Complex number

Definition

A complex number is a number that has both a real part and an imaginary part, typically expressed in the form $a + bi$ where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property that $i^2 = -1$. Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane.

5 Must Know Facts For Your Next Test

1. The real part of a complex number $a + bi$ is $a$, and the imaginary part is $b$.
2. The complex conjugate of a complex number $a + bi$ is $a - bi$.
3. The modulus (or absolute value) of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$.
4. Complex numbers can be added, subtracted, multiplied, and divided using specific algebraic rules.
5. The set of all complex numbers forms a field under addition and multiplication, denoted by $\mathbb{C}$.

Review Questions

• What are the real and imaginary parts of the complex number $7 - 3i$?
• Calculate the modulus of the complex number $4 + 5i$.
• Find the product of $(2 + i)$ and $(3 - i)$.

Related terms

Imaginary Unit:

Denoted as $i$, it is defined by the property that $i^2 = -1$.

Complex Conjugate:

For a given complex number $a + bi$, its conjugate is $a - bi$. Used in simplifying division involving complex numbers.

Modulus:

Also known as absolute value; for a complex number $a + bi$, it is $\sqrt{a^2 + b^2}$. Represents its distance from the origin in the complex plane.