Complex Calculator is a Java program that implements the functions of a usual desk calculator, but accepts input and displays output both symbolically and geometrically. In addition, the program can display the action of complex functions interactively. The program is to be used in teaching at high school and university level.
Complex Calculator makes complex numbers simple!
You need a browser that has Java 1.1 to run the program. Suitable browsers are Netscape 4 and Internet Explorer 4.
Above this section you should see a button labeled "Click here to get a Complex Calculator.". If you do not see it, your browser does not have Java 1.1, or Java is not switched on. If you see the button, but no calculator window appears, your browser might have Java, but not Java 1.1. The status bar at the bottom of the window may report "class AppletButton not found", and there will be an error message in the Java Console saying that the class "ActionListener", which is part of Java 1.1, but not Java 1.0, was not found. In particular, Netscape Communicator 4.05 and 4.5 on the Macintosh do not have full Java 1.1, although their Java Consoles proclaim to have "Java 1.1.2", respectively "Java 1.1.5". The Java console will report "class ActionListener not found".
Depending on your system, some of the interactive mapping functions are slow. In a future version of the program they may be disabled. On some systems, entering formulas into the calculator by keyboard causes the characters to appear twice. I will fix this soon. In any case, entering formulas by using the buttons works. Some systems (e.g. on Macintosh) don't give the program the chance to redraw the screen during a mouse drag. Only the first and last mouse position will be recognized by the program then. Sadly, this makes the program a lot less entertaining. This is a fault in the implementation of Java on those system and not of the program. If you have a Macintosh, consider the native Mac version of the program.
I'd be grateful if you'd email me with any other problems. It would be useful if you tell me what system and browser you use, and any error messages displayed in the Java console.
In this mode, there are two windows: the calculator and the Calculation World. The calculator may be used to enter numbers, operators, and functions in reasonably standard mathematical notation. When = is entered, the expression on the display is evaluated and the result is displayed both on the calculator and as a point on the calculation world. Numbers may also be entered geometrically, by clicking on the calculation world. The Calculation World may be presented as a plane or as a sphere. In either case, the world may be shifted or rotated by selecting "Move" from the menu on the Calculation World and then dragging on the window. The effects of moving are undone by pressing button "Reset". Select "Draw" to enter more numbers geometrically. The button "Clear" erases previously marked numbers.
In this mode, there are three windows: the calculator, the z World, and the f(z) World. The calculator is used to define a function f. (Press "C" in case you need to clear the display.) Define a function by entering in the calculator's display "f(z) = sin z", or any other function. What is displayed must start with "f(z) = " to make sense. There is a space before and another after the = symbol. Once the function has been defined, points, lines, circle, square, rectangles, and grids may be drawn on the z World. They will be mapped via the defined function f to points on the f(z) world. Select "Draw" and drag the mouse on the z World to enter an arbitrary line. It will be mapped interactively. Select "Line" and drag to enter a straight line between two points (On the sphere a straight line does not usually look straight.). Select "Circle" and drag to enter a circle. Drag from the center of the desired circle to any point on its circumference. Similar functions are available for squares and rectangles. Select "Grid" and drag to select a rectangle over which a rectangular grid or net of lines will be drawn. Each of the lines is then mapped to the f(z) world. The effect of f on a neighbourhood becomes visible. As usual, select "Move" and press "Reset" to move the worlds and make movements undone. Dragging on the f(z) World moves it.
Since the shapes are sampled at finitely many points, they may sometimes appear jagged rather than smooth. The sampling may miss interesting points at which the function is undefined or infinite.
There are three windows: the calculator, the z World, and the |f(z)| World. The calulator is used to define a function f as before. The z World is used to specify a square. The |f(z)| World will display the modulus of the function over the specified square. The image on the |f(z)| World may be rotated by dragging on it. A new square may be specified in the z World by selecting "Square" and dragging.
However, since the function is only sampled at finitely many points, the picture is only an approximation. If the function is sampled at a point where the function is undefined (such as ln(0)), the display will represent this by displaying a negative value (clearly indicates something is wrong, because the modulus of any number is non-negative). Try the function f(z) = ln 0 to observe this. If the function is infinite at a sampled point, the |f(z)| World will reflect this by showing a value that exceeds the cube in which the maps are normally fitted. Try the function f(z) = inf to observe this, or f(z) = 1/z over the (initial) square centered exactly on 0. However, in general, even though the function may have undefined points and infinite points on the specified square, the sampling may miss them.
