Perform Operations with Complex Numbers Section 4.6: Perform Operations with Complex Numbers

Not all quadratic equations have real-number solutions Not all quadratic equations have real-number solutions. For example, x2 = -1 has no real- number solutions because the square of any real number x is never a negative number. To overcome this problem, mathematicians created an expanded system of numbers using the imaginary unit i , defined as The imaginary unit i can be used to write the square root of any negative number.

The Square Root of a Negative Number 1. If r is a positive real number, then . Example: 2. By property (1), it follows that .

A complex number written in standard form is a number a + bi where a and b are real numbers. The number a is the real part of the complex number, and the number bi is the imaginary part.

If b ≠ 0, then a + bi is an imaginary number If b ≠ 0, then a + bi is an imaginary number. If a = 0 and b ≠ 0, then a + bi is a pure imaginary number. Two complex numbers a + bi and c + di are equal If and only if a = c and b = d.

Sums and Differences of Complex Numbers To add (or subtract) two complex numbers, add (or subtract) their real parts and their imaginary parts separately.

Example 1: Solve 2x2 + 18 = -72

Example 2: Write the expression as a complex number in standard form. a) (12 – 11i) + (-8 + 3i) = 4 – 8i (15 – 9i) – (24 – 9i) = -9 35 – (13 + 4i) + i = 22 – 3i

HOMEWORK (Day 1) pg. 279; 4 – 20 even

Multiplying Complex Numbers To multiply two complex numbers, use the distributive property or the FOIL method just as you do when multiplying real numbers or algebraic expressions.

The complex numbers of the form a + bi and a – bi are called complex conjugates. The product of complex conjugates is always a real number. You can use this fact to write the quotient of two complex numbers in standard form.

Absolute Value of a Complex Number The absolute value of a complex number z = a + bi, denoted |z|, is a nonnegative real number defined as This is the distance between z and the origin in the complex plane.

Example 3: Write the expression as a complex number in standard form. -5i(8 – 9i) = -40i + 45i2 = -40i + 45(-1) = -40i – 45

(-8 + 2i)(4 – 7i) = -32 + 56i + 8i – 14i2 = -32 + 64i – 14(-1) = -32 + 64i + 14 = -18 + 64i

Example 4: Write the quotient in standard form.

Example 5: Find the absolute value of 5 – 12i and 17i. |5 – 12i| b) |17i|

HOMEWORK (Day 2) pg. 280; 22 – 32 even, 42 – 48 even