In this Newsletter:
1. New on IntMath: Quadratrix, quadratic
2. Resources: Desmos, e-book
3. Math in the news: Review and teaspoon
4. Math movies: Coin, key
5. Math puzzle: Race
6. Final thought: Real cost
Challenge: Using only a straight edge and compasses,
Don't work too hard on it, since neither is possible. The ancient Greek mathematicians put quite a bit of effort into solving these two problems.
One of the approaches Hippias and Dinostratus used was to construct a quadratrix curve. |
The page has animations of how the curve is constructed, and some background.
I noticed a lot of my students would rather use the Quadratic Formula rather than solving a quadratic by factoring. One big reason for this is factoring can involve an amount of guesswork (finding 2 numbers that add to a certain number, and whose product is another number).
Mathematician Po-shen Loh recently outlined a method (not entirely original, and somewhat similar to completing the square method) which does away with the guesswork.
I added his method as alternative solutions in IntMath. See: Solutions for Examples 4 and 5, and Exercise 2 on the page: |
Po-Shen Loh is associate professor of mathematics at Carnegie Mellon University.
Desmos, the excellent free online graphing site, is offering some free Webinars over the next few weeks.
These ones are mostly for teachers who want to create activities on Desmos. You can see details here: Desmos Webinars |
There are sessions on 14 Jan, 21 Jan and 28 Jan. Don't miss the links to recordings of previous seminars on the same page.
This is an interesting book, targeted at programmers, which aims to "show how to engage with mathematics".
The book is by Jeremy Kun, engineer at Google. See: Pimbook |
You'll find:
Mathematics is not a static thing. New discoveries and ways of thinking about old problems occur all the time.
Here's a review by Quantamagazine of the main developments during 2019. |
The discussion about replacing "equality" with "equivalence" is interesting, and reminded me of my rant in The equal sign - more trouble than it’s worth?
We're heading for a "storage crisis". The amount of data generated each year (around 400 zettabytes by one estimate) will mean we won't have enough raw materials to manufacture all the optical and magnetic storage devices needed by the end of the century. We need to find other ways to store data.
One possible solution is to follow nature, and store the massive amounts of data in synthetic DNA. Scientists in Singapore are working on how to reduce the massive costs in such an approach.
Proability has a knack for throwing up counterintutitive situations.
This TED-Ed video is an exploration of extensions to the simple coin flip. |
(This video is by Poh-shen Loh whose quadratic equation approach I mentioned above.)
I've been fascinated with savantism for a long time. Savants demonstrate extraordinary mathematical, artistic or musical abilities.
Derek was born prematurely and is blind, had learning difficulties, autism but also amazing musical abilities, including perfect pitch and improvisation skills. |
So it begs the question - are we all born with such abillities and it just happens that some of us get to express them, while most of us don't?
The puzzle in the last IntMath Newsletter involved a substitution cipher. No-one actually had a go at it (I know there are a lot of distractions in December...) so I'll leave it there and give you another chance to solve it.
Alana, Bob and Chaitali are running against each other in pairs over a 2 km distance. If Alana can beat Bob by 100 m, and Bob can beat Chaitali by 200 m, by what distance could Alana beat Chaitali (assuming their speeds are constant)?
You can leave your responses here.
The "real" cost of any commodity should include the negative impacts the extraction and processing of that commodity have on all of us - the environment, our health, our future.
The IMF estimated that, in 2017, global fossil fuel subsidies grew to $5.2 trillion, or 6.5% of global GDP. [Source]
Why are we subsidising fossil fuel companies at all?
There are so many better things governments could be doing with our tax money...
Until next time, enjoy whatever you learn.
]]>In this Newsletter:
0. Season's greetings
1. New on IntMath: Putnam
2. The year in review
3. Math movies: Captive
4. Math puzzle: Cipher
5. Final thought: Imagine
'Tis the season for special days. If you celebrate any of these, enjoy.
Festival | 2019/20 date | Notes |
---|---|---|
Bodhi Day | 8 Dec, 2 Jan | "8th day of the 12th month" - so there's 2 of them |
Pancha Ganapati | Hindu festival celebrated from 21 through 25 Dec | Based on the solstice |
Solstice | 22 Dec | Celebrated in many cultures |
Hannukah | 22 Dec to 20 Dec | Movable. Could be late November to early January. |
Christmas | 25 Dec | "12 Days" finish on 6 Jan |
Chinese New Year | 25 Jan (new moon) to 8 Feb (full moon) | Movable. Could start 21 Jan to 20 Feb |
The William Lowell Putnam Mathematical Competition is held each year in early December. It's aimed at undergraduate students and has quite a reputation. Reader Pat Lachapelle participated in the competition this year and kindly provided a series of guest articles outlining the solution for an earlier competition question.
See how to go about solving a word problem. Most of the concepts involved are around the grade 9 school level (age 15). There are 7 parts to the solution. |
Thousands of readers have found the following useful or interesting during 2019. In case you missed any of them...
I wrote some of these years ago, but they are still popular - and/or causing controversy.
Here's an approach that could make quite a difference when students are learning about 3-D graphs.
The device, called CAPTIVE, offers six degrees of freedom (6DoF) for users - with applications ranging from video gaming to medical diagnostics to design tools. See: Device allows users to manipulate 3-D virtual objects more quickly |
The article includes a video of the process in action. (It's a 2017 article, but nonetheless interesting.)
