August 2015 – Page 2 – Mathemafrica

UCT MAM1000 lecture notes part 26 – complex numbers part iv

OK, so we saw something pretty interesting last time when we multiplied together complex numbers using the modulus argument form.

Remember that for two complex numbers which we will write as $z_1=r_1(\cos\theta_1+\sin\theta_1 i)$ and $z_2=r_2(\cos\theta_2+\sin\theta_2 i)$ , where $r_i$ are the moduli, and $\theta_i$ are the arguments of $z_i$ . If we multiply them together then we get:

$z_1 z_2=r_1 r_2 (\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$

Well, what would happen if the two complex numbers were the same? ie. if we have $z=r(\cos\theta+i\sin\theta)$ and we want $z^2$ ?

Well, then clearly:

$z^2=r^2(\cos 2\theta+i\sin 2\theta)$ .

What if we then multiplied this by $z$ one more time:

$z^3=z^2 z=r^3(\cos (2\theta+\theta)+i\sin(2\theta+\theta))=r^3(\cos 3\theta+i\sin 3\theta)$

hmm, do we already see a pattern emerging? Let’s say that we have a complex number with modulus 1. Complex numbers of the form:

$z=\cos\theta+i\sin\theta$

Are clearly modulus 1. We know that the modulus is the square root of the sum of the squares of the real and imaginary parts of a complex numbers so $|z|=\sqrt{\cos^2\theta+\sin^2\theta}=1$ .

ok, so how about if we have $z^n$ where $n$ is an integer?…

By Jonathan Shock| August 23rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|5 Comments

UCT MAM1000 lecture notes part 25 – complex numbers part iii

So we saw last time that we can take a complex number and put it in a 2 dimensional plane called the complex plane, where its horizontal distance from the origin is given by its real part, and the vertical distance from the origin is given by its imaginary part. We can thus think of the real and imaginary parts as the Cartesian coordinates of that point.

It turns out that there is another way to represent a complex number, but rather than using the real and imaginary parts to specify it, we will use two other pieces of information.

If I tell you that a complex number is a distance $|z|$ away from the origin in the complex plane, then this leaves you with a whole circle of possibilities. All the points on the circle of radius $|z|$ about the origin are the same distance from the origin. But if I also give you an angle subtended between the x-axis and the line joining the complex number and the origin, read anti-clockwise from the x-axis, this will completely pin down the point in the complex plane.…

By Jonathan Shock| August 23rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|5 Comments

UCT MAM1000 notes part 24 – complex numbers part ii

So, last time we discovered that numbers are maybe not quite as real as we thought that they were, and that we can have numbers which don’t obviously correspond to something in the real world (though we’ll discover later that they are a way to jump between islands of reality).

In the resources on Vula you will find some great notes on complex numbers, so I want this to be an additional resource, and not an alternative. This means that sometimes we will look at things from a slightly different perspective than in the resource book.

Let’s start off discussing a bit more about the complex plane.

When you learnt about integers, one of the first things that you learnt to do was to put them in order. 3 came after 2 and 7 came after 6. You could put them all in a line. When you learnt about the negative numbers, it was quite clear that this line which had previously started with zero simply went backwards in the other direction, and you could count backwards to whatever large negative integer you wanted.…

By Jonathan Shock| August 23rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|4 Comments

Could these have been the first computer games?

It is a genre of games that originated in Africa between the 6^th and 7^th century AD. It has many variations, and is still enjoyed today in Africa and many other places around the world.

I am referring to a family of games called Mancala. These are mathematical games requiring significant strategic thinking, so much so that it has been reported that in some areas of Ghana, it was customary for kings to use such games to demonstrate strategic acumen and mental dexterity. The word Mancala originates from the Arabic word naqala meaning “to move” (wikipedia).

Mancala games are believed to have been first developed by the ancient Kush Civilization of the Upper Nile when accountants and engineers there began using counters on a tablet with depressions to carry out mathematical calculations about 3600 years ago (Andy Rabagliati, www.wizzy.com). In time, such computing devices were used for recreational purposes.…

By Tapiwa Chadenga| August 22nd, 2015|Uncategorized|2 Comments

Africa counts: Africa’s history with numbers

On Friday the 21st of August, I will be giving a presentation on the history of mathematics in Africa. The event will be hosted by the UCT Mathematics Society, in room 212 Mathematics Building, starting at 13hrs00. The talk will briefly trace the development of mathematical practice and thought through out the continent. However, an emphasis will be placed on sub Saharan Africa, as there is little recognition for this region’s mathematical achievements. That not withstanding, I will include some of the more recognised, though not well promulgated, contributions from other parts of the continent. As an African, I believe it is incumbent upon us to recognise our own achievements before expecting anyone else to. Lets meet and share information. Africa counts!

