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Interactive Mathematics by Murray - 2M ago

17 Jan 2018

1. Math art in code
2. Resource: Making graphs easier to understand
3. Math in the news: Truly random numbers
4. Math movies
5. Math puzzle: Sum the digits
6. Final thought: Truth

1. New on IntMath (a) Math art in code

This is a new section in IntMath. There are many beautiful images and animations created using computer code with a mathematical basis. Some of them are quite simple to produce and could make good class exercises (or mini-projects) at the intersection of mathematics, coding and art. (STEAM, in other words!)

 Delaunay Triangulation is a specific kind of triangulation (the important mathematical concept used in everything from engineering to computer graphics). See some math, art and coding at:
(b) IntMath accessibility

During my university days I volunteered to help a group of blind people with their daily tasks. Ever since, I've been interested in the alternative ways blind people can learn - especially in the area of mathematics. For a long time I've wanted to make IntMath more accessible, but have never found the time until recently.

To make a Website "accessible", you need to add (and sometimes rearrange) the underlying code so that it makes more sense for screen readers. (For example, you may have seen a "Skip to main content" link at the top of some Web pages - that saves the blind person having to listen to all the links and image descriptions at the top of most page.)

 Anyway, I've recently begun to work on accessibility of IntMath. If you - or someone you know - uses a screen reader when browsing the Web, please give me feedback on how I can improve the accessibility of IntMath. (I know I still have a loing way to go...)
2. Resource: Making charts and graphs easier to understand

The ability of presenting data in meaningful ways becomes more of a marketable skill with every passing day.

Usability experts, Nielsen Norman wrote a great article on how to present graphs so that users experience the least "cognitive load" - and hence can figure out what it means quicker.

 Math students and teachers alike should find this useful. See:
3. Math in the news: Truly random numbers

Almost all coding languages have a "random" function, which is meant to produce a random number. (There's an example in the Delaunay Triangulation page referred to before.)

However, it never is truly random, since it is often based on some "seed" value, and can be reporduced if the seed is known. Truly random numbers are really important for effective cybersecurity.

 This article describes a 2016 breakthrough where University of Texas researchers claim to have achieved it in a quite practical way. See:
4. Math Movies

I have a backlog of math videos to share, so I will include 2 in each Newsletter for a while.

(a) Can a robot pass a university entrance exam?
 There are a lot of implications for education in the machine learning realm. This video explores something I've been saying for a long time - we should get computers to do what computers are good at (calculation and memory), and get humans to do what they are good at (problem-solving and creativity).
(b) A brief history of banned numbers

The Pythagoreans were very strict about the numbers you could - and could not - talk about. (There is a story about one of his followers being killed for proving the existence of irrational numbers.) To this day we have numbers that are thought of as "special" (like unlucky 13, or 666, or 4).

 Here's a look at some of the numbers that have been banned throughout history:
5. Math puzzles

The puzzle in the last IntMath Newsletter asked about the shaded portion of a triangle.

Correct answers with explanation (covering a wide variety of approaches) were given by: Ihage, Francis, Rick, Aled, Danesh, Thomas, Chris, JDK, Tomas, Vijay, Eamon, and Gerard.

My incorrect addition: In the last Newsletter, I added an extra example involving degenerate conics. Unfortunately, I was "off on a tangent", and some of you (Chris, Alan, Eamon) questioned me on it. You were right and I corrected that part of the post. (Whenever we make a mistake, we should immediately admit to it and correct it. More on this in the "Final Thought" below.)

New math puzzle: Sum the digits

What is the sum of the digits of the number (1025 − 25)?

You can leave your responses here.

6. Final thought: Truth

It should concern any educated person in current times that the most effective way to convince people of a certain point of view is to say it often enough, and to brush aside criticisms — not by presenting solid evidence — but by sowing doubt. There many examples thoughout history of this working well, especially to allow big business to get away with scandalous things (e.g. the tabacco industry, leaded fuel, nuclear power in Japan).

Late in 2017 there were reports that officials at the (US) Centers for Disease Control and Prevention had been asked to refrain from using the words "evidence-based" or "science-based" in their funding submissions. This kind of thing has never ended well in the past, and won't this time. Beware those who think the "post-truth" era is acceptable, or even a good thing.

Words matter and truth matters.

Bertrand Russell had this to say about mathematics, a field of science (like all of them) that relies on proof and evidence.

“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.” [Bertrand Russell]

[Hat-tip to PiPo for alerting me to some of the items in this Newsletter.]

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Math art code, graphs, random numbers appeared first on SquareCirclez.

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Interactive Mathematics by Zac - 3M ago

12 Dec 2017

0. Season's greetings
1. New on IntMath: Shell method
2. Resource: Mathpix
3. Math in the news and Math movie: Bitcoin
4. Math puzzle: Triangles
5. Final thought: No more cars?