The puzzle in the last IntMath Newsletter asked whether median or mean would be the best way to describe a typical value.
There were two parts to the puzzle, and correct answers with sufficient explanation were given by Dritan, Danish and Bob (who gets a special mention for pointing out it could be different depending on who the audience for the statistic is!).
A good way to see what's happening in this puzzle is to draw graphs.
Person A "typically" could expect to earn around $90 (the median) per day. Clearly the mean (just over $75) is not "typical" of his earnings, so we would take the median to be the best statistic.
Person B, on the other hand, usually earns around $82 (the mean), not $72 (the median), so in this case, the mean would represent the "typical" earnings.
A popular article at this time of the year is: The 12 Days of Christmas - How many Presents?
This cryptogram also involves 12. What is the value of each letter?
T W O T H R E E + S E V E N ----------- T W E L V E
You can leave your responses here.
Next year marks the 40th anniversary of John Lennon's murder.
My hope for 2020 is that more of us will get out of our echo chambers, start to listen to each other, and "live life in peace".
Imagine there's no countries
It isn't hard to do
Nothing to kill or die for
And no religion, tooImagine all the people
Living life in peaceJohn Lennon, Imagine
All the best to everyone for the holidays, and for 2020.
Until next time, enjoy whatever you learn.
]]>1. New on IntMath: Witch of Agnesi
2. Resources: Hypertools
3. Math in the news
4. Math movies
5. Math puzzle
6. Final thought
In this Newsletter:
1. New on IntMath: Witch of Agnesi
2. Resources: Hypertools, tech
3. Math in the news: Light, Etalumis
4. Math movies: Mod, culture
5. Math puzzle: Mean, mode
6. Final thought: 20
I've started a series of articles on historical math curves.
The first one is the "Witch of Agnesi", which is not a witch at all, and was included in a math handbook written by a rarity of the 18th century: a female mathematics academic. See: Witch of Agnesi |
The page has some background to the curve and some animations explaining how it is constructed. There's even a connection between this curve and calculus.
Data will be the most valuable commodity in the 21st century, and those able to make sense of it will do well. This tool could help.
Dartmouth College has released the open-source package HyperTools, which can transform data into visualizable shapes or animations for data comparisons and testing theories. See: HyperTools transforms complex data into visualizable shapes |
Most of you will know I'm a fan of technological approaches to teaching and learning of mathematics.
Here are some resources (some free, some paid) that can help with this. |
Image processing is at the core of augmented reality, autonomous driving and facial recognition. But because it's currently achieved using electronic circuits, it's too slow.
A new optical computing and imaging technique operates at the speed of light and the mathematical operation itself consumes no energy. |
Simulations are developed to try to understand and predict the future of such things as climate change, disease transmission and cosmic events. But how do we interpret and make best use of new data?
A new system, called Etalumis ("simulate" spelled backwards) developed by a group of scientists from the University of Oxford and others, makes use of Bayes' Theorem (which is a way to calculate probabilities based on probabilities for events that have already occurred) to solve such problems.
Etalumis uses Bayes inference to improve existing simulators via machine learning. |
This video (by Brilliant) explains modular arithmetic. It moves fairly quickly, but that's the beauty of videos - you can pause and go over parts. |
This is an interesting way to visualize the spread of culture throughout history. The animation distils hundreds of years of culture into just five minutes. See: Charting Culture |
The comments rightly point out some of the cultural biases in the video, but it probably can't be helped when compressing such a highly complex data set into just 5 minutes.
Clarification: The puzzle in the September IntMath Newsletter asked about the probability involved in a given octagon. There were conflicting answers given, and I provided a solution. However, it turned out my attempt had a problem, and Tomas' answer was correct. You can see the discussion here: Octagon discussion
The puzzle in the last (October) IntMath Newsletter asked about "square ages".
There were two parts to the puzzle, and Thomas was the only one who got both parts correct. (Special mention to Don who got the first part, and the range of years correct for the second part.)
The table below shows the daily earnings for two workers for 5 days and the mean and the median salary.
Person A | Person B | |
---|---|---|
Day 1 | $100 | $72 |
Day 2 | $87 | $97 |
Day 3 | $90 | $70 |
Day 4 | $10 | $71 |
Day 5 | $91 | $100 |
Mean | $75.60 | $82 |
Median | $90 | $72 |
(a) Which statistic, the mean or the median, would be best to describe the typical daily salary for the 5 days for Person A? How about for person B?
You can leave your responses here.
Jean-Francois Rischard, ex-World Bank vice president wrote the book High Noon: 20 Global Problems, 20 Years To Solve Them in 2003. Sadly, the scorecard for his 20 Problems is rather abysmal as we approach the 20th year of the century.
I wrote a summary review of the book in 2008. I concluded by saying:
Mathematics is involved in the solutions for all these problems. So instead of doing algebra with no purpose other than to pass an examination, let's put our mathematical efforts into coming up with ways to solve (at least some of) these problems.
Governments should be placing many of the 20 Global Problems in the center of their policy frameworks, but they're not, and too many governments are actively doing the opposite.
As a counterpoint to the destruction of the Amazon, here's what one Brazilian photographer achieved in 18 years, something that should be happening worldwide:
Please be careful who you vote for...
Until next time, enjoy whatever you learn.
