The imaginary unit i is defined as Date: 2.5 Complex Numbers (2.5) Complex Numbers The imaginary unit i is defined as 81 - Example

Equality of Complex Numbers The set of all numbers in the form a+bi with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part, of the complex number: a+bi Equality of Complex Numbers a+bi = c+di if and only if a = c and b = d

Adding and Subtracting Complex Numbers (a+bi) + (c+di) = (a+bi) - (c+di) = (a+c) + (b+d)i (a-c) - (b-d)i Perform the indicated operation, writing the result in standard form. (-5 + 7i) - (-11 - 6i) =(-5-(-11)) - (7-(-6))i = (-5+11) - (7+6)i = 6 - 13i

i 8 6 2 3 - + 5 - = i 4 - i ) 3 1 )( 2 ( + - 2 = i 6 + i - i 3 - 2 = i Examples i 8 6 2 3 - + 5 - = i 4 - i ) 3 1 )( 2 ( + - remember i2=-1 2 = i 6 + i - i 3 2 - 2 = i 5 + 3 + i 5 + =

Conjugate of a Complex Number The complex conjugate of the number a+bi is a-bi, and the conjugate of a-bi is a+bi. distribute Example: foil

Principal Square Root of a Negative Number For any positive real number b, the principal square root of the negative number -b is defined by 9 16 - × Example

Example

Solve using the quadratic formula: standard form for complex numbers

The Complex Plane A complex number z = a + bi is represented as a point (a, b) in a complex plane, shown below. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane. Imaginary axis Real axis a b z = a + bi

Plot in the complex plane: Text Example Plot in the complex plane: a. z = 3 + 4i b. z = -1 – 2i c. z = -3 d. z = -4i Plot the complex number z = 3 + 4i the same way we plot (3, 4) in the rectangular coordinate system. Imaginary axis -5 -4 -3 -2 1 2 3 4 5 z = 3 + 4i Real axis z = -1 – 2i -1 -1 The complex number z = -1 – 2i corresponds to the point (-1, -2) in the rectangular coordinate system. -2

Plot in the complex plane: Text Example Plot in the complex plane: a. z = 3 + 4i b. z = -1 – 2i c. z = -3 d. z = -4i Imaginary axis z = -3 = -3 + 0i, this complex number corresponds to the point (-3, 0). -5 -4 -3 -2 1 2 3 4 5 z = 3 + 4i z = -3 Real axis z = -4i = 0 – 4i, this complex number corresponds to the point (0, -4). z = -4i -1 -1 -2 z = -1 – 2i

The Absolute Value of a Complex Number The absolute value of the complex number a + bi is

Determine the absolute value of z = 2-4i Example Determine the absolute value of z = 2-4i

Complete Student Checkpoint Perform the indicated operation. a. (-5+7i)-(-11+6i) Divide and express the result in standard form: =(-5-(-11))-(7-6)i =6 + i b. (5+4i)(6-7i) = 30 - 35i + 24i - 28i2 = 30 - 11i + 28 = 58 - 11i

Complete Student Checkpoint Solve using the quadratic formula: Perform the indicated operation and write the result in standard form:

Complete Student Checkpoint Determine the absolute value of the following complex number: 2 - 3i Plot in the complex plane. -1 -2 2 1

2.5 Complex Numbers