INTRO Complex numbers are numbers that consist of two parts — a real number and an imaginary number. Complex numbers are the building blocks of more intricate math, such as algebra. They can be applied to many aspects of real life, especially in electronics and electromagnetism. The standard format for complex numbers is a + bi, with the real number first and the imaginary number last. Because either part could be 0, technically any real number or imaginary number can be considered a complex number. Complex does not mean complicated; it means that the two types of numbers combine to form a complex, like a housing complex — a group of buildings joined together. Real numbers are tangible values that can be plotted on a horizontal number line, such as fractions, integers or any countable number that you can think of. Imaginary numbers are abstract concepts that are used when you need the square root of a negative number.
modulus and argument of a complex number
We already know that r = sqrt(a 2 + b2) is the modulus of a + bi. We write this modulus as |a+bi|. We also know that the point p(a,b) in the Gauss-plane is a representation of a + bi. The intersection point s of [op and the trigonometric circle is s( cos(t) , sin(t) ). That number t, a number of radians, is called an argument of a + bi. We say an argument because, if t is an argument so t + 2.k.pi is an argument too. Here and in all such expressions k is an integer value.
A representation of a complex number
A complex number has a representation in a plane. Simply take an x-axis and an y-axis (orthonormal) and give the complex number a + bi the representation-point P with coordinates (a,b). The point P is the image-point of the complex number (a,b). The plane with all the representations of the complex numbers is called the Gauss plane. With the complex number a + bi corresponds just one vector OP or P. The image points of the real numbers 'a' are on the x-axis. Therefore we say that the x-axis is the real axis. The image points of the 'pure imaginary numbers' 'bi' are on the y-axis. Therefore we say that the y-axis is the imaginary axis. We calculate the 6-th roots of (-32 + 32.sqrt(3).i) The modulus is r = 64. The argument is (2.pi/3). The roots are 2( cos(pi/9 + 2 k pi/6) + i sin(pi/9 + 2 k pi/6) ) with k = 0,1,..,5 In the Gauss-plane these roots are the vertices of a regular hexagon.
the point p(a,b) in the Gauss-plane is a representation of a + bi. So conjugate complex numbers have the same modulus and opposite arguments.
Lesson Summary
There is only one practical method for finding the power or roots of a complex number. Before raising a complex number to a power or finding a root or roots of a complex number, the complex number must be in polar or cis form, and then De Moivre’s Theorem can be utilized.
Graphing both of these complex solutions on the same polar axis reveals an important feature of the roots of complex numbers:
What this reveals is that both solutions are the same distance from the origin, and but they are rotated by π radians when graphed. This suggests is that the roots of complex numbers, when graphed, are spaced apart in such a manner, that the difference between their angles is equal. In this case, both solutions are separated by π radians. Finding the cube root of a complex number would have the three solutions be \begin{align*}\frac{2\pi}{3}\end{align*} radians or 120 º apart when graphed. Finding the fourth root of a complex number would have the four solutions be \begin{align*}\frac{2\pi}{4} = \frac{\pi}{2}\end{align*} radians or 90 º apart. The general rule would be to find the first root by the method seen in the above example. Leave the answer in cis form. To find the additional roots use rotations of \begin{align*}\frac{2\pi}{n}\end{align*} radians or \begin{align*}\frac{360^\circ}{n}\end{align*} to find the angle difference between the roots. Add this angle “ n - 1” times to the first angle, until a complete circle has been formed. The procedures, when combined, are known as De Moivre’s Theorm.
