Algebra II
Advanced algebraic concepts including polynomials, rational expressions, and complex numbers.
Complex Numbers and Imaginary Units
Enter the Imaginary World
Sometimes, equations don't have real-number solutions—like \( x^2 + 1 = 0 \). That's where complex numbers come in!
What Are Complex Numbers?
A complex number is written as \( a + bi \), where:
- \( a \): real part
- \( b \): imaginary part
- \( i \): imaginary unit, where \( i^2 = -1 \)
Operations
You can add, subtract, multiply, and even divide complex numbers. Just treat \( i \) like a variable, but remember \( i^2 = -1 \).
Why Use Complex Numbers?
Complex numbers help solve equations that have no solutions in the real number system and are used in fields like engineering and physics.
Visualizing Complex Numbers
You can plot complex numbers on a special graph called the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.
Key Formula
\[i^2 = -1\]
Examples
Adding \( (2 + 3i) + (1 - 4i) = 3 - i \)
Multiplying \( (1 + i)(1 - i) = 1 - i^2 = 2 \)
In a Nutshell
Complex numbers extend the real numbers by including solutions to equations like \( x^2 + 1 = 0 \).