Higher Concepts of MathematicsIMAGINARY AND COMPLEX NUMBERSIMAGINARY AND COMPLEX NUMBERSThis chapter will cover the definitions and rules for the application ofimaginary and complex numbers.EO 2.1STATE the definition of an imaginary number.EO 2.2STATE the definition of a complex number.EO 2.3APPLY the arithmetic operations of addition, subtraction,and multiplication, and division to complex numbers.Imaginary and complex numbers are entirely different from any kind of number used up to thispoint. These numbers are generated when solving some quadratic and higher degree equations.Imaginary and complex numbers become important in the study of electricity; especially in thestudy of alternating current circuits.ImaginaryNumbersImaginary numbers result when a mathematical operation yields the square root of a negativenumber. For example, in solving the quadratic equation x^{2} + 25 = 0, the solution yields x^{2} = -25.Thus, the roots of the equation are x = +. The square root of (-25) is called an imaginary25number. Actually, any even root (i.e. square root, 4th root, 6th root, etc.) of a negative numberis called an imaginary number. All other numbers are called real numbers. The name"imaginary" may be somewhat misleading since imaginary numbers actually exist and can beused in mathematical operations. They can be added, subtracted, multiplied, and divided. Imaginary numbers are written in a form different from real numbers. Since they are radicals,they can be simplified by factoring. Thus, the imaginary number equals ,25(25) (1)which equals . Similarly, equals , which equals . All imaginary519(9) (1)31numbers can be simplified in this way. They can be written as the product of a real number and. In order to further simplify writing imaginary numbers, the imaginary unit i is defined as1. Thus, the imaginary number, , which equals , is written as 5i, and the12551imaginary number, , which equals , is written 3i. In using imaginary numbers in931electricity, the imaginary unit is often represented by j, instead of i, since i is the commonnotation for electrical current.Rev. 0Page 11MA-05

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