0. Season's greetings

'Tis the season for Hanukkah (starts today), Christmas (25th Dec or 7th January), New Year (1st January and many other dates in other countries) and other celebrations. Whatever you celebrate, I hope next year is full of health, peace and good learning.

1. New on IntMath: Shell method

There have been a lot of reader requests for a page on Shell Method. Some of the 3D mental gymnastics you need to do for this topic can be quite challenging, but there are some great applications using this method of finding volumes using integration.

 See: Shell method for the volume of a solid of revolution
2. Resource: Mathpix

Mathpix is a clever math solver that can read hand-written math using your mobile's camera.

 It makes use of a math solver, Desmos and some other tools to solve problems and graph results. See: Mathpix (available in Android and iOS versions)

It's quite well done and has a lot of promise, but I found the results somewhat "clunky" at times. (It quite often misread math, even printed examples.)

This provides us with yet another dilemma, similar to the one discussed when I mentioned Photomath last year. Such tools can help students learn math, but there is always the fear students will just use the tool to do their homework, and will learn very little.

What such tools really should inspire is a completely different way of going about math education. Rather than it being all about "learning how to do the algebra", it should be about "learning how to use the many available tools to solve real problems."

Disclaimer 1: IntMath has no connection with Mathpix.

Disclaimer 2: Like all technologies, Mathpix can be used for good, and not-so-good.

3. Math in the news and Math Movie: Bitcoin

As I write, the value of bitcoins has surged. To give an idea of how much, here's a quote from Bloomberg:

