There is a new “number” that we associate negative numbers with under the square root. These are called COMPLEX NUMBERS, and they are indicated with the letter i, where:* \(i=\sqrt{-1}\)*

We can do regular operations like addition, subtraction, multiplication, and division with complex numbers. Keep the following in mind, to make complex multiplication simpler.

\(i=\sqrt{-1}\)

\(i^{2}=i\times i=\sqrt{-1}\times \sqrt{-1}=-1\)

\(i^{3}=i\times i\times i=\sqrt{-1}\times \sqrt{-1}\times \sqrt{-1}= -1\times \sqrt{-1}=-i\)

\(i^{4}=i^{2}\times i^{2}=-1\times -1=1\)

\(i^{5}=i^{4}\times i=i\)

\(i^{6}=i^{4}\times i^{2}=-1\)

\(i^{7}=i^{4}\times i^{3}=-i\)

\(i^{8}=i^{4}\times i^{4}=1\)

*Solved Example 1*

Find the value of \(i^{203}\)

*A.* i

*B.* -i

*C.* 1

*D. *-1

Solution: Now, we need to first break up \(i^{203}\: to\: i^{202+1}\)

\(i^{202}\times i^{1}\)

\((i^{2})^{101}\times i\)

\((-1)^{101}\times i\)

-1 x i

= -i

The correct answer is B.

### Add and Subtract Complex Numbers

Adding and subtracting complex numbers is similar to adding and subtracting polynomials. Just add or subtract the real and imaginary parts.

**Example:** Solve: (-6 - 5i) - (3 - 4i)

(-6 - 5i) - (3 - 4i) = (-6 - 5i) + (-3 + 4i)

= [-6 + (-3)] + (-5 + 4)i

= -9 - i

### Multiply Complex Numbers

Multiplying complex numbers is similar to multiplying polynomials.

**Solve:** (2 + 5i) (6 + 4i)

(2 + 5i) (6 + 4i) = 12 + 8i + 30i + 20i2

= 12 + 38i + 20(-1)

= 12 + 38i - 20

= -8 + 38i

### Divide Complex Numbers

For complex division, we need to know what a conjugate is. For starters, all complex numbers take the form of a + bi, where a is the x-coordinate and b is the y-coordinate. A complex conjugate is changing the + in a + bi to a − bi or vice versa. When we have complex division, we multiply by a clever form of “one,” which will be the conjugate of the denominator.

**Example: **Write each quotient in the form \(a+bi:\frac{6}{7-4i}\)

\(\frac{6}{7-4i}=\frac{6}{7-4i}\times \frac{7+4i}{7+4i}\)

\(\frac{42+24i}{49-16i^{2}}\)

\(\frac{42+24i}{49-16(-1)}\)

\(\frac{42+24i}{65}\)

\(\frac{42}{65}+\frac{25}{65}i\)