SECTION A.7 Complex Numbers; Quadratic Equations in the Complex Number System A59 DEFINITION The Complex Number System Complex numbers are numbers of the form a bi, + where a and b are real numbers. The real number a is called the real part of the number +a bi; the real number b is called the imaginary part of +a bi; and i is the imaginary unit, so = − i 1. 2 Two complex numbers are added by forming the complex number whose real part is the sum of the real parts and whose imaginary part is the sum of the imaginary parts. To subtract two complex numbers, use this rule: Equality of Complex Numbers + = + = = a bi c di a c b d if and only if and (1) Sum of Complex Numbers ( ) ( ) ( ) ( ) + + + = + + + a bi c di a c b d i (2) Difference of Complex Numbers ( ) ( ) ( ) ( ) + − + = − + − a bi c di a c b d i (3) Adding and Subtracting Complex Numbers (a) ( ) ( ) ( ) [ ] ( ) + +−+ =+−++ =+ i i i i 3 5 2 3 3 2 5 3 1 8 (b) ( ) ( ) ( ) ( ) ( ) + −+ =−+− =+− =− i i i i i 6 4 3 6 6 3 4 6 3 2 3 2 EXAMPLE 1 Now Work problem 15 For example, the complex number − + i 5 6 has the real part −5 and the imaginary part 6. When a complex number is written in the form +a bi, where a and b are real numbers, it is in standard form . However, if the imaginary part of a complex number is negative, such as in the complex number ( ) + − i 3 2 , we agree to write it instead in the form − i 3 2 . Also, the complex number +a i0 is usually written merely as a . This serves to remind us that the real numbers are a subset of the complex numbers. Similarly, the complex number +bi 0 is usually written as bi. Sometimes the complex number bi is called a pure imaginary number . 1 Add, Subtract, Multiply, and Divide Complex Numbers Equality, addition, subtraction, and multiplication of complex numbers are defined so as to preserve the familiar rules of algebra for real numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. FUN FACT The term imaginary number was originally coined by Rene Descartes in 1637 and was meant to be derogatory. At that time, even negative numbers were poorly understood, just as the number zero once was. Today, imaginary numbers (complex numbers) are used in fields such as fluid dynamics and electrical networks. j Figure 23 shows the results of Example 1 using a TI-84 Plus CE graphing calculator. Figure 23
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