5.4 Complex Numbers

Objectives Solve quadratic equations with complex solutions and perform operations with complex numbers. Assignment: 17-79 odd

Not all quadratic equations have real-number solutions: Has no real-number solution because the square of any real number x is never negative. We create an “expanded” number system by introducing the imaginary unit i. i = So, x x

Simplify: x x x

Solve the equation:

Complex Numbers 2 + 3i 5 - 5i a + bi These are complex numbers: A complex number written in standard form that has a real part a and an imaginary part b. a + bi Real part imaginary part If the real part a = 0, then a + bi is called a pure imaginary number. These are pure imaginary numbers: -6i, 2i

Just as every real number corresponds to a point on the real number line, every complex number corresponds to a point in the complex plane. -6 + 3i 2 + 3i (-6,3) (2,3) -2i (0,-2) Notice these are just like the ordered pairs we are familiar with.

(a + bi) + (c + di) = (a + c) + (b + d) i To add complex numbers (in standard form): Add the real part to the real part And the complex part to the complex part: (4 – i) + (3 + 2i) = 7 + i (a + bi) + (c + di) = (a + c) + (b + d) i Subtraction of complex numbers: (a + bi) - (c + di) = (a - c) + (b - d) i

Multiplying Complex Numbers Recall that 5i(-2 + i) (7-4i)(-1+2i)

We will call these two complex numbers: Notice (1+6i)(1-6i)=37 1 - 6i Complex Conjugates. The product of two complex conjugates (a+bi) & (a – bi) is always a REAL NUMBER.

Dividing two complex numbers Multiply numerator and denominator by the complex conjugate of the denominator… Answer left in standard form.

Absolute Value of a Complex Number The absolute value of a complex number z = a + bi, denoted |z|, is a non-negative real number defined as follows: Geometrically, the absolute value of a complex number is the numbers distance from the origin in the complex plane. Z = 3 + 4 i Z = -1 + 5 i