]]>In this Newsletter:
1. New on IntMath: Polynomial equations graph solver
2. Resources: Tilt Brush, charts
3. Math in the news: Lenses, integral equations
4. Math movies: Potential
5. Math puzzle: Square age
6. Final thought: Water
After many reader requests, I rewrote a lot of the Solving Polynomial Equations chapter recently. I'm not a big fan of the way this topic is normally handled (via the Remainder and Factor Theorems) mostly because those techniques only work for low degree polynomials with "nice" numbers and involve an amount of guesswork. But I agree there is some benefit to learning the concepts.
The revised pages are:
Roots of a Polynomial Equation
I also added one new interative page:
We can always use numerical approaches to finding the roots of equations (like Newton's Method). But I've always found (for the last 30 years while we've had such tools) that zooming in on the x-intercepts in a graph application is easier and quicker. We can get whatever accuracy we need by zooming in some more. |
Has anyone played with this? What's your experience been?
Charts of complex data can be hard to comprehend. This new method from Columbia Engineering and Tufts University aims to develop easier to read data visualizations. See: New data science method makes charts easier to read at a glance |
Greek mathematician Diocles first reported on a problem most lenses have - spherical aberration. Here's a summary of a recent fix from Popular Mechanics: See: How One Mathematician Solved a 2,000-Year-Old Camera Lens Problem |
And here's the actual paper for those interested:
General formula for bi-aspheric singlet lens design free of spherical aberration (link no longer available)
Solving integral equations has been a staple of science and engineering for hundreds of years. A new optical approach makes computer solutions significantly quicker. |
Most of us fit in the "forgotten middle" - neither exceptional nor problematic. This talk gives some insights - for both students and teachers - on how such people can reach their full potential. See: How we can help the "forgotten middle" reach their full potential |
Everyone has a story - a reason why they have not met their own expectations or hopes for their lives. How we approach our obstacles can make a huge difference. |
The puzzle in the last IntMath Newsletter asked about the probability involved in a given hexagon.
There were five attempts at an answer, and all were different. My approach was as follows.
Choose 1 for the length of each side of the octagon. That means the triangles and the square also all have side length 1.
The octagon's area is 2(1 + √2) (a well-known formula, or it can be derived from splitting it up into triangles).
The area of each (equilateral) triangle is √3/4 (half the base times the height) and we have 4 of them, so the total area of the triangles is √3.
The area of the square is 1.
The area of the blue parallelograms (kites) is the hexagon's area minus the area of the trangles and square.
Area blue = 2(1 + √2) − √3 − 1 = 1 + 2 √2 − √3.
So the probability is
If I said I was x years old in the year x^{2}, it would mean for example, in the year 900 CE (AD), I was 30.
How old would somebody be if they could say that in the 21st century?
When is be the next century when nobody will be able to say the year is the square of their age?
You can leave your response here.
Parts of Australia are undergoing the worst drought in recorded history. Some towns have already effectively passed "day zero" since trucking in water has become necessary. In many places, "day zero" will come sometime next year. Meanwhile, Capetown in South African recently averted a "day zero" situation.
For years scientists have been predicting such scenarios, and for years all we've seen is political football games.
Yes, the rains will come again. But it's more likely the next drought will be even longer. Many Australian farmers are talking about giving up, and this is in the country that promised to be the "food bowl of Asia".
You can't eat coal, and you can't drink oil.
So how did Capetown do it? They imposed personal water restrictions of 50 liters per person per day and tough industrial restrictions ("Level 6B" water restrictions). By comparison, US citizens consume around 500 liters per day, New Zealanders consume 227 liters a day and in India, it's 120 liters per day.
See the state of your country in this Water Risk Atlas data map.
Until next time, enjoy whatever you learn.
]]>In this Newsletter:
1. Popular on IntMath: Calculators
2. Resources: Music lab, Webinars
3. Math in the news: AI, atomic clocks
4. Math movies: Tensors, irrationals
5. Math puzzle: Hexagon
6. Final thought: They knew
Today is equinox (equal day and night), where the center of the sun is directly above the equator (very near where I live). In the Southern hemisphere, it's the beginning of spring.
The various calculators on IntMath continue to be popular among visitors.
Some of the other calculators are:
There are now over 200 interactive applets on IntMath. You can see a list of all of them here:
Interactive applets on IntMath
Most music has elements of rhythm, melody and harmony, which are all mathematical in nature.
Chrome Music Lab contains several interactive experiments that let you explore harmonic series, spectrogram and a melody generator. In the process, you'll see math in action. See: Chrome Music Lab |
Here's another round of free Webinars from the people who bring us Wolfram|Alpha where you'll learn what's new in Mathematica and Wolfram Language 12. Each of these is on a Wednesday, 5-6pm GMT (3pm in Sydney, 10pm Tue in LA).
Driven by neural networks (mathematical systems that learn by analyzing huge amounts of data), AI has much potential, but also poses dangers.
The Allen Institute for Artificial Intelligence has produced a system that passed an 8th grade science test with a score of 90%, and a 12th-grade exam at 80%. See: A Breakthrough for A.I. Technology: Passing an 8th-Grade Science Test |
Note:
These are significant limitations in my view. Scientific thinking cannot be adequately assessed via multiple choice.
However, this is still quite an achievement and is yet another step forward in the inevitable advancement of AI.
Did you know that not only do we need leap years (the Earth's rotation around the sun is very nearly 365.25 days), we also need leap seconds (there has been around 1 leap second added to UTC (aka GMT) every 2 years since 1972)?
From accurate GPS to the timing needed for coordinating computers across the world, super-accurate atomic clocks are essential. This readable summary gives some good background. |
Daniel Fleisch of Wittenberg University specializes in electromagnetics and space physics.
