Complex Numbers: (lesson 1 of 2)

## Complex number arithmetic

### Definitions:

1.

2. A complex number is any number of the form a + bi where a and b are real numbers.

### Addition and Subtraction of complex numbers

To add or subtract two complex numbers, you add or subtract the real parts and the imaginary parts.

(a + bi) + (c + id) = (a + c) + (b + d)i.

(a + bi) - (c + id) = (a - c) + (b - d)i.

Example 1:

(3 - 5i) + (6 + 7i) = (3 + 6) + (-5 + 7)i = 9 + 2i.

(3 - 5i) - (6 + 7i) = (3 - 6) + (-5 - 7)i = -3 - 12i.

Exercise 1: Addition and Subtraction

### Multiplying complex numbers

Example 2:

Let's take specific complex numbers to multiply, say 2 + 3i and 2 - 5i.

(2 + 3i)(2 - 5i) = 4 - 10i + 6i - 15i^{2} = 4 - 4i - 15i^{2}

The definition of i tells us that i^{2} = -1 . Therefore,

(2 + 3i)(2 - 5i) = 4 - 4i -15(-1) = 19 - 4i.

If you generalize this example, you'll get the **general rule for multiplication**

(x + yi)(u + vi) = (xu - yv) + (xv + yu)i

Exercise 2: Multiplying complex numbers

### Conjugate complex numbers

We define the conjugate of a + bi as

Example 3:

Conjugates are important because of the fact that a complex number times its conjugate is real.

Example 4:

### Modulus of a complex number

We define modulus or absolute value of complex number a + bi as
. We write modulus of a + bi as |a + bi|.

Example 4:

|3 + 4i| =

Exercise 3: Conjugate and modulus

### Division of complex numbers

The process of division of complex numbers:

step 1: Find the conjugate of a denominator.

step 2: Multiply the complex fraction, both top and bottom complex number.

Here is the complete division problem:

**Now, we can write down a general formula for division of complex numbers **

Exercise 4: Divide complex numbers

### Fundamental Theorem of Algebra

Every nth - order polynomial possess exactly n complex roots.