Imaginary number

Definition

An imaginary number is a complex number in which the real part is zero and the imaginary part is non-zero. It is typically written in the form $bi$, where $b$ is a real number and $i$ denotes the imaginary unit, defined as $i = \sqrt{-1}$.

5 Must Know Facts For Your Next Test

1. Imaginary numbers are used to extend the real number system to solve equations that have no real solutions, such as $x^2 + 1 = 0$.
2. The basic unit of imaginary numbers is $i$, where $i^2 = -1$.
3. Any complex number can be written as a sum of a real and an imaginary number: $a + bi$.
4. Operations with imaginary numbers follow similar rules to those for real numbers but include the additional property that $i^2 = -1$.
5. Imaginary numbers are plotted on the vertical axis of the complex plane, while real numbers are plotted on the horizontal axis.

Review Questions

• What is the value of $i^2$?
• How would you express an imaginary number using its standard form?
• Why are imaginary numbers necessary in solving certain quadratic equations?

Related terms

Complex Number: A number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Real Number: A value representing a quantity along a continuous line, including all rational and irrational numbers.

$i (Imaginary Unit)$: $i$ is defined as $\sqrt{-1}$, and it satisfies the equation $i^2 = -1$. It forms the basis for all imaginary numbers.