In this video, he gives a good, simple explanation of tensors and their relation to vectors, using simple household objects. See: What's a Tensor? |
His passion for explaining things to others is obvious.
The Pythagoreans had their fundamental beliefs questioned by the philosopher Hippasus, who showed √2 could not possibly be a rational number.
Ganesh Pai gives us the background in this TED-Ed video. |
The puzzle in the last IntMath Newsletter asked about a mystery object.
There was only one attempt at an answer (by Nicola, who guessed it was an abacus).
One possible approach to image search: The easiest way to find out what an image could be is to take a picture of it with Google Lens (on your phone). This brings up many sites talking about – and naming – the object.
Another approach: Right clicking on the image (on the blog copy of the Newsletter) and choosing "Copy image address" and pasting that into the search box on Google Images also brings up many results.
The object was a kleroterion, used by the ancient Greeks to randomize the selection of citizens (male, of course) to the boule (local council), to political roles, and as court jurors. (See more: Kleroterion)
It was basically like a modern-day lotto machine, that spits out numbered balls.
Find the probability for a point chosen randomly within the octagon to be blue. The red triangles are equilateral.
You can leave your response here.
I was watching a documentary about the 1980s recently, and it included the tragic events of the explosion of space shuttle Challenger, which held the first teacher in space, Christa McAuliffe.
The engineers knew there were safety concerns (it was a very cold day and vital O-rings failed, a highly probable outcome known for over a decade previously), but went ahead with the launch.
Later this century, when several tipping points have occured and climate change becomes irreversible, kids will ask their grandparents, "You knew it was a risk, so why did you let it happen?"
Until next time, enjoy whatever you learn.
]]>1. New on IntMath: CSS matrix math
2. Resources: Humble Pi, AnswerThePublic
3. Math in the news: Proof
4. Math movies: Parker, van Gogh
5. Math puzzle: Mystery object
6. Final thought: Dry leaves
In this Newsletter:
1. New on IntMath: CSS matrix math
2. Resources: Humble Pi, AnswerThePublic
3. Math in the news: Proof
4. Math movies: Parker, van Gogh
5. Math puzzle: Mystery object
6. Final thought: Dry leaves
I recently gave a talk to a local meetup group on the mathematics behind CSS matrix. CSS stands for "cascading style sheets", and is the system where Web designers can set font sizes, colors, and also set sizes and vary shapes for objects like images and videos.
Matrices are used to transform geometric objects (scale, skew, rotate, translate and so on.) Computer games make extensive use of matrices to simulate depth, 3D objects and so on.
I also developed the following interactive graph applet that demonstrates the concepts in the talk.
This is an interactive graph where you can vary sliders to see how CSS matrix changes the size, shape and location of an object. |
When I was teaching a group of engineering students some years ago, I went to my boss with an idea. I suggested we incorporate examples of cases where things went wrong, so students would learn the importance of accuracy and the safety issues that can arise out of sloppy and inaccurate mathematics.
He wasn't enthusiastic and squashed the idea, saying it would scare off students from choosing the degree. I felt it was a lost opportunity.
So I was interested when I came across the book "Humble Pi - A Comedy of Maths Errors" by Matt Parker.
This readable book was exactly what I had in mind when I approached my boss. Let's learn from cases where people made math errors, and see what the consequences were – not to apportion blame, but to learn what can go wrong.
We all make math errors, but usually the worst outcome is a drop in grade, or momentary embarrassment. The people working in science and engineering fields should be aware of why their math teachers insisted on accuracy.
So I suggested the book for the local library, and was pleasantly surprised how long I had to wait before I could read it (it turned out to be quite popular).
I recommend this book for any math student or teacher.
One of the videos below features Parker, covering some of the same errors detailed in the book.
Disclaimer: I have no connection with Matt Parker (other than through Twitter) and receive no commission.
Some teachers see their job as simply giving out information, but there is no "value add" in that approach, especially since students can easily access such information in abundance.
One thing we can do better while teaching a topic is to actually address the questions students really have about that topic. One approach is to simply ask students what their questions are, and there are a lot of apps and sites that facilitate this process (e.g. Google Forms and Survey Monkey are both easy to use).
Another thing to consider is to look at the questions students are likely to ask, before you even start planning the lessons. AnswerThePublic is a good resource for this.
AnswerThePublic is a database of common questions that people ask about topics. It provides a rich source of ideas on how we might go about introducing a topic, and pre-empting the stumbling blocks. |
Topics to try are:
You can choose the "Data" tab at the top of each visualization to get easier-to-read lists of questions, and download the lot as a CSV (for Excel).
Sometimes in this resource you see questions that may seem quite odd, like "Does calculus cause kidney stones?", but "calculus" means "stone" and in medicine, it refers to a build-up of hard substances in the body. I get it on my teeth.
The title "Gaussian correlation inequality" sounds scary, but it goes something like this.
I have a dart board sitting on a rectangular shape, and assume I'm a good dart player. When I throw a lot of darts at it, I expect the accuracy of my throws to form a somewhat bell-shaped curve distribution. That is, most of the darts land somewhere close to the middle, and there are less dart holes as I go out from the middle.
The greater the circle overlaps the rectangle, the probability of striking both goes up.
This is a one-hour presentation by Matt Parker at the Royal Institution, London. It covers some interesting examples that are worth considering. |
Parker is the Public Engagement in Mathematics Fellow at Queen Mary University of London.
Turbulence is one of the most tricky phenomena to model using mathematics. It is complicated and chaotic.
This video by Natalya St. Clair explores how Van Gogh incorporated turbulence in his art to give the impression of movement. |
The puzzle in the last IntMath Newsletter asked about radii of mutually tangent circles. In fact, it turned out to be a 3x3 system of equations - it wasn't really a geometry question.
Correct answers with sufficient reasons were submitted by Russell, Nicola, Tomas and Thomas.
This time, some investigation may be involved.
The above object was used to achieve a particular mathematical outcome – one that is still vitally important to this day. What is the object, where was it used, and what was the mathematical outcome?
(If you can't actually find it or figure it out, your speculation will prove interesting!)
You can leave your response here.
Equatorial Singapore, where I live, is normally lush and green, and doesn't experience leaf falls as is normally the case for most places in Autumn.
However, this year we've had the driest and second hottest July on record, and practically no rain so far in August causing my local park to look like this:
Such dry and hot conditions are caused by a positive Indian Ocean dipole, the situation where the Western Indian Ocean is hotter than the Eastern part, causing hot droughts over Australia and most SE Asian countries.
These make ideal conditions for farmers in Indonesia to set off forest fires in order to plant more palm oil, so there's been many fire hot spots reported there.
Meanwhile, the dry season in the Amazon has been the excuse, along with President Bolsanaro's encouragement, for farmers there to burn vast amounts of the Earth's lungs for ever-expanding methane-producing cattle farms.
In the Arctic, fires across the tundra continue to burn, spewing even more carbon into the atmosphere.
These events are giving us an insight into how things will be if governments, companies and all of us fail to address our land use, our consumption, and our "economic growth at all costs" mentality.
We can stop it, but will we?
Until next time, enjoy whatever you learn.
]]>1. New on IntMath: Eigenvalues and eigenvectors
2. Resources: Desmos and 3D simulator
3. Math in the news: News and science
4. Math movies: Graphics and addresses
5. Math puzzle: Tangent circles
6. Final thought: Culture
In this Newsletter:
1. New on IntMath: Eigenvalues and eigenvectors
2. Resources: Desmos and 3D simulator
3. Math in the news: News and science
4. Math movies: Graphics and addresses
5. Math puzzle: Tangent circles
6. Final thought: Culture
Eigenvectors are very important in many science and engineering fields, including compuer science. (Google's search algorithm is based on eigenvectors.) It was one of the many topics I learned about as a math student where I could find the answers, but I really had no idea what I'd found, what the concept really meant, or what they were good for.
I wrote some new pages incorporating interactive applets which I hope gives the reader a better idea of what this interesting topic is all about. (It's usually taught at university level, but the mathematics involved is not that challenging - mostly multiplying and adding.)
The applet on this next page allows you to explore the physical meaning and geometric interpretation of eigenvalues and eigenvectors.
Like most matrix operations, it's very easy to make simple mistakes when calculating eigenvalues and eigenvectors.
This calculator allows you to check your work, and/or to explore what happens with eigenvectors for everything from 2x2 and 3x3 up to 9x9-size matrices. |
You may also be interested in:
How to find eigenvalues and eigenvectors? (outlining the algebraic steps for finding them - this was the only part I learned at university).
Applications of eigenvalues and eigenvectors (which includes a highly simplified description of how Google search works).
Here's a recent tweet by teacher Liz Caffrey:
I know from my own experience how much richer my understanding of topics is after I've created similar interactive explorations. Doing it with Desmos is really quite easy, and I encourage you (whether student or teacher) to give it a go. You don't need a lot in the way of programming skills.
This is a 3D simulation of the solar system, built on an "open source gravity simulator".
It also has some "What-if" scenarios. This example shows what would happen if Earth actually orbitted Saturn:
You can drag left-righ, up-down to explore and use the mouse wheel to zoom. Press the "Play" button at the top to animate the scene. |
You can also change the physics parameters, and add masses. It's very nicely done. (Probably best on a laptop.)
Last week marked the 50th anniversary of the moon landings. This was an extraordinary human achievement, but the misguided belief that it was all a hoax continues to grow, fuelled by ignorant people (and bots) through social media. Where and how we find our news, and how we critically examine the sources and conclusions, becomes increasingly important as the growth of fake news has the potential to rip apart our societies.
In the US, more people get their news from social media than traditional news sources, while in other countries it's the reverse. See a short summary: Where do people find news on their smartphones?
From the original Digital News Report, by Reuters and University of Oxford: Executive Summary and Key Findings of the 2019 Report
Those of us involved in science education would like to think most students graduate with a reasonable general knowledge of the various branches of science, and will have a good grounding in how science actually works. But it appears that is not always the case.
The (US) National Science Board's Science and Engineering Indicators 2018 makes for some sobering reading.
Science and Technology: Public Attitudes and Understanding
Here's a sample of the misunderstandings gleaned from the results of various quizzes.
No wonder we have the growth of drug-resistant superbugs, rampant species destruction, and climate change deniers.
When I first arrived in Japan where I lived for 4 years, I was amazed at how different their address system was compared to what Iwas used to. Rather than sequential numbers (usually odd on one side and even on the other) along roads, the Japanese system is built around regions and sub-regions of a suburb.
So I found the pretext of this next talk quite interesting.