In 2010, programmer Laszlo Hanyecz exchanged 10,000 bitcoins for two Papa John's pizzas. With the chain's current limited-time offer of any large or pan pizza for $10, that many bitcoins would now buy more than 16 million large Papa John's pizzas.  So what's the mathematics behind bitcoin? This video is from the brilliant 3Blue1Brown: 4. Math puzzles The puzzle in the last IntMath Newsletter asked about intersecting parabolas. Correct answers with explanation were provided by Tomas, Chris, and C Trenor, who rightly said we should make use of all the tools available. (He suggests Desmos, Symbolab, Wolfram|Alpha and GeoGebra.) In fact, each respondent used a graphics tool to see what was going on, which makes a lot of sense to me. Also, see: Conic Sections 3D Interactive, where you can investigate in 3 dimensions what's happening when a plane intersects a cone, producing parabolas, hyperbolas and so on. (Move the top slider to 90°, and you'll see one of the "degenerate forms" appear). You only really understand a math problem if you can do it algebraically, numerically and graphically. Let's look at it all 3 ways. Numerically We can easily substitute each of the 5 coordinate pairs into the given equations, and will see that they "work". However, that doesn't answer the "how is it possible?" part of the question. Algebraically Our aim is to solve the following 2 equations simultaneously. 2x2 − xy − y2 − 4x + 4y = 0 ... (1) 6x2 + xy − y2 − 16x + 2y + 8 = 0 ... (2) We can factor the left side of each of the above. This first involves some observation (there appears to be a common term, (x − y) throughout the first one). We use polynomial division to give: (x − y)(2x + y − 4) = 0 ... (3) The second one doesn't have the common term (x − y), so we try the second term from the first factorization, (2x + y − 4) and obtain: (3x − y − 2)(2x + y − 4) = 0 ... (4) Since there is a common term between (3) and (4), we only need to solve: x − y = 3x − y − 2 With some further thought, we can conclude solutions lie along the lines: y = x y = 3x − 2 y = −2x + 4 These are just the expressions we found by factoring. Graphically This is what Desmos gives us for equation (1). It's actually two intersecting lines, which in this case are degenerate parabolas (which occur when our parabolas are infinitely thin): And here's what we get for (2), another set of straight lines: (The above lines are the ones we found algebraically, earlier.) Here's the graph of both degenerate parabolas on the one set of axes. The line through (0, 4) and (2, 0) is y = −2x + 4, the common term in our algebraic solution given above (and is quite dark because the 2 graphs are coinciding). The graph also shows the 5 intersection points mentioned in the puzzle question: Another example The parabolas do not need to be "degenerate" in order to produce 5 intersection points. A parabola, when considered as the intersection of a plane and a cone, actually has 2 arms, like this: If our second parabola has 2 arms and is rotated on its axis, we can also get more than 4 points of intersection, in this case 6, as shown here: New math puzzle: Shaded area It's not exactly the star on top of a Christmas Tree, but this problem reminded me of it. The first two triangles are combined as shown in the third image. What fraction of the large triangle is the shaded area? You can leave your responses here. 6. Final thought: No more car growth in Singapore Singapore is a small island state and it's been my home for over 20 years. I've never owned a car here (they are prohibitively expensive, and I've never wanted one since the public transport is mostly excellent). Singapore has one of the highest rates of luxury car ownership in the world [Image credit: Inquirer] Singapore's 2013 Transport Master Plan (PDF) says Roads already take up 12 per cent of Singapore’s total land area, compared to 14 per cent for housing. Most cities worldwide are being choked to death by cars and it's crazy that cars could take up almost the same land area as people. Singapore announced in October this year that it was cutting the growth rate of all private passenger vehicles to zero. It coincides with a big increase in spending on public transport infrastructure. Are cars choking your city? Perhaps we need a different approach, and Singapore's policies on this issue mostly make sense to me. See you again in 2018. Until next time, enjoy whatever you learn. The post IntMath Newsletter: Shells, resource, Bitcoin appeared first on SquareCirclez. Related posts: 1. IntMath Newsletter: z-Table interactive graph, CK-12 resource, range hood In this Newsletter: 1. New z-Table interactive graph 2. Resource:... 2. IntMath Newsletter: Newton’s Method, Desmos activities In this Newsletter: 1. New interactive graph: Newton's Method 2.... 3. IntMath Newsletter: Halloween, GeoGebra resource In this Newsletter: 1. What happened to the IntMath Newsletter?... 4. IntMath Newsletter: squaring circle, 17 equations In this Newsletter: 0. PhotoMath 1. Squaring the circle with... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Zac - 3M ago 28 Nov 2017 In this Newsletter: 1. New applet: Calculus first principles 2. Resource: ImmersiveMath 3. Math in the news: Proof School, GPU problem solvers 4. Math movie: Inspiration 5. Math puzzle: Intersecting parabolas 6. Final thought: Tenants 1. New applet: Calculus first principles Sometimes the early concepts (usually called "first principles") of calculus can be a bit confusing.  This new applet allows you to explore the concepts of differentiation and integration from first principles. Calculus first principles interactive applet 2. Resource: ImmersiveMath This is "the world's first linear algebra book with fully interactive figures".  Immersive Linear Algebra contains some excellent animations and manipulatives that will help you to understand vectors, dot products, matrices, and determinants. The pages take a while to load, but it's worth it! 3. Math in the news: (a) Proof School A few years back I did some mathematics curriculum consultancy work for the newly-established School of Science and Technology, Singapore. Their approach is to incorporate various technologies into an interdisciplinary mix of art, design, media and environmental studies, on top of the national curriculum. I was reminded of that experience when I came across this article on Proof School, in California.  Proof School is designed for "students who love math". Here is some background about the school, which has roots in the "math circle" concept, where kids can feel comfortable in a supportive and challenging math environment: A School Built Entirely Around the Love of Math (b) Innovative approach to solving massive problems on an ordinary PC Russian scientists realized that if they used the GPU (graphics processor unit) of an ordinary PC, they could solve "260 million complex double integrals on a desktop computer within three seconds only". The best a supercomputer can achieve when solving the same problems is 2 to 3 days - and at a much greater expense.  The GPU used a Nvidia GPU designed for use in game consoles. 