Numbers are so familiar to us that it might seem unimaginable that there was a time when the very concept didn’t exist. Indeed the invention of numbers is lost in antiquity. Historians of mathematics speculate that the origin of numbers was probably connected with real problems of life at the time, like describing whether there was one animal, or more than one animal as food source (or a threat). A certain level of abstraction was required to use numbers. Three rabbits, three stars and three rocks only share the property of three-ness. Manipulation of number – with no connection to physical objects – was a great intellectual leap.
Negative numbers arrived on the scene much later. Trading and commerce meant that profit and loss should be accounted for properly. Negative numbers were used to represent an absence or a loss. Despite this the negative numbers were not immediately accepted by mathematicians. Early practitioners of algebra would often discard negative values when they appeared as solutions. After all it’s easy to picture three people in a room. Or two. Or one. Or even none. But what does minus one person in a room look like? One of my students recently suggested it would be like a ghost! There may be grounds for rejecting negative numbers as the solution to a particular problem but in other situations their use may be perfectly acceptable.
Negative numbers eventually found their place in our number system because they can be solutions of equations – just as valid as their positive namesakes. Likewise the history of zero is just as fraught with controversy and confusion. Zero initially served as a placeholder in the representation of number. For example, it is the zeros which tell you about the size of the numbers 15 and 105 and 1005. But zero as a number in its own right took a long time to gain acceptance. Just like negative values, the solutions to some equations can be zero.
The negative and positive numbers (integers and all the values between them) along with zero can be represented on a numberline stretching infinitely in both directions.
For most people that’s the end of the story – we usually don’t need other types of number to survive in life. Or do we?
If the Earth suddenly stopped moving around the Sun - how long would it take to fall into the Sun?
First of all, this is a situation that will never, ever happen! It would take a phenomenal amount of energy to stop planet Earth in its tracks. A passing malevolent alien attack fleet imparting that much energy to the Earth would likely destroy it! (Another problem for another day :-p ) Nevertheless it's an interesting maths problem to think about (maybe only to me) on a lazy afternoon.
We can use conservation of energy to solve this. The Earth has kinetic energy (from its motion) and gravitational potential energy because of its position in the gravitational field of the Sun. Ignoring the effects of other planets, the sum of those energies is constant and we can use that fact to figure out the fall time. Click below if you want to avoid the maths and just to skip to the answer!
The Cassini spacecraft captured this breathtaking view of Saturn on May 4th 2014 but I just noticed it today.
Cassini captured this view at a distance of approximately 2 million miles (3 million kilometers) from Saturn using a near infrared filter. Cassini was high above the ring plane and this is a view of the planet we never get from Earth. From our position near the centre of the solar system Saturn (and most of the planets beyond Earth) look fully illuminated all of the time.
One of the features that immediately catches the eye is that hexagon shape around the north pole of Saturn. The hexagon at Saturn's north pole isn't a new feature. It was seen in images taken by the Voyager 1 and 2 probes back in the 1980s.
Here are some closer Cassini views of it:
For some sense of scale: each side of the hexagon is a bit wider than the Earth.
Saturn is a rapidly rotating gas giant planet. How can a regular and seemingly long-lived feature like this arise in the atmosphere of Saturn? Astronomers don't have the definitive explanation yet although there is experimental evidence from laboratories on Earth which might give a clue.
The video shows an experiment to simulate conditions that might lead to a regular structure being set up in the atmosphere of Saturn. They built a cylindrical tank capable of varying the fluid flow within concentric regions inside. A hexagon appeared at the chaotic boundary between fluids moving at very different speeds. A number of vortices formed in the region separating the fluid flow and distributed themselves evenly around the pole at centre. Why a hexagon? Actually the experimenters could fine tune the spin rates to produce a hexagon but they could also generate other regular polygons too. You can see some them here.
The experiment showed how stability and order could arise from chaotic conditions induced by large differences in wind speeds at different latitudes on Saturn. There are still questions to be answered. For instance why is there no comparable feature at the south pole of Saturn? There is a huge, long-lived storm at the south pole, but no polygonal structure.
I love trawling through the raw image section of the Cassini website. It can lead to serious distraction no matter if they're images of the rings, the moons or Saturn itself. But I'm aware that this mission won't last forever; Cassini's time is running out and sometime in 2017 NASA scientists will place it in a final series of orbits which will send it crashing into Saturn.
Week 5 on the course was devoted to the mid-term exams in all the academic subjects; so no lectures or seminars.
In Week 6 we got stuck into more algebraic tools and methods; namely polynomial division, factor theorem and remainder theorem. Read on for more details.
This was the second week of geometry (following on from straight lines last week) and we looked at some of the properties of circles.
This week has been all about quadratic functions and equations. Read on below the fold for more details.
The first formal week of teaching has ended for my 14th cohort of Foundation students at INTO Newcastle University. The first week often feels like a bit of a drag; we're often skimming through topics that they should be completely familiar with. Not all of my students have arrived in the UK so I'm not launching straight into stuff that'll be assessed on the exams immediately.
Details of each session below the break.
On the eve of a new academic year of teaching and as usual I'm worried about getting the balance right. It's the usual conflict. Students are required to know the subject well enough to pass exams at various points during the next 8 months. There are various methods and techniques I can employ to ensure this happens. But if that was was the only focus of the course then I'd be guilty of "teaching to the test". Students would probably pass the exams but they'd be ill-equipped to handle new material or cope with even small variations on what they've already seen.
As a maths teacher I want my students to be independent learners. For me that means they are able learn from their mistakes and that they have strategies for thinking about and solving mathematics problems in a very general way. The intensity of my course - the amount of time allocated and the material that the students are expected to know - means that I definitely won't get enough sessions emphasising skills required to be an independent learner.
In practice what will probably happen is that I build some kind of activity into most classes where I get the students thinking about how to solve a problem that isn't textbook. There are plenty of teachers blogging about how to do this and lots of real world scientific examples to draw on.
Anyway, I'll record the best examples of my strategies here!
Well my plan to post a blog entry every day was derailed by the rubbish wifi in each hotel. Honestly! Don't promote your hotel as having wifi if the service is this bad. Better to say you don't have it and not disappoint your guests! Rant over :-) All this travelling seems to blend the days together in my memory. Here's a quick catchup on what's happened so far...
On Tuesday I was at Chung Hwa High School in Miri. Only time for one session so I did the Buffon's needle experiment. After several hundred needle throws we got pi to be 2.9. We talked about reasons why the experiment might give a different value to the actual value (3.1, to one decimal place).
Just three days before I'm on a plane to Malaysia. Finally, I got to try the Buffon's needle experiment with some student volunteers today. Instead of needles, I'm using 15mm panel pins from Homebase. (How I can get hundreds of these through airport security is another matter!) Cotton buds would have been ideal but (a) they're too big for A4 paper (in the configuration I want to use) and (b) they're tough to cut down to a uniform size. Hmmm...
I went through the lesson with my students. They're almost at the end of their Foundation year at INTO and they have a good background in trigonometry and calculus. Between them - 200 needles - and an eventual estimate of pi which came out at 2.9. Not too bad for such a small number of pins.
Hopefully I'll collect enough data from classes in Malaysia to hit one decimal place of accuracy. Maybe. We will see :-)
Dr Adrian Jannetta
Guitar strummin' explorer of the universe. Mild mannered maths teacher by day and astronomer by night.