Many houses do not have any kind of address system, and to find anyone, you need to rely on local knowledge. Chris Sheldrick has an interesting solution - use three-word addresses! |
I like the basic concept, but surely numbers are more universal? Wouldn't addresses based on latitude and longitude make more sense, especially as that's how our phones know where we are already?
The puzzle in the last IntMath Newsletter asked about the possible dimensions of 2 pencils.
A correct answer with sufficient reasons was submitted by Georgios. Special mention to Nicola who worked on a generalized solution involving circles and ellipses, which she explored using Desmos.
Circles that are "mutually tangent" touch at one point only. We have 3 such circles, with centers P, Q, R and radii p, q, r respectively. We are given the lengths PQ = 15, QR = 21 and PR = 14. Find p, q and r.
You can leave your response here.
Pangolin (image credit)
Living in different countries gives you fresh insights about culture. A lot of the cultural practices that seemed vital to our existence have faded as time goes on, and others we question due to their environmental impacts. While they may not have had much effect hundreds of years ago when there were less people in the world, perhaps some aspects of the following don't make a lot of sense any more?
Maintaining the traditions while minimizing their impacts is surely the best way to move forward.
Until next time, enjoy whatever you learn.
]]>1. New on IntMath: Bubble sort rainbow
2. Resources
3. Math in the news
4. Math movies
5. Math puzzle
6. Final thought
In this Newsletter:
1. New on IntMath: Bubble sort rainbow
2. Resources: Symbols, study skills
3. Math in the news: Number sense
4. Math movies: Stress and learning
5. Math puzzle: Pencils
6. Final thought: No environment, no economy
One of the beauties of mathematics is a downside for new learners. Mathematicians love to be precise and concise, so they make use of a vast range of symbols, which can make things very confusing for the novice.
In the image at left, there are 12 math symbols. Can you name each one, and give an example of its use? |
Here are some places you can find out the meaning of all these symbols (and more), with examples of their use:
Basic Math Symbols (RapidTables)
List of mathematical symbols (Wikipedia)
Exhaustive List of Mathematical Symbols and Their Meaning (ScienceStruck)
Common Symbols in Mathematics: Maths Glossary (SkillsYouNeed)
You may also be interested in my discussion here:
Towards more meaningful math notation
Many students are ineffective and inefficient when it comes to study time. It gets worse as the number of distractions — mostly from devices — gets worse. |
Here's a list of study tips I came across recently. We can't expect too many students to read and act on such a list (that's not realistic), but they could form a good starting point for a class activity.
25 Scientifically Proven Tips For More Effective Studying
Exercise: The title says the tips are "scientifically proven". Do you agree there's enough evidence in the article to conclude this?
Researchers from the University of Tübingen were studying how an artificial neural network classified objects from images, and they noticed a curious and unexpected outcome — the network began to develop a 'number sense'. For example, while examining the image of 4 dots (top), the neural network had peak activity at the number 4 (bottom) |
According to the original paper, Number detectors spontaneously emerge in a deep neural network designed for visual object recognition:
Although our model was merely trained to classify natural images in a task that was unrelated to numerosity, its spontaneously emergent numerosity-tuned units allowed reliable categorization of the number of items in dot displays. These findings suggest that the spontaneous emergence of the number sense is based on mechanisms inherent to the visual system.
For an accessible summary of the research, see: 'Number sense' arises from the recognition of visible objects
It's summer holidays for many of you now and your stress levels are probably lower now than the last time you took a math test.
It's a good time to think about how to address your stress, before the next semester starts. Here are 2 videos covering the issue, each from a different point of view.
The puzzle in the last IntMath Newsletter asked about averages in a bowling game.
Correct answers with sufficient reasons were submitted by Rebecca, Michael, Tom, Gerard, Nicola (who rightly questioned one of the assumptions of the question), Eamon, and Deborah.
We have two cylindrical pencils where the ratio of the height to the diameter is the same for the first pencil as for the second. If the sum of their heights is 1, the sum of their surface areas is 8π and the sum of their volumes is 2π, find all possibilities for the dimensions of each pencil.
You can leave your response here.
Request: If you present your answer with a full explanation of how you obtained it, then others can follow it and learn something from it.
There is understandable concern that jobs (mining, manufacturing, energy) will disappear if nations actually get serious about mitigating climate change.
In We Can't Have A Thriving Economy Without Thriving Ecosystems, David Festa makes the point that destroying ecosystems destroys jobs, and more importantly, destroys food sources.
It seems to me we all need to make our lifestyles more sustainable. That means some jobs will disappear, but it's possible that many more — ones better for the planet and our enjoyment of our planet — will be created.
Are we willing to lose a bit, or do we all lose the lot eventually?
Until next time, enjoy whatever you learn.
]]>In this Newsletter:
1. New on IntMath: Logarithm function from its graph
2. Breakthrough Junior Challenge
3. Resources: Wolfram, MathGraph32
4. Math in the news: Computers and equations
5. Math movies: Calculus, Probability
6. Math puzzle: Bowling
7. Final thought: extinctions
This article explains how to observe the characteristics of a logarithmic graph to determine its equation. See: How to find the equation of a logarithm function from its graph? |
One of the best ways to learn something is to explain it to someone else.
The Breakthrough Junior Challenge encourages students to create a short explanatory video. Prizes include a $250,000 post-secondary/college scholarship. |
The overall idea, from the site:
Explain a big scientific idea in Physics, Life Sciences or Mathematics with a short video. You will have until June 15, 2019 at 11:59 PM PDT to submit your video (3:00 minutes max).
