What Are Complex Numbers?

Complex numbers extend our idea of numbers to include solutions to equations like \(x^2 + 1 = 0\). A complex number has a real part and an imaginary part: \(a + bi\), where \(i = \sqrt{-1}\).

Parts of a Complex Number

• Real Part: The \(a\) in \(a + bi\)
• Imaginary Part: The \(b\) in \(a + bi\)

Operations with Complex Numbers

• Addition: \((a + bi) + (c + di) = (a + c) + (b + d)i\)
• Subtraction: \((a + bi) - (c + di) = (a - c) + (b - d)i\)
• Multiplication: \((a + bi) \cdot (c + di) = (ac - bd) + (ad + bc)i\)

Why Use Complex Numbers?

Complex numbers help us solve equations that have no real solutions and are essential in physics, engineering, and more.

Visualizing Complex Numbers

You can graph complex numbers on a plane, where the x-axis is the real part and the y-axis is the imaginary part—a bit like plotting points!