Higher Concepts of Mathematics IMAGINARY AND COMPLEX NUMBERS IMAGINARY AND COMPLEX NUMBERS This chapter will cover the definitions and rules for the application of imaginary and complex numbers. EO  2.1 STATE  the  definition  of  an  imaginary  number. EO  2.2 STATE  the  definition  of  a  complex  number. EO  2.3 APPLY  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 this point.   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 the study of alternating current circuits. Imaginary  Numbers Imaginary  numbers  result  when  a  mathematical  operation  yields  the  square  root  of  a  negative number.  For example, in solving the quadratic equation x² + 25 = 0, the solution yields x² = -25. Thus, the roots of the equation are x = + . The square root of (-25) is called an imaginary 25 number.   Actually, any even root (i.e. square root, 4th root, 6th root, etc.) of a negative number is  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  be used 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 imaginary 5 1 9 (9) (  1) 3 1 numbers 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 as 1 .    Thus,  the  imaginary  number, ,  which  equals ,  is  written  as  5i,  and  the 1 25 5 1 imaginary  number, ,  which  equals ,  is  written  3i.    In  using  imaginary  numbers  in 9 3 1 electricity,  the  imaginary  unit  is  often  represented  by  j,  instead  of  i,  since  i  is  the  common notation for electrical current. Rev. 0 Page 11 MA-05