See these previous prize-winning videos made by students.
The number of submissions for math topics is quite low compared to the other categories. So, give it a go!
Wolfram has released a new version of the Wolfram Language and Mathematica, and is offering 4 free Webinars to explain the new features. |
I'm particularly looking forward to the June 12 one, which will cover:
Mathematics and Scientific Visualization
Calculus • Algebra • Complex Visualization • Geographic Visualization • Molecular Visualization
You can register here.
Of course, their main aim is commercial, but you can still learn a lot of worthwhile things by seeing how the experts make use of a powerful mathemtaical tool.
You may also be interested in Wolfram U, a series of videos which explores a range of math topics using Mathematica.
I've featured MathGraph32 a few times before on IntMath. It's a cross-platform, cross-browser interactive graphing solution, with some very nice features. It's similar in many ways to GeoGebra, but I find it easier to use for many cases.
From the developer, Yves Biton:
MathGraph32 can be used:
It also works on mobile devices.
Check out these tutorial videos in English, and here are some graph interactives.
Throughout history, a lot of mathematical research ended up in the "too hard" basket, because it was just impossible to do all the calculations necessary to draw a conclusion.
One such example is that of Diophantine Equations, the most famous of which was x^{n} + y^{n} = z^{n}, which has no whole number solutions when n is greater than 2, finally proved by Andrew Wiles after 300 years.
The article Using computers to crack open centuries-old mathematical puzzles by Christopher Rasmussen, Associate Professor of Mathematics at Wesleyan University, explains in layman's language what the researchers aimed to do by using computers.
The actual paper talks about how they used Sagemath, an open-source computer algebra system, to solve cubic Ramanujan-Nagell equations, for example:
x^{3} + 3^{k} = q^{n}, where q is an odd prime.
One such solution is x = 2, k = 4, q = 89 and n = 1, giving:
2^{3} + 3^{4} = 89^{1}
An example of a simple linear equation written in matrix form is:
(See Matrices and Linear Equations for background.)
A new circuit can solve large cases of such a system of linear equations in a single operation within a few tens of nanoseconds, much faster than using current or quantum methods. |
It uses an analog method of in-memory computing, where one equation coefficient is stored as a unique resistance in one memristor (an electrical component that regulates the flow of current in a circuit and remembers the amount of charge that has previously flowed through it, retaining this memory even without power.)
The circuit was tested on a wide set of algebraic problems including:
See: Speeding up artificial intelligence
I've featured 3Blue1Brown's brilliant videos in earlier Newsletters. This time it's the series, Essence of Calculus.
If you are about to embark on studying calculus, this would be a great introduction: |
Mining unobtainium is hard work – the rare mineral appears in only 1% of rocks in the mine. But your friend Tricky Joe has something up his sleeve (involving conditional probability). |
The puzzle in the last IntMath Newsletter asked about a general formula for a given sequence. There were an interesting range of responses.
Correct answers with sufficient reasons were submitted by Michael (who used modulus arithmetic) and Russell. Correct solutions (but with missing reasoning or details) were presented by: Gayathri, Christopher, Larry, Arun, and Nicola (who used a recursive approach).
In his latest bowling game, Raj scored 199 and this raised his average over a number of games from 177 to 178. What must he score in the next game to raise his average to 179?
You can leave your response here. Hint: If you present your answer with full explanation of how you obtained it, then others can follow your thinking and learn something from it.
Some years ago I was on a retreat in Malaysia. I enjoy going for walks in the early morning, and one day I followed a track into the (secondary) jungle. I was quite surprised to come across recent evidence (just hours old) of elephants using the same track. It could have been dangerous had I met those elephants, but I was excited to know wild elephants were still in the area. (Most of the virgin jungle in that area has been cleared for palm oil production, so it was doubly surprising.)
So I was saddened by the death this week of the last captive male Sumatran rhino in Malaysia, the last in a country that used to have plenty of them.
There is a real insanity of animal destruction going on in order to feed the irrational ivory, rhino horn, pangolin scale and shark fin markets.
Those who buy these products will need to consume fingernails when the animals are all gone. It will have the same effect on their libido and health.
Until next time, enjoy whatever you learn.
]]>A logarithm graph of the form y = log_{10}(x) has the following shape:
Notice the graph passes through the point (1, 0) (since
log(1) = 0, no matter what base we are using) and the point (10, 1), since log_{10}(10) = 1.
The following graph shows the cases when the base is:
We can find the base of the logarithm as long as we know one point on the graph. Here, we assume the curve hasn't been shifted in any way from the "standard" logarithm curve, which always passes through (1, 0).
Example: A logarithmic graph, y = log_{b}(x), passes through the point (12, 2.5), as shown. What is the base, b?
Answer: We substitute in our known values y = 2.5 when x = 12.
2.5 = log_{b}(12)
To find the base, we just need to apply the basic logarithm identity:
If y = log_{b}(x), then
b^{y} = x
Applying this, we have:
b^{2.5} = 12
This gives
So the base of the given logarithm equation is 2.7.
Let's now see some "non-standard" ways the logarithm graph can appear.
The general form for this curve is:
y = d log_{10}(x)
If we multiply the log term, we elongate (or compress) the graph in the vertical direction. In this next graph, we see from the top-most curve:
d = 5, 4, 3, 2;
y = log_{10}(x) (dark green);
y = 0.5 log_{10}(x)
While this may look similar to Figure 2 above, it is quite a different situation.
