sábado, 28 de febrero de 2015

Roots of complex numbers

Consider $z=a+ib$ a nonzero complex number. The number $z$ can be written in polar form as
\[z=r(\cos \theta +i \sin \theta)\]
where $r=\sqrt{a^2+b^2}$ and $\theta$ is the angle, in radians, from the positive $x$-axis to the ray connecting the origin to the point $z$.

Now, de Moivre's formula establishes that if $z=r(\cos \theta +i\sin \theta)$ and $n$ is a positive integer, then
\[z^n=r^n(\cos n\theta+i\sin n\theta).\]
Let $w$ be a complex number. Using de Moivre's formula will help us to solve the equation $z^n=w$ for $z$ when $w$ is given. Suppose that $w=r(\cos \theta +i\sin \theta)$ and $z=\rho (\cos \psi +i\sin \psi)$. Then de Moivre's formula gives $z^n=\rho^n(\cos n\psi+i\sin n\psi)$. It follows that $\rho^n=r=|w|$ by uniqueness of the polar representation and $n\psi = \theta +k(2\pi)$, where $k$ is some integer. Thus
\[z=\sqrt[n]{r}\left[\cos\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right) \right]\]
Each value of $k=0,1,2,\ldots ,n-1$ gives a different value of $z$. Any other value of $k$ merely repeats one of the values of $z$ corresponding to $k=0,1,2,\ldots ,n-1$. Thus there are exactly $n$th roots of a nonzero complex number.

The complex number $z$ can also be written in exponential form as
because $e^{i\theta}=\cos \theta +i \sin \theta$.

Thus, the $n$th roots of a nonzero complex number $z$ can also be expressed as
where $k=0, 1, 2, \ldots , n-1$.

In the following applet you can see a geometrical representation of the $n$th roots for a family of complex numbers. Change the values of the real and imaginary parts of $z$ and the $n$th root. Some examples are:
  1. Re(z)=1/2, Im(z)=1.72, n=3; 
  2. Re(z)=sqrt(2), Im(z)=pi, n=7.

External link to GeoGebra applet: http://tube.geogebra.org/student/m298919

sábado, 25 de octubre de 2014

Relative velocity: Boat problems

Problem 1. 

A river  flows due East at a speed of 1.3 metres per second. A girl in a rowing boat, who can row at 0.4 metres per second in still water, starts from a point on the South bank and steers due North. The boat is also blown by a wind with speed 0.6 metres per second from a direction of N20ºE.

Figure 1: The red arrows represent the velocities of the boat (b), wind (w) and flow (r).

  1. Find the resultant velocity of the boat and its magnitude.
  2. If the river has a constant width of 10 metres, how long does it take the girl to cross the river, and how far upstream or downstream has she then travelled?

Problem 2. 

A river  flows due West at a speed of 2.5 metres per second and has a constant width of 1 km. You want to cross the river from point A (South) to a point B (North) directly opposite with a motor boat that can manage to a speed of  5 metres per second.

  1. If you head out pointing your boat at an angle of 90 degrees to the bank. How long does it take to get from point A to point B?
  2. After crossing the river you realised that it took  longer than expected. In what direction should you point you motor boat in order to reduce the time to cross the river? How long will it take you to get from point A to point B? Is it a better time?

Applet GeoGebra

The following applet shows a representation of the problem 2, considering that the boat starts from a point A. It also shows the velocities (vectors) and their magnitudes (speeds) of the boat and current.

  1. Move the sliders to change the magnitude and direction of vectors.
  2. Click the 'Start' button to activate the motion of the boat.
  3. Click the 'Reset' button to put back the boat to its original position.
  4. You can also change the width of the river. Chose a number between 5 and 1000.
  5. All velocities can be considered either as metres per second or km per second. 

Open this applet in an external window: Relative Velocity: Boat Problem