Complex numbers are numbers that have two parts: a real part and an imaginary part. They are written in the form:

$z = a + bi$

where:

• $a$ is the real part.
• $b$ is the imaginary part.
• $i$ is the imaginary unit, which is defined by $i^2 = -1$.

Key Concepts:

1. Addition/Subtraction: Complex numbers can be added or subtracted by combining their real and imaginary parts separately. For example:

$(3 + 4i) + (1 + 2i) = (3 + 1) + (4i + 2i) = 4 + 6i$

2. Multiplication: When multiplying complex numbers, you apply distributive properties and simplify using $i^2 = -1$. Example:

$(3 + 4i)(1 + 2i) = 3(1) + 3(2i) + 4i(1) + 4i(2i) = 3 + 6i + 4i + 8i^2$ Since $i^2 = -1$, we simplify: $3 + 10i + 8(-1) = -5 + 10i$

3. Complex Conjugate: The conjugate of a complex number $z = a + bi$ is denoted as $\overline{z}$ and is $a - bi$. It is useful for dividing complex numbers and finding magnitudes.

4. Magnitude (Modulus): The magnitude of a complex number $z = a + bi$ is given by:

$|z| = \sqrt{a^2 + b^2}$

For example, the magnitude of $3 + 4i$ is:

$|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$

5. Polar Form: Complex numbers can also be represented in polar form as $z = r(\cos \theta + i \sin \theta)$, where:
   • $r = |z|$ is the magnitude.
   • $\theta$ is the argument (angle) of the complex number.

Would you like further details or any specific questions answered?

Related Questions:

1. How do you divide complex numbers?
2. What is Euler's formula and how does it relate to complex numbers?
3. How can complex numbers be represented on the Argand plane?
4. What are the applications of complex numbers in physics and engineering?
5. What is De Moivre's Theorem?

Tip: When performing operations with complex numbers, always remember to treat $i^2 = -1$ carefully during simplification.