4. Math Movie: Exceptional teachers (This is a bit late for last month's Teachers' Day, but still...) Here's Stephen Hawking talking about his inspirational mathematics teacher: "I have to admit – I wasn’t the best student – but with Dikran Tahta's support I became a professor of mathematics at Cambridge, in a position once held by Isaac Newton."  See the short video: Stephen Hawking On The Teacher That Changed His Life 5. Math puzzles The puzzle in the last IntMath Newsletter asked about different formulas for producing a given sequence of numbers. Correct answers with explanation were provided by Phil, Karam and Chris. New math puzzle: Intersecting parabolas A quadratic graph with vertical axes has the general form: y = ax2 + bx + c Two parabolas with vertical axes can intersect in at most 2 points. Now, consider the following graph. Here, I've rotated one of the parabolas: The equation of that rotated parabola has the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 We can see the two parabolas intersect in 4 points, which is the maximum number of intersection points. Or is it? How is it possible for the coordinates of the five points (−1, 6), (3, −2), (1, 1), (0, 4) and (2, 0) to satisfy both of these quadratic equations? 2x2 − xy − y2 − 4x + 4y = 0 6x2 + xy − y2 − 16x + 2y + 8 = 0 You can leave your responses here. 6. Final thoughts: Tenants Here's an apt quote from Rose Bird, Chief Justice of the California Supreme Court: "We have probed the earth, excavated it, burned it, ripped things from it, buried things in it, chopped down its forests, leveled its hills, muddied its waters, and dirtied its air. That does not fit my definition of a good tenant. If we were here on a month-to-month basis, we would have been evicted long ago." [Hat-tip to PiPo for alerting me to some of the items in this Newsletter.] Until next time, enjoy whatever you learn. The post IntMath Newsletter: First principles applet, resources appeared first on SquareCirclez. Related posts: 1. IntMath Newsletter: Free resources, derivative from first principles In this Newsletter: 1. Free resource: Study Tips from Cornell... 2. IntMath Newsletter: 3D planes, resources, 200 TB math answer In this Newsletter: 1. 3D intersection of planes interactive applet... 3. IntMath Newsletter: Fourier graph, resources In this Newsletter: 0. Top 100 Math Blog 1. Fourier... 4. IntMath Newsletter: trig differentiation applet, art, music In this Newsletter: 1. Differentiation applet - trigonometric functions 2.... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Zac - 5M ago 25 Oct 2017 In this Newsletter: 0. Hiatus 1. Revised applet: Double Springs Interactive Graph 2. Resource: Wolfram Data Repository 3. Math in the news: Going Viral 4. Math movie: Maps 5. Math puzzle: Sequence formulas 6. Final thought: Learners 0. Hiatus I've been swamped over the last few months with consultancy jobs, essential IntMath site maintenance and back pain, so I haven't had a chance to write an IntMath Newsletter in quite a while. Thank you for your patience - there's only 24 hours in a day! 1. New applet: Double springs I recently revised the double springs applet, which is an example of a composite trgionometric curve. From a coding point of view, it's one of the most complicated applets in all of the ones on the site and was quite a challenge.  You can find the applet about 1/3 way down this page: Here's some mathematical background: 2. Resource: Wolfram Data Repository A promising future exists for those who can understand, interpret, manimpulate and visualize big data. The folks at Wolfram have made publicly available several datasets, which you can experiment with using your preferred code. (Python comes to mind, and Wolfram's own programming language is designed for easy data manipulation.)  Here is the background to the project from the Wolfram blog: Data Publishing that Really Works And here is the data repository: 3. Math in the news: (a) Catching the Great American Eclipse at 35,000 feet In August there was a total eclipse of the sun passing over the entire USA. AlaskaAir took a group of eclipse watchers up to get a good view, and they had to make sure they maximized their customers' experience. The math behind the flight is quite interesting. Threading the celestial needle: Catching the Great American Eclipse at 35,000 feet (b) Going Viral These days, it gets harder to capture peoples' attention, and to "go viral" is the holy grail of most publishers. There are plenty of companies that will help you to get there (for a fee). SearchLaboratory (a search optimization company) has an interesting mathematical take on how they applied a model developed for epidemic prediction in human health to the issue of "how much" and "how many" places you need to promote your content in order for it to go "viral".  See: Going Viral - The Mathematics Behind Content and Online PR See especially the section, "Step 3: Applying the Stochastic General Epidemic Model". 4. Math Movie: Data maps  I've been a lover of maps from way back, and so I enjoyed this TED talk where Danny Dorling talks about several different data visualization maps. See: Maps that show us who we are not just where we are 5. Math puzzles The puzzle in the last IntMath Newsletter asked about the angles of the hour, minute and second hands of a clock. Correct answers with explanation were provided by Rick, Chris, Don, and Thomas. As usual, there were an interesting variety of approaches used, including "brute force VBA". Update: A year ago (almost exactly) the Newsletter puzzle concerned a sheep in a circular field. Chris had another go at solving it by finding a few clever integrals, using "SciPy" (Scientific Python) software, which I mentioned earlier in this Newsletter. New math puzzle: Sequence formulas Example: The n-th term of the sequence 3, 5, 7, ... can be expressed using the formula an = 2n + 1, where n = 1, 2, 3, ... Now try these two. (a) We have a sequence which alternates between 1 and −1 as follows: 1, −1, 1, −1, ... Propose as many formulas as you can to represent the n-th term for this sequence. (b) Bonus: Similarly, find formulas for the n-th term of the sequence: 3, −2, 3, −2, ... You can leave your responses here. 6. Final thoughts: Learners Here are two great quotes from moral philosopher Eric Hoffer (1898-1983): "In a time of drastic change it is the learners who inherit the future. The learned usually find themselves equipped to live in a world that no longer exists." "The hardest arithmetic to master is that which enables us to count our blessings." Until next time, enjoy whatever you learn. Related posts: 1. Double springs interactive graph The revised double spring applet now allows exploration on... 2. IntMath Newsletter: squaring circle, lens fillet, data visualization In this Newsletter: 1. Squaring the circle - a reader's... 3. IntMath Newsletter: Reflecting graphs, data and efficient lawn mowing In this Newsletter: 1. How to reflect a graph through... 4. IntMath Newsletter: trig differentiation applet, art, music In this Newsletter: 1. Differentiation applet - trigonometric functions 2.... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Zac - 5M ago I recently updated the double springs graphs interactive applet (you'll see it about 1/3 way down the Composite Trigonometric Curves page, here): The applet demonstrates one example of composite trigonometric curves. That is, the result we get from adding together different periodic (and non-periodic) signals. Applications Very few signals in electronics are pure sine curves. Instead, they are a combination of DC (direct current) and AC (alternating current) signals and the job of any receiver (like your phone, TV or radio) is to split apart those combined signals into something useful. Apart from electronics, we come across composite trigonometric graphs wherever waves are combined (like ocean waves with smaller ripples caused by the wind, or eathquakes). Background to the applet The composite trigo applet was one of the most complex ones I've ever written. There are many variables involved, and I needed to rewrite parts of my SVG script to allow for drag events on mobile devices. (I actually developed it some years ago for Flash, but have revised it a few times since since Flash doesn't run on mobile devices.) The applet uses numerical integration to calculate the mass positions and spring extension amounts. The method used is Runge-Kutta (RK4). The basic idea with Runge-Kutta 4 is to solve a differential equation by starting with a given initial value and then following a path using appropriate slopes, thus tracing out the solution curve. (This is an extension of the thinking behind Newton's Method, which we use to find roots of equations.) In the case of the double springs applet, it is a system of differential equations (the motion of each spring affects the motion of the other one) and so each step needs to allow for the different forces and velocities that are going on. So for each step, the Runge-Kutta function accepts the current 2 velocities of the 2 masses, and outputs the next 2 velocities for the subsequent step. Other math used when developing the applet included: • Trigonometric graphs (of course) • Varying amplitude based on the stretch or compression of the spring • Time • Transformation geometry (translation and scaling of springs and masses) • Matrices (the position of the masses and the springs are internally tracked via some matrices) • Scalable vector graphics (images that are produced by telling the browser to draw objects via a script) • Spring theory (mass, spring constants, extension, etc) Related posts: 1. IntMath Newsletter: Double springs, data, going viral In this Newsletter: 0. Hiatus 1. Revised applet: Double Springs... 2. Euler’s Method and Runge-Kutta RK4 An updated spring applet uses Runge-Kutta Order 4 Method... 3. Frequency of notes on a piano - interactive learning object Explore piano note frequencies in this interactive learning object,... 4. Conic sections interactive applet This interactive applet allows you to explore the concept... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Murray - 7M ago 31 Jul 2017 In this Newsletter: 1. New applet: Piano note frequencies interactive 2. Resource: Continued fractions 3. Math in the news: Space filling 4. Math movie: One in a million 5. Math puzzle: clock 6. Final thought: Really worth the doing 1. Frequencies of notes on a piano - interactive learning object  Explore piano note frequencies in this new interactive learning object, combining trig graphs and exponential growth. Frequency of notes on a piano - interactive learning object Here's some background on what's going on: What are the frequencies of music notes? Frequency of notes on a piano - interactive learning object 2. Resource: Continued Fractions You may recall the Euclidean Algorithm for finding the greatest common divisior of 2 numbers. For example, the GCD of 93 and 42 would be found as follows: 93 = 2 × 42 + 9 42 = 4 × 9 + 6 9 = 1 × 6 + 3 6 = 2 × 3 + 0 We stop at that last line because there is a remainder of 0, and we conclude the GCD is 3. Continued fractions are a special form of fraction and can be used when finding GCDs, as well as in many other applications like finding square roots and solving quadratic equations.  This extensive resource covers continued fractions. It contains many worked examples and several online calculators and it's by Richard Knott, Surrey University: 3. Math in the news: Space filling Fruit shop owners know a thing or two about space filling. They need to display their wares attractively, safely and in a space-efficient way. Mathematicians have studied space filling for some years, especially since Lord Kelvin (the temperature guy) asked in 1887: How can space be partitioned into 3D structures of equal volume in a way that minimizes the total surface area of each structure? Some Belgian mathematicians had some breakthroughs on this issue last year. From Phys.org:  Researchers in 2016 discovered a new 3D structure that divides space into 24 regions, and have shown that it is the best solution yet to a modified version of a geometrical space-partitioning problem that has challenged researchers for more than a century. See more: New 3-D structure shows optimal way to divide space 4. Math movie: How to visualize one part per million A lot of people struggle with very large (or very small numbers). It's not surprising as most of us don't use them every day, but as we do hear about them constantly on the news, it's important to get a good conception of them.  In this short TED-Ed video we get some good pointers on what one million really looks like. How to visualize one part per million 5. Math puzzles The puzzle in the last IntMath Newsletter asked how far a person walked given some (seemingly) insufficient conditions. Correct answers with explanation were provided by Chris, Thomas and Nash. The walking problem was based on one by Lewis Carroll, author of Alice in Wonderland, and Through the Looking Glass. It appeared in his book A Tangled Tale. Apart from being a well-known author, Carroll (real name Charles Lutwidge Dodgson) was a mathematician, logician, Anglican deacon, and photographer. New math puzzle: Clock True or false? Consider a clock at 11:40 AM. When the second hand reaches somewhere close to 20 seconds past the minute, the clock face will be divided into 3 equal parts. How about for any other cases (like close to 01:45:30, or 02:50:35)? Is it true? Can you prove your conclusion for the clock at any time? You can leave your responses here. 6. Final thought: Really worth the doing Today's quote is from Lewis Carroll: One of the secrets of life is that all that is really worth the doing is what we do for others. Until next time, enjoy whatever you learn. The post IntMath Newsletter: Piano trig applet, continued fractions appeared first on SquareCirclez. Related posts: 1. Frequency of notes on a piano - interactive learning object Explore piano note frequencies in this interactive learning object,... 2. IntMath Newsletter: trig differentiation applet, art, music In this Newsletter: 1. Differentiation applet - trigonometric functions 2.... 3. Math movie: How to play a Rubik’s Cube like a piano This movie explores the connections between Rubik's cube and... 4. A logarithmic music scale This musician's use of a logarithmic musical scale reminds me... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Murray - 7M ago Background An interesting mathematical challenge for musicians throughout the centuries has been the issue of tuning instruments so they sound good when played together. Pythagoras and his followers worked on such problems over 2500 years ago. For stringed instruments (like violin or cello), this is not such a big concern, because slight changes in finger position can produce a pleasing sound. But for keyboard players, such flexibility does not exist. They have to hope their piano tuner can get things just right. There is a catch though. On any piano, only 8 notes (out of the 88 notes on a piano) are actually tuned exactly right. Those are all the A notes, from A-27.5 (which means it has a frequency of 27.5 Hz) up to A-3520 (with frequeny 3,520 Hz). All the other notes are actually slightly "off" - a compromise so that pianos can sound OK when playing in any key. This is due to the oddities of exponential growth and equal-tempered tuning. You can read more on the background of this at: What are the frequencies of music notes? The new applet The new applet follows on from the above considerations and allows you to explore the graphs of piano note frequencies. There is a virtual piano that you can play (one note at a time) and after playing each note, you can see the graph of that note and its frequency. You can find the new applet here: You can also see a composite signal, which is the sum of the fundamental note A-220 and the note just played. Here's a screen shot: Discussion A-440 and A-200 When you play A-440, the graph looks as follows. They grey one is A-220, and the green one is A-440. You'll notice the frequency of A-440 is twice the frequency of A-220, and the wavelength is half as long. The two curves coincide at each major time point (at 1 and 2 on the t-axis). E - the "perfect 5th" A perfect 5th above A-220 (in theory) is 220× 3/2 = 330 Hz. On a piano, E is actually 329.63 Hz, just a little bit "flat". You can see in the graph there are 1.5 wavelengths for E (the green curve) in the one wavelength of A-220. It's extremely close, and it appears the two curves coincide at t = 1 and t = 2. C# - the "major 3rd" To sound right, a major 3rd "should" have frequency 275 Hz (that is, 5/4 × 220), but is quite "sharp" at 277.18 Hz on a piano. You can see the green curve is not exactly passing through exactly t = 2. As the frequncies get higher, it becomes increasingly "off". This was unacceptable to many musicians during the Baroque era, and they would use other tuning methods to make sure the major 3rd sounded better. The downside was they couldn't play in certain other keys at all. Signal combinations In the applet, you can also see the combined signal of the fundamental (A-220 in this case), with the note just played. A-220 and B♭ - 233.08 These two notes are right next to each other on the piano, and sound discordant (bad) when played together. The combined signal gives some insight why. You can see a regular combined signal which is very close to the original two (grey) signals, resulting in "beating" - a kind of shimmering which is not always pleasant. (It can be put to good use in music, but that's for another discussion.) A-220 and C#-275 The major 3rd is one of the most common and "pleasing" sounds in music. On a piano tuned by equal temperament, however, you can see the 3 curves doen't line up so well (the grey A-220, C#-275 and the green combination of the two) because actually, C# is 277.18, which is too sharp for a totally pleasing sound. It can't be helped if pianos need to play in any key (which of course, they nearly always need to do). Conclusion Piano note frequencies are an interesitng application of trigonometric graphs, exponential growth and music perception. Once again, here's the link to the applet: The post Frequency of notes on a piano - interactive learning object appeared first on SquareCirclez. Related posts: 1. A logarithmic music scale This musician's use of a logarithmic musical scale reminds me... 2. IntMath Newsletter: Piano trig applet, continued fractions In this Newsletter: 1. New applet: Piano note frequencies interactive... 3. Friday math movie: pentatonic scale The pentatonic scale is popular across all cultures. Some math... 4. Friday math movie: The math of sound, frequency and pitch Vi Hart gives us an enthusiastic overview of how... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Murray - 7M ago 30 Jun 2017 In this Newsletter: 1. New applet: Quadratic function graphs 2. Resource: Cymath 3. Math in the news: Math is not static 4. Math movie: Food and your brain 5. Math puzzles 6. Final thought: Age First, two quick apologies: • There's been a growing time gap between Newsletters. My consultancy work has been very busy lately and I haven't had as much time to devote to the Newsletter. • The last edition, 11th May, was plagued by some technical issues and some of you received it twice, while others missed out. I did my best to rectify it. 1. New applet: What does b do in a quadratic function?  The quadratic function y = ax2 + bx + c is a parabola. In this new applet, you can vary each of the parameters a, b and c using sliders. Changing the variables a and c are fairly simple to understand, but what does b actually change? See: [Hat-tip to PiPo for the idea behind this applet.] 2. Resource: Cymath  Cymath is a free math solver, covering a broad range of topics including logarithms, trigonometry, calculus and of course, algebra. Similar to Wolfram|Alpha or the commercial Mathway solver, you enter your math question in the box and it provides an answer. But in Cymath's case, you get a full step-by-step solution for free. Cymath is quick, and solutions for the most part appear to be mathematically sound. As they say on their About page: Our math solver is powered by a combination of artificial intelligence and heuristics, so that it solves math problems step-by-step like a teacher would. The site also provides a useful Reference section (containing common formulas with examples) and a Practice section (with common text book style questions, with solutions). There are Android and iOS mobile apps as well. Cymath raises some interesting issues. Unfortunately, most students will use such a facility to complete their math homework for them (thus gaining marks, but not knowledge), but some will find the step-by-step solutions useful for their learning. However, there are more issues involved here. a. Why do math? The Cymath solver (like the other two mentioned) does quite a good job of giving the correct mathematical answer (the "what") and to some extent, the process for getting that answer (the "how"), but doesn't provide any indication of why we are doing the particular piece of mathematics or how to interpret the answer. (To be fair, I wouldn't expect them to do so - I'm just pointing out an issue.) Going back to Cymath's own statement, if a math teacher just provides students with the algebraic steps for finding a solution, then they're doing a poor job of teaching that topic. b. What math should we do? Since these tools are now readily available, why do we still make students spend most of their time doing the algebraic steps needed for the final answer? Surely it would be better to allow students to use the available algebra solvers, and spend most of their time learning how to solve actual application problems? For example, one of the example topics in Cymath is polynomial division. It doesn't really make sense to get students to learn how to find (x^3-1)/(x+2), when one of the main reasons for doing so is to solve polynomial equations. Such questions are usually very contrived (with "nice" numbers) and quite unrealistic anyway. We can ask the solver directly to solve such things using a statement like: solve x^3+x^2-17x+13=0 It finds the 3 solutions numerically in an instant and frees us up to think about the implications of the answer. 