on Mathstodon, LinkedIn and Twitter to ascertain the logic behind this plot: