IntMath aims to interest and educate people in the joys of mathematics. It does so by providing clear examples, relating things to the "real world" and providing interactive applets that allow the user to explore mathematical concepts.
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).
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:
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.
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).
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:
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)?
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.]
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.
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:
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?
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]
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.
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.
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:
(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.
(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."
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.]
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:
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:
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.
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".
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:
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."
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)
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.
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:
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:
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.
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?
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.
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 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.
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.
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...
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?
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.
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?
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
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.
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.
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.
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?
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.