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Unformatted text preview: UNIT 4 COMPLEX NUMBERS INTRO: This Unit is about complex numbers. Unlike real numbers, which fill a one-dimensional line (in fact, we frequently say the real number line ), the complex numbers fill an entire plane, the so-called complex plane . We wish to understand complex numbers geometrically, their locations and transformations in the complex plane, their powers and their roots. 1. Complex Numbers The most general complex number z can be written as z = a + ib where a and b are real numbers and i = - 1 is an imaginary number (since no real number squares to give a negative number). The product ib is called a pure imaginary number , or often, simply an imaginary number , and signifies the product of b times - 1. We usually represent the complex numbers as filling the x- y plane, with the points on the x-axis representing the real number line, and the points on the y-axis representing the pure imaginary numbers , ie., the complex numbers with zero real part. Plotted on the complex plane, the complex number a + bi would then be located at the point with x-component a and y-component b . A graph of the complex plane is below. You can see the real numbers 1, 2, 3, etc., located along the horizontal axis, and the pure imaginary numbers i, 2 i, 3 i, etc., located along the vertical axis. Can you locate where the number 3+2 i would be located, or the number- 2- 2 i ? Actually, if you look carefully, the graph has dots located at these numbers. Remember, each point on the complex plane is a (complex) number, not a pair of coordinates.-3-2-1 1 2 3 Minus 3 i Minus 2 i Minus i i 2 i 3 i The complex plane Because you have been dealing with complex numbers since they were introduced in first year high school algebra in order to solve the quadratic equation, we will not deal with the interesting question of what exactly a complex number is , that is to say, how we can multiply a real number b times a symbol i , or add a real number to it. Instead, we will go directly to a VECTOR GEOMETRY 48 GREENBERG review of terminology and of how to perform operations on complex numbers. We will write the complex number z as z = a + bi . This is the form used in nearly every field of science and engineering. One particularly important exception is in electrical engineering, where - 1 is often called j , in order to reserve the symbol i for the current. Thus, in this field of engineering, one will commonly see a complex number written as a + bj . We will, of course, maintain the tradition most widely used. The real part of z is defined by Re ( a + bi ) = a, and the imaginary part of z is defined by Im ( a + bi ) = b. This latter terminology may seem very strange to you, because the imaginary part of any complex number is, by definition, real. Indeed, it might seem more reasonable to define the imaginary part of a + bi to be bi , rather than b . However, this definition was chosen for convenience. In engineering, for example, the solution of a potential problem is often foundconvenience....
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