Powers and roots of complex numbers pdf

To work effectively with powers and roots of complex numbers, it is helpful to use a polar representation for complex numbers, as shown in figure 8. Now for that, first of all, ill write it in the polar form. Powers and roots of complex numbers demoivres theorem. The expression under the radical sign is called the radicand. Square roots of other negative numbers then could be defined such as. In particular, we are interested in how their properties di.

As a consequence, we will be able to quickly calculate powers of complex numbers, and even roots of complex numbers. Raising complex numbers, written in polar trigonometric form, to positive integer exponents using demoivres theorem. Let z and q be complex numbers, and let n be a positive integer. The reasons were that 1 the absolute value i of i was one, so all its powers also have absolute value 1 and, therefore, lie on the unit circle, and 2 the argument arg. We have already studied the powers of the imaginary unit i and found they cycle in a period of length 4 and so forth. The complex logarithm, exponential and power functions. Th every complex number has exactly n distinct nth roots. In this case, the power n is a half because of the square root and the terms inside the square root can be simplified to a complex number in polar form. Powers of complex numbers are just special cases of products when the power is a positive whole number. Here, we recall a number of results from that handout. If you are looking for more in complex numbers, do check in. This section shows how to find powers and root of complex numbers. In many cases, these methods for calculating complex number roots can be useful, but for higher powers we should know the general fourstep guide for calculating complex number roots.

Well start with integer powers of \z r\bfei\theta \ since they are easy enough. The complex logarithm, exponential and power functions in these notes, we examine the logarithm, exponential and power functions, where the arguments. The trigonometric form of a complex number provides a relatively quick and easy way to compute products of complex numbers. Frequently there is a number above the radical, like this. And i have to find out the second roots of this complex number. In order to use demoivres theorem to find complex number roots we should have an understanding of the trigonometric form of complex numbers. This discovery led to the initial definition of the imaginary number i.

Finding the roots of a complex number examples, solutions. In this section were going to take a look at a really nice way of quickly computing integer powers and roots of complex numbers. Consider the following example, which follows from basic algebra. Powers and roots of complex numbers read trigonometry. A number u is said to be an nth root of complex number z if u n. Complex numbers and powers of i the number is the unique number for which. Lecture 4 roots of complex numbers characterization of a.