In this mode, there are two windows: the calculator and the Re(f(x)) World. Use the calculator to define a function f as before. The Re(f(x)) will display the function mapped over the real numbers in the standard way. The Re(f(x)) World may be shifted by dragging on it, and you may zoom it. Try the function f(x) = sin x to get the familiar picture of the sin curve. The picture is not always the picture you expect from real analysis: Try f(x) = ln x. ln applied to a negative real numbers yields a good complex number, and the Re(f(x)) World displays its real part. When the function is infinite or undefined at a point, the picture may reflect this by not drawing a image point. However, since the function is only sampled at finitely many points, such points may be missed. Try f(x) = tan x to see an example of this.
The complex numbers are the points
on the surface of the sphere. In the starting position of the sphere (press
"Reset" to achieve it in case you have rotated the sphere), 0 is the South
pole of the sphere, infinity the North pole, and the equator represents
the circle of radius 1 around 0. Check mathematics text books, for example
"Complex Analysis" by Ian Stewart and David Tall (Cambridge University
Press) for more about the sphere.
Introducing complex numbers, the most basic questions are: What do the functions Re, Im, mod, and arg do? Using the z -> f(z) mode, the students can be left to discover the answers by themselves. Similarly, they will discover that addition and subtraction on the plane are just like 2d-vector addition and subtraction.
The reciprocal function f(z) = 1/z has the interesting property of mapping the unit circle onto itself and swapping the inside and the outside of it. On the sphere, this means that the equator stays fixed, while the northern and souther hemisphere are swapped over.
In cartography, the world sphere is mapped onto a sheet of paper by the reverse process of the way the plane is mapped onto the Riemann sphere. To remove the distortion, the plane is then mapped onto itself by the natural logarithm. This method is called "stereographic projection".
The sin function maps vertical lines to confocal hyperbolae and horizontal lines to confocal ellipses. After seeing the picture on the f(z) World, the student can form some theorems like these and prove (or disprove) them.
The squaring function f(z) = z^2 maps the unit circle onto itself, but twice. Try drawing the unit circle by hand, not using the circle button, and observe how fast the image circle develops. After only half a revolution in the z World, the f(z) World circle will already be complete. With higher polynomials like f(z) = z^3 + z^2 + z + 1, simple curves can map onto curves with crossings.
The Joukovsky aerofoil transformation has been used to make aeroplane wings. The function is f(z) = 0.5 * (z + 1/z). Map a circle of roughly radius 1, sitting near, but not quite centered on 0, to get an image that could be the section of an aeroplane wing.
The Riemann sphere is always useful to observe a neighbourhood of infinity, but there are some functions that are special in relation to the sphere. The opposite function opp maps each point to the point oppsite it on the sphere. The students could try to devine its definition. It is opp(z) = conj(-1/z).The reciprocal function f(z) = 1/z maps the outside of the unit circle to its inside, but also performs a conjugation.
A Möbius transformation is a function of the form f(z) = (az + b)/(cz + d) for complex numbers a,b,c,d. It may be rewritten as f(z) = a/d * z + b/d, if c = 0, or f(z) = -(ad -bc)/c * 1/(cz + d) + a/c, if c is not 0. Möbius transformations map circles or straight lines onto circle or straight lines. On the sphere, a straight line is no more than a circle crossing infinity, so it's useful to consider them one concept.
The exponential function f(z) = exp z is the inverse of the natural logarithm ln. It can be shown that exp z = e^z and therefore we have exp(x + iy) = exp x * exp(iy), where x and y are real numbers. This conforms to the polar form of a complex number z = r e^(theta i) where r is z's radius (modulus), and theta is its angle in radians. Therefore vertical lines go to circles around 0 and horizontal lines go to radii from 0 when mapped by exp. In particular, exp is periodic, the period is 2pi.
The natural logarithm of a complex number can be expressed in terms of the radius and argument. ln(r e^(theta i)) = ln r + theta i. This conforms to the Cartesian form x + iy of a complex number. Therefore circles around 0 map onto vertical lines and straight lines through 0 map onto horizontal lines.
These are just some ideas, the teacher may adapt the use of Complex Calculator
to the curriculum. As soon as the students playfully realize that complex
numbers needn't be thought of as complex entities, but simple points, their
difficulty evaporates!