### Imaginary Unit

**real number**is a number in the set of all rational and irrational numbers. The equation $x^2 = -1$ does not have a solution in the set of real numbers because the square of any real number is positive. To solve certain types of problems, mathematicians defined a number to be the solution of this equation. The

**imaginary unit**is the square root of –1, written as $i$.

### Pure Imaginary Numbers

An **imaginary number**, also known as a pure imaginary number, is a number of the form $bi$, where $b$ is a real number and $i$ is the imaginary unit. Some examples are $12i$ and $i\sqrt{19}$.

### Defining Complex Numbers

**complex number**is the sum of a real number and a pure imaginary number. The

**standard form of a complex number**is $a+bi$ for any real numbers $a$ and $b$. The real part of the complex number is $a$, and the imaginary part is $b$.

Complex Number | Real Part | Imaginary Part |
---|---|---|

$6+i\sqrt3$ | $6$ | $\sqrt3$ |

$\sqrt7 -5i$ | $\sqrt7$ | $-5$ |

$12i$ | $0$ | $12$ |

$\sqrt{19}$ | $\sqrt{19}$ | $0$ |

Real and imaginary numbers are both subsets of complex numbers:

- When $a=0$, $a+bi=bi$, which is a pure imaginary number.
- When $b=0$, $a+bi=a$, which is a real number.

### Complex Plane

A coordinate plane is used to locate points in terms of distance from the $x$- and $y$-axes. The **complex plane** is used to locate points that represent complex numbers in terms of distance from the real axis and the imaginary axis. The **real axis** is the horizontal axis in the complex plane and represents the set of real numbers. The **imaginary axis** is the vertical axis in the complex plane and represents the set of pure imaginary numbers.

- The number $a+bi$ is located at the point $(a,b)$.
- If $a=0$, then the number $bi$ is located on the imaginary axis.
- If $b=0$, then the number $a$ is located on the real axis.

Graph each number in the complex plane:

1) $9i-3$

2) $\sqrt{-4}$

3) $-\sqrt{-64}+5$

4) $\sqrt{3}+\sqrt{-2}$

5) –3

Identify the coordinates of each point, and write them in the form $(a,b)$.

For $-3+9i$, the value of $a$ is –3. The value of $b$ is 9. The coordinates of the point are $(-3,9)$.

For $0+2i$, the value of $a$ is zero. The value of $b$ is 2. The coordinates are $(0,2)$.

For $5-8i$, the value of $a$ is 5. The value of $b$ is –8. The coordinates are $(5,-8)$.

For $\sqrt{3}+i\sqrt{2}$, the value of $a$ is $\sqrt{3}$. The value of $b$ is $\sqrt2$. The coordinates are $(\sqrt3,\sqrt2)$, or about $(1.7,1.4)$.

For $-3+0i$, the value of $a$ is $-3$. The value of $b$ is zero. The coordinates are $(-3,0)$.