3. Math in the news: Math is not static In school mathematics, we often get the feeling that mathematics is some fixed, static entity that has not changed in hundreds of years. However, new math is being developed all the time. Here are two examples. a. New statistics  A drug researcher usually assumes the response to a new drug will be the same for the experimental group as for the control group. But what if that is not the case? Also, how should we deal with outliers in our experimental results? Usually they are discarded, but is this the best approach? In New statistical methods would let researchers deal with data in better, more robust ways Rand Wilcox, Professor of Statistics, University of Southern California points out that a number of new statistical techniques have been developed over the last 3 decades to better handle issues that arise when comparing groups of individuals or things, and the relationships between those groups. The problem is, he says, researchers aren't using those new techniques becuase they don't know about then, and the techniques are not being taught in undergraduate statistics courses. b. A new 'branch' of math  Ever notice the branches of a river look similar to the branches of a leaf, the roots of a plant or our blood vessels? They all tend to branch out at similar angles. In A new 'branch' of math, we read that MIT researcher Dan Rothman used mathematical time modelling to examine how groundwater erodes surrounding hills, evnetually splitting a valley into 2 branches. Rothman's team determined there is a special angle (72°) at which streams branch. They confirmed this by examining 5,000 stream junctions in the Florida Panhandle. Here's a related scientifc paper explaining how this process works in blood vessels: Quantification of Blood Flow and Topology in Developing Vascular Networks. 4. Math movie: How the food you eat affects your brain  Many students have a terrible diet and it probably makes learning more difficult for them. In this short TED-Ed video, Mia Nacamulli explores the effects on our brains (and our learning abilities) depending on what we eat. Food for thought... 5. Math puzzles The puzzle in the last IntMath Newsletter asked how many ancestors a male bee has. Correct answers with explanation were provided by Isabel, Francis, and Jagmeet. This was a case of Fibonacci Series, of course. New math puzzle: Walking A woman sets out on a "power walk" from her home and returns 6 hours later. If she walks 8 km/h on level land, 6 km/h uphill, and 12 km/hr downhill, how far did she walk? You can leave your responses here. 6. Final thought: Age Here's an age equation that's pretty accurate: When you're 16, old people are 40. When you're 50, old people are 70, and when you're 100, you know you’re old! Until next time, enjoy whatever you learn. The post IntMath Newsletter: Quadratic graphs, Cymath, food and learning appeared first on SquareCirclez. Related posts: 1. New applet: What does b do in a quadratic function? y = ax2 + bx + c is a parabola.... 2. IntMath Newsletter: 7 billion people, blog carnival, food In this Newsletter: 1. 7 billion people next week 2.... 3. IntMath Newsletter: quadratic, resources, pentatonic and MathJax In this Newsletter: 1. Math tip - Quadratic formula by... 4. How to find the equation of a quadratic function from its graph A reader asked how to find the equation of a... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Murray - 7M ago The parabola is one of the most common curves we come across in engineering and science, as it is often an appropriate choice for modelling portions of curves because of its simplicity. We've seen elsewhere how to Draw the Graph of a Quadratic Equation (a parabola), and about Parabolas with Vertical and Horizontal Axes. Moving in the reverse direction, we learned how to Find the equation of a quadratic function from its graph. In this new applet, we learn the effects of changing each of the a, b and c variables in the quadratic form of a parabloa, y = ax2 + bx + c Changing a and c Changing variables a and c are quite easy to understand, as you'll discover in the applet. You'll see something like the following as you move the sliders: Changing b The effect of changing variable b is not so clear. The original curve seems to move around a new curve. What is that new curve and how much does the original curve move by? The new applet Time to check out the applet at: The post New applet: What does b do in a quadratic function? appeared first on SquareCirclez. Related posts: 1. How to find the equation of a quadratic function from its graph A reader asked how to find the equation of a... 2. Conic sections interactive applet This interactive applet allows you to explore the concept... 3. New applet: Domain and range investigation This interactive applet allows you to explore the concepts... 4. New Riemann Sums applet Here's some background on the new Riemann Sums applet... Read Full Article Visit website • Show original • . • Share • . • Favorite • . • Email • . • Add Tags Interactive Mathematics by Murray - 7M ago 11 May 2017 In this Newsletter: 1. New applet: Domain and range exploration 2. Resource: Azure Machine Learning Studio 3. Math in the news: Riemann hypothesis 4. Math movie: Is math discovered or invented? 5. Math puzzles: Bees 6. Final thought: Bees 1. New applet: Domain and range exploration  This interactive applet allows you to explore the concepts of domain and range for several different mathematical functions. Here's the applet: Domain and range exploration And here's some background to it:New applet: Domain and range investigation 2. Resource: Azure Machine Learning Studio  For those of you wondering what future jobs will involve, here's a resource that can give you some insights. Microsoft Azure Machine Learning Studio is a collaborative, drag-and-drop tool you can use to build, test, and deploy predictive analytics solutions on your data. It integrates with Python and R - software that math students should learn. There's a free option to try it out. 3. Math in the news: Riemann hypothesis  Question: How many primes are there less than one million? Answer: 78,498 Question: Who cares?Answer: All of us, every time we log in to a bank account, or visit any secure Web site (and if we want to earn a million dollars). Cryptography (the science of secret security codes) depends heavily on large primes, and the distribution of those primes is an important consideration for better security. Mathematicians have tried to approximate the number of primes less than a given large number, and in 1859 the German mathematician Bernhard Riemann conjectured that it was related to the solutions of the Riemann zeta function, ζ(s). This appears to work (and it has been used ever since), but no one has ever proved it. There's a$1 million prize waiting for anyone who does manage to solve it.

