### Adding and Subtracting Complex Numbers

A **term** of an algebraic expression is a part of the expression that is separated from other parts by addition or subtraction. **Like terms**, such as $3x$ and $-8x$, are terms that contain the same variable (or variables) with the same exponents. Adding and subtracting complex numbers is very similar to adding and subtracting algebraic expressions by combining like terms.

- For addition, add the real parts and add the imaginary parts.
- For subtraction, distribute the negative, and then add.

The letter $z$ is often used to represent a complex number. If an expression or equation includes two complex numbers, $z_1$ can represent the first complex number, and $z_2$ can represent the second complex number.

1) $z_1+z_2$

2) $z_1-z_2$

Write the addition and subtraction expressions. For subtraction, distribute the minus sign.

For $z_1+z_2$:Reorder so that like terms are next to each other.

For $z_1+z_2$:Combine like terms.

For $z_1+z_2$:### Multiplying Complex Numbers

### Powers of *i*

It is important to remember that $i$ is not a variable; it has a value of $\sqrt{-1}$. So, unlike powers of variables, such as $x^2$, powers of $i$ need to be simplified. When calculating powers of $i$ starting from $i^1$ and onward, a pattern emerges. Analyzing the pattern involves two rules of exponents: the product of powers property and the power of a power property.

The product of powers property states that two powers with the same base can be multiplied by adding their exponents. For example:### Simplifying Powers of $i$

Power of $i$ | Expanded Form | Simplified Form |
---|---|---|

$i^1$ |
$i^1=i$ |
$i$ |

$i^2$ |
$\begin{aligned}i^2&=\left(\sqrt{-1}\right)^{2}\\&=-1\end{aligned}$ |
$-1$ |

$i^3$ |
$\begin{aligned}i^3 &= i^2 \cdot i\\&=-1\cdot i\\&=-i\end{aligned}$ |
$-i$ |

$i^4$ |
$\begin{aligned}i^4&=i^2 \cdot i^2 \\&= -1 \cdot (-1)\\&=1\end{aligned}$ |
$1$ |

$i^5$ |
$\begin{aligned}i^5 &= i^4 \cdot i\\&=1\cdot i\\&=i\end{aligned}$ |
$i$ |

$i^6$ |
$\begin{aligned}i^6 &= i^4 \cdot i^2\\&=1\cdot i^2\\&=-1\end{aligned}$ |
$-1$ |

$i^7$ |
$\begin{aligned}i^7 &= i^4 \cdot i^3\\&=1\cdot i^3\\&=-i\end{aligned}$ |
$-i$ |

$i^8$ |
$\begin{aligned}i^8 &= i^4 \cdot i^4\\&=1\cdot i^4\\&=1\end{aligned}$ |
$1$ |

$i^9$ |
$\begin{aligned}i^9 &= \left(i^{4}\right)^{2} \cdot i\\&=1^{2} \cdot i\\&=i\end{aligned}$ |
$i$ |

There are only four possible values for a power of $i$, and they repeat in the pattern: $i$, -1, $-i$, 1.

Analyzing the calculations of the powers of $i$ shows that for every power greater than $i^4$, it is possible to factor out a power of $i^4$ and write it as 1.

### Products of Complex Numbers

**complex conjugate**is the number with the same real part and the opposite imaginary part, $a-bi$. For $z=a+bi$, the complex conjugate is written as $\bar z=a-bi$. The product of a complex number and its complex conjugate, or $z\bar{z}$, is a real number:

### Dividing Complex Numbers

**radical expression**is an expression that contains at least one radical sign, $\sqrt{\phantom{0}}$. Rationalizing the denominator of a fraction is the process of rewriting the fraction without a radical expression in the denominator. A denominator with two terms can be rationalized by multiplying the numerator and denominator of the fraction by the conjugate of the denominator. For the radical expression $2+\sqrt{3}$, the conjugate is $2-\sqrt{3}$. The product of a radical expression and its conjugate is a rational number.

1. Write the two numbers as a fraction: