Had long walk today up on Simonside Hills near Rothbury. Fantastic view across a lot of Northumberland. Made a panorama to try and convey what the view is like from the top:
Hope to go back in the future to visit the solstice stone.
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?
A copy of the PDF used in my NASTRO talk at The Trap this evening can be downloaded here.
Another day, another diagram. This one is for a forthcoming talk about quasars and active galactic nuclei. Synchrotron radiation plays a big part in powering jets emerging from the centres of active galaxies and quasars. I needed a simple picture and used PSTricks to rustle one up.
Details of how this was coded are below.
I needed a diagram for an article on noctilucent clouds. They range from pretty (and copyrighted) to pretty ugly online so I had a go at making my own. As usual this was done with LaTeX (via LyX) and PSTRICKS.
Here's the finished picture:
The diagram is simple: part of a circle, some tangent lines, cloud symbols and labels. Then shade between the various curves and lines. Finally, a multi-do loop to draw the Sun.
Details below the fold.
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!
Dr Adrian Jannetta
Guitar strummin' explorer of the universe. Mild mannered maths teacher by day and astronomer by night.