Dorje Brody, a mathematical physicist at Brunel University London has recently come close to doing just that. Their possible solution involves eigenvalues of matrices.

This Prime Counting article gives some good background.

For other unsolved math problems which also hold out the promise of $1 million, see: Unsolved Math Problems from Wolfram's MathWorld. 4. Math movie: Is math discovered or invented?  I quite like the idea that mathematics is simply "out there" and our job is to discover it. But others have another view. In this short TED-Ed video, Jeff Dekofsky examines the two sides of the argument. 5. Math puzzles The puzzle in the last IntMath Newsletter asked how many solutions there were for an inequality. Correct answers with explanation were provided by Jeel, Sachin, Gerard and Colin, with a variety of approaches used from pseudocode to listing as a group, or counting by one integer (plus/minus) at a time. New math puzzle: Bees Female bees have two parents (one male and one female), but the drone (male bee) has only one parent, his mother. What is the maximum number of genetic ancestors a male bee would have in the ninth generation back? You can leave your responses here. 6. Final thought: Bees When I was a young boy, my father (a research agronomist) used to get my brothers and me to pollinate some of his experimental plots. We'd place bags over the flowers and shake them, so the pollen would do its magic. I think of that experience whenever I read about bee colony collapse disorder. In Bee decline threatens US crop production, we learn that each year$3 billion of the U.S. economy depends on pollination from native pollinators like wild bees. As bee population drops, the demand for their services rises.

[Image source]

You can read how differential equations are used to map bee populations in this paper (PDF).

I don't think most of us want to be employed as plant pollinators in the future, and since we like to eat, this is a story that's worth following.

Until next time, enjoy whatever you learn.

The post IntMath Newsletter: Domain, range, Azure, Riemann appeared first on SquareCirclez.

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