2 Complex NumbersIf , then A complex number is a number of the form where a and b are real numbers. The real part of the complex number is a and the imaginary part is bi.
3 Ex 1. State the real and imaginary parts of the following complex numbers. -2
4 Operations on Complex Numbers To add complex numbers, add the real parts and add the imaginary parts. (a + bi) + (c + di) = (a + c) + (b + d)I To subtract complex numbers, subtract the real parts and subtract the imaginary parts. (a + bi) – (c + di) = (a – c) + (b – d)i
5 Multiply complex numbers like binomials, using (a + bi) . (c + di) = (ac – bd) + (ad + bc)i
6 Ex 2. Perform the indicated operation and write the result as a + bi. (3 + 5i) + (4 – 2i)(3 + 5i) – (4 – 2i)(3 + 5i)(4 – 2i)
7 Powers of Any power of i can be reduced to one of the following: To reduce a power of i, just divide by 4 and the remainder will correspond to one of these. For example,Because 23/4 has a remainder of 3 so the answer corresponds to
9 Class WorkPerform the indicated operation and write your answer in a + bi form.(2 + 5i) + (4 – 6i)(5 – 3i)(1 + i)
10 Square Roots of Negative Numbers Just as every positive real number r has two square roots , every negative number has two square roots as well.If -r is a negative number, then its square roots are , because:and
11 If –r is negative, then the principal square root of –r is: For example:
13 Ex 5. Evaluate and write the result in the form of a + bi.
14 Class WorkEvaluate the expression and write your result in the form of a + bi
15 Complex Roots of Quadratic Equations We know that if b2 – 4ac < 0 then the quadratic has no real solutions. However, in the complex number system, the equation will always have solutions. The solutions will be complex numbers and will have the form a + bi and a – bi. The solutions will always come in pairs, called conjugates.
16 Ex 6. Solve each equation.x2 + 9 = 0x2 + 4x + 5 = 0