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
- The real part of a complex number $a + bi$ is $a$, and the imaginary part is $b$.
- The complex conjugate of a complex number $a + bi$ is $a - bi$.
- The modulus (or absolute value) of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$.
- Complex numbers can be added, subtracted, multiplied, and divided using specific algebraic rules.
- 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 $(3 - i)$.
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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.
