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High School Mathematics - 22.1 Complex Numbers - Introduction

 Introduction We know that the solution of a quadratic equation ax2 + bx + c = 0, a not equal to 0-------------1 with real coefficients a,b,c is given by real numbers x1 and x2 where x1 = -b +Ö(b2-4ac)/2a x2 = -b - Ö(b2-4ac)/2a only if the discriminant b2 - 4ac >= 0. For b2 - 4ac < 0, we do not have solution of equation 1 in the set of real numbers because square of every real number is non-negative. The mathematical need to have solution for negative discriminant led us to extend the real number system to new kind of numbers, namely complex numbers that allow the square root of negative numbers. We assume that the square root of -1 is denoted by the symbol i, called imaginary unit. Thus, for any two real numbers a and b we can form a new number a+ib. This number a+ib is called complex number. The set of all complex numbers is denoted by C. Complex Numbers A complex number is a number of the for a+ib where a and b are real numbers and i is the imaginary unit which has the property i2 = -1 A complex number is denoted by a single letter such as z, w etc. The real and imaginary parts of z = a+ib are denoted a = Re z and b = Im z If z = a+ib , the number a - ib is called the complex conjugate of a+ib and is denoted as z bar. Example 1: Write the complex conjugate of -6-i7 Answer: -6+i7 Example2: Write 4 - Ö-5 as a complex number. Solution: Step 1: 4-Ö-1x5 Step 2: 4- Ö-1Ö-5 Step 3: 4 - iÖ5 Answer: 4 - iÖ5 Directions: Answer the following questions. Also write at least 5 examples of your own.
 Q 1: Find the value of y if (3y-2) + i(7-2x) = 01/31/42/3 Q 2: Find the conjugate of 3+i1/(3+i)3-i3+i Q 3: Write the real part of 2 - iÖ22-Ö22i Q 4: Find b such that 2a + i4b and 2i represent the same complex number.1/401/2 Q 5: Write the imaginary part of the number Ö5/7 iÖ5i5/7 iÖ5/7 Q 6: Find the conjugate of 4/55/41/34/5 Q 7: Write Ö-27 as complex number.iÖ2727iÖ27 Q 8: 4 - Ö-5 as a complex number.4 - iÖ54 - 5iÖ5 Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!