PRESENTATION: The Number Line
The idea of negative numbers is something that most students are aware of, predominantly through being exposed to temperature. Especially if they live in a cold part of the world (like Sweden where I currently live), they will be very familiar with representing temperatures with negative numbers. However, if you ask some students, especially young students, if they can subtract a larger number from a smaller number they will reply “You can’t do that!”.
In order to introduce negatives it makes sense for students to try and realise where numbers came from and the logic of how they were discovered. They should be able to realise that we must have started with the number 1 and then the number line grew to the right as more and more types of numbers were discovered – natural numbers, rational numbers, irrational numbers etc – before we started to fill in the number line to the left of zero as well.
The presentation above walks students through one possible, logical order in which the number line may have been populated. Each slide poses a question which should encourage them to discuss with each other how the number line grew. You may find that they populate the number line in a slightly different order (maybe zero comes earlier in their thinking) so that is worth keeping in mind. It is a good exercise to get them to think about how numbers came to be and how we can extend what we know about positives to think about negatives.
PRESENTATION: Colour by Numbers
The concept of negativity does not just apply to the number line. The idea of negativity can also be applied to colour, both digitally on screens and also on paper in the form of paint or ink. The physics behind colour is something that many people are unaware of and it has a fascinating link with the idea of negativity.
It is good for students to think about how concepts they learn in mathematics apply to other areas of their lives in addition to the actual content itself. Many students will expect their new smartphones, tablets and computers to have brighter and more colourful screens and may not fully appreciate the relevance of mathematics to allowing this to happen.
The end of the presentation mentions the number system hexadecimal very briefly. If you have the opportunity, take the time to explore the ideas of different number systems, including hexadecimal and binary (and others if you can!). Numberphile have a great video that explains hexadecimal, which you could use to aid your discussions.
PRESENTATION: Complex Numbers
The negative numbers are an extension of the number line in a direction that some students may not have considered possible before being introduced to them. As a further extension, we can extend the number line in another direction – perpendicularly from the real number line.
Complex numbers open up a world of possibilities that was not available prior to their discovery. They allow us to solve a number of problems that were previously unsolvable; namely, problems in fluid dynamics, quantum mechanics and use of the Riemann zeta function in number theory.
In addition to allowing us to solve previously unsolvable problems, complex numbers allow us to discover things which we would not have been aware of otherwise. The Mandelbrot Set is a fantastic example of an amazing discovery made using complex numbers. It is a set of numbers that follow a particular rule and they are coloured in a particular coloured based on how quickly they converge. The wonderful thing about this set of numbers is that the pattern it creates is a fractal, which repeats infinitely as you zoom in.
Numberphile do an excellent job of explaining the Mandelbrot Set in a video which you could show.