Here is the graph of y = log_{10}(−x).
The minus sign before the "x" has the effect of reflecting the curve in the y-axis. We get a vertical mirror image of the curve we saw in Figure 1.
Here is the graph of y = −log_{10}(x).
The minus sign before the "log(x)" has the effect of reflecting the curve in the x-axis. We get a mirror image horizontally.
Figure 5: Graph of y = −log_{10}(x)
Here is the graph of y = −log_{10}(−x).
We get a mirror image across both the vertical and horizontal axes.
Figure 6: Graph of y = −log_{10}(−x)
NOTE: Compare Figure 6 to the graph we saw in Graphs of Logarithmic and Exponential Functions, where we learned that the exponential curve is the reflection of the logarithmic function in the line y = x. This is not the same situation as Figure 1 compared to Figure 6.
Next, we'll see what happens when we come across logarithm graphs that do not pass neatly though (1, 0) or (−1, 0) as in the cases we've just seen.
We introduce a new formula,
y = c + log(x)
The c-value (a constant) will move the graph up if c is positive and down if c is negative.
For example, here is the graph of y = 2 + log_{10}(x). Notice it passes through (1, 2). It is the curve in Figure 1 shifted up by 2 units.
Similarly, this is the graph of y = −2 + log_{10}(x). It is the result of shifting the curve in Figure 1 down by 2 units, and it passes through (1, −2).
We introduce another new formula, where we've replaced x with (x + a):
y = log(x + a)
The "a" will have the effect of shifting the logarithm curve right if a is negative, and to the left if a is positive.
For example, the graph of y = log_{10}(x − 4) looks like:
Example 4: Graph of y = log_{10}(x − 4)
The curve in Figure 1 has been shifted to the right by 4 units.
Also, the graph of y = log_{10}(x + 5) gives us the curve in Figure 1 moved left by 5 units:
Example 5: Graph of y = log_{10}(x + 5)
Let's pull what we've learned so far into an example.
Question: Find the equation of the logarithm function (base 10) for the following curve.
Answer: We observe the shape of this curve to be closest to Figure 4, which was y = log_{10}(−x).
We'll assume the general equation is:
y = c + log_{10}(−x + a).
We also observe the (almost) vertical portion of the graph is at x = 2.5, so we replace −x with −(x − 2.5) and conclude a = 2.5. (Try a few values for x to see why a ends up being positive.)
Here's an interim graph, where I've moved the curve 2.5 units left so I can easily see how high or low the graph is:
We know the graph needs to pass through (−1, 0), and we observe we are 1.5 units too high. So we conclude c = 1.5.
So the required equation is
y = 1.5 + log_{10}(−x + 2.5).
There are many computer packages (SPSS, Excel, etc) that can determine a "best fit" curve for a given set of data. But for the sake of explaining how to determine an unknown logarithmic function from its data plot, let's take a look at this example (one which is close to my heart).
The experiment: A group of people are asked to learn a list of random words. They are tested immediately, then again after some time has elapsed, and then repeatedly over longer time spans. As expected, the number of words they remember correctly diminishes over time.
The average data is plotted over time as follows, where the horizontal axis is time, and the vertical axis is the proportion of the words they got right. (This is the famous Ebbinghaus Forgetting Curve.)
Example 7: Ebbinghaus Forgetting data
The question: You can see a "best fit" curve has been drawn through the data points. What is the equation of that curve?
The answer: We observe it is most like Figure 5 , which had the formula y = −log_{10}(x).
We assume our logarithmic fundtion will have the form:
y = c − d log_{10}(x).
We now aim to find the values of c and d.
We know from the section on Graphs on Logarithmic and Semi-Logarithmic Axes that we can turn a logarithmic (or exponential) curve into a linear curve by taking the logarithm of one of the variables.
To do so, we take the original data set (the columns "time" and "proportion") and find the logarithm of the independent variable, time. (Normally we would use natural log for such analysis, but I'm sticking with base 10 to be consistent with the earlier examples.)
time (t) |
log_{10}(t) | proportion |
---|---|---|
1 | 0 | 0.84 |
5 | 0.69897 | 0.71 |
15 | 1.17609 | 0.61 |
30 | 1.47712 | 0.56 |
60 | 1.77815 | 0.54 |
120 | 2.07918 | 0.47 |
240 | 2.38021 | 0.45 |
480 | 2.68124 | 0.38 |
720 | 2.85733 | 0.36 |
1440 | 3.15836 | 0.26 |
2880 | 3.45939 | 0.2 |
5760 | 3.76042 | 0.16 |
10080 | 4.00346 | 0.08 |
Now we can easily deduce the linear equation for this curve (I'm taking the first data point and the second last, since the last one is clearly not that close to the smooth curve.)
I use the point-slope formula for a line:
y = 0.79872 − 0.16985 x
Next, we simply replace the "x" with log_{10}(x) and achieve the required equation:
y = 0.79872 − 0.16985 log_{10}(x)
NOTE: The Ebbinghaus Forgetting curve is usually modelled using an exponential curve, but this data quite nicely fits a logarithmic curve for the given values. It's not a good model if we extrapolate it into the future, since this logarithmic curve doesn't flatten out (like the appropriate exponential one would, and it eventually goes off into negative territory.)
[Data credit: Penn State]
Hopefully, whatever logarithmic graph you are trying to find the equation for will be covered by one or more of the cases above.
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