How clear is this post?

…

By Tapiwa Chadenga| August 20th, 2015|Uncategorized|0 Comments

Hamilton and the extension of complex numbers to higher dimensions

Hamiltonian, the Quarternions and the Octonians

We’ve just extended the number system we can deal with by saying that we are perfectly at liberty (with certain important qualifications) to take the square root of a negative number and it gives you a multiple of $i$ . We will see in the following sections that this is incredibly useful for being able to extend our mathematical machinery to new domains. In fact, this extension of the number system we are dealing with is not at all unique. The idea of irrational numbers was for a very long time believed not to be true. The possibility that you could have a number which was not a fraction seemed absurd. In fact even more basic than that, the idea of 0 as an important concept was not conceived for a long time in the history of mathematics. Zero was introduced by Indian mathematicians in the 9th century AD and the idea of irrational numbers was only dreamt up by the Greeks in around the 5th century BC.…

By Jonathan Shock| August 20th, 2015|Uncategorized|0 Comments

On constants of integration in our polynomial approximations and Ramanujan’s remarkable formula

You might have noticed something slightly strange happening when we made our approximation for $\pi$ using the Maclaurin polynomial for $\arctan x$ . We were slightly sneaky in that we performed a definite integral, but we didn’t seem to have any constant of integration.

The sneaky line was this one:

$\arctan x\approx\int \sum_{i=0}^n(-1)^i x^{2i} dx= x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}+...=\sum_{i=0}^n \frac{(-1)^{i}x^{2i+1}}{2i+1}$

Where we have written the $\arctan x$ function as an integral and not written the constant of integration.

The point is that we should really be saying:

$\int \frac{1}{1+x^2}dx=\arctan x+c$

and so there should be a constant in the expression on the left. Then, when we perform the integration we are left with another constant (let’s call the original one above $c_1$ and the second one $c_2$ . So what we really should have written was:

$\arctan x+c_1\approx\int \sum_{i=0}^n(-1)^i x^{2i} dx= x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\frac{x^9}{9}+...+c_2=\sum_{i=0}^n \frac{(-1)^{i}x^{2i+1}}{2i+1}+c_2$

Then we can write:

$\arctan x\approx\sum_{i=0}^n \frac{(-1)^{i}x^{2i+1}}{2i+1}+c_2-c_1$

We can then fix our constants of integration by knowing that $\arctan 0=0$ and thus $c_2=c_1$ and thus we don’t actually have any constants to worry about.…

By Jonathan Shock| August 19th, 2015|Uncategorized|0 Comments

Mathematics or dreams, which is more real? Izibalo okanye amaphupha, yeyiphi eyona iyinyani?

Translated from here.

Mathematics or dreams, which is more real? Izibalo okanye amaphupha, yeyiphi eyona iyinyani?

Mathematics can sometimes seem dream-like, at least on first encounter. Later on, one gets
used to a new mathematical object, and it seems everyday. I remember how strange the idea of
a group was to me, how mysteriously it grew from three almost trivial axioms to a forest
of subgroups and quotient groups and equivalence classes and so on. Of the few dreams I
now recall, there was one with a huge hall full of people, perhaps a giant cave, and I was descending a long,
rickety staircase — or was I sliding down a cable? — feeling myself among a heretofore
completely unsuspected part of humanity, who perhaps nobody from above ground had ever seen.
Groups were a bit like that, and saying that a square had a symmetery group did not
make them appear any less unexpected.

…

By Portia Mabedla| August 17th, 2015|Uncategorized|1 Comment

Some animations of Taylor approximations

Here are some simple animations which might help to understand what we were doing in class today. In each case the function which we are approximating is in red, and the polynomial approximation is in blue.

The following are the different approximations to the function $(1+x)^5$ starting at a constant, then a straight line with non-zero gradient, then a quadratic etc. Each one matching the higher and higher derivatives of the function at $x=0$ . You see that by the time we get to a fifth order polynomial, the match is exact. We know this because we know that there is an exact expansion of this function which is a fifth order polynomial.

This is the same plot, but now with the function $(1+x)^{5.2}$ . This time the polynomial approximation doesn’t stop at the fifth order, but can keep going to any power of $x$ .

This is the same function approximated but now approximated about the point $x=0.25$ .…

By Jonathan Shock| August 17th, 2015|Uncategorized|0 Comments

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