Computing the Logarithm and the Value of the Argument of a Gauss Integer

Abstract

1. The purpose of this study and the idea of involution
   Generally speaking, it is difficult to calculate the value of the logarithm and argument of a Gauss Integer for high school students because they do not learn enough calculus. Here, I will explain how to calculate these values without using calculus and a number table. We can obtain these values using the idea of involution, which will help the students understand the concept easily.
2. The program to obtain the logarithm value
   I wrote the CAI (Computer Assisted Instruction) program in Visual Basic which obtains the value of a logarithm. See Figure 1. Involutions of the bases 10, 2, 3 and 6 are shown on the screen. If we click one of these involutions, we can see its value in the lower right frame. In addition, we can see the exponent in Japanese and English. In this case, we can see that 2100 is between 1030 and 1031. So the value of log102 is close to 0.30. To see the big numbers or small numbers, the scroll bar is moved. Similarly, we can calculate the values of log103 and log106 by comparing with the involutions of 10. Of course, we can calculate other logarithm values by changing the bases.
3. The program to obtain the argument value
   The CAI program also obtains the value of arg(a+bi), a and b integers. See figure 2. The involutions of z=2+3i and their values are shown in the frame. At the same time, the argument of zn is indicated every 180 degrees. We can see this by the change of sign of Im(z). If we want to see the lower part or the right part of the frame, we can do it by moving the vertical or horizontal scroll bar respectively. In figure 2, arg(z16) nearly equals 900 degrees. So, we can see that the approximate value of arg(z) is 56.25 degrees (=900/16). In fact, arg(z) is 56.3099 degrees. Of course, we can calculate other Gauss Integer arguments by changing Re(z) and Im(z). The idea of involution can be applied to both logarithm and argument.