Partial preview of the text
Download complex_numbers in detail and more Study notes Mathematics in PDF only on Docsity!
2.4+Complex Numbers 125 50. Use the formula from the 51. Solve for r:V = ar?h 52. Use the formula from the previous question to find previous question to find the height of a cylinder the radius of a cylinder with a radius of 8 anda with a height of 36 and a volume of 16z volume of 3242. 53. The formula for the 54. Solve the formula from the circumference of a circle is previous question for z. C = 2ar. Find the Notice why z is sometimes circumference of a circle defined as the ratio of the with a diameter of 12 in. circumference to its (diameter = 2r). Use the diameter. symbol z in your final answer. 2.4 Complex Numbers Learning Objectives In this section, you will: > Add and subtract complex numbers. > Multiply and divide complex numbers. > Simplify powers of i Figure1 Discovered by Benoit Mandelbrot around 1980, the Mandelbrot Set is one of the most recognizable fractal images. The image is built on the theory of self-similarity and the operation of iteration. Zooming in on a fractal image brings many surprises, particularly in the high level of repetition of detail that appears as magnification increases. The equation that generates this image turns out to be rather simple. In order to better understand it, we need to become familiar with a new set of numbers. Keep in mind that the study of mathematics continuously builds upon itself. Negative integers, for example, fill a void left by the set of positive integers. The set of rational numbers, in turn, fills a void left by the set of integers. The set of real numbers fills a void left by the set of rational numbers. Not surprisingly, the set of real numbers has voids as well. In this section, we will explore a set of numbers that fills voids in the set of real numbers and find out how to work within it. Expressing Square Roots of Negative Numbers as Multiples of i We know how to find the square root of any positive real number. In a similar way, we can find the square root of any negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number i is defined as the square root of —1. So, using properties of radicals, We can write the square root of any negative number as a multiple of i. Consider the square root of —49. 
