Complex numbers, integral to advanced mathematics, are numbers consisting of a real part and an imaginary part. They are pivotal in various scientific and engineering fields, offering deep insights into mathematical concepts.

Addition and Subtraction of Complex Numbers

Complex Number Form:

• $z = a + bi$, where '$a$' is the real part, '$bi$' is the imaginary part.

Addition:

• For $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, add real and imaginary parts separately:
• $z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i$.

Subtraction:

• $z_1 - z_2 = (a_1 - a_2) + (b_1 - b_2)i$.
• Used in various mathematical and engineering contexts.

Multiplication of Complex Numbers

Distributive Property:

• Multiply $(a_1 + b_1i)(a_2 + b_2i)$ to get $a_1a_2 + a_1b_2i + b_1a_2i + b_1b_2i^2$.
• Simplify using $i^2 = -1 : a_1a_2 - b_1b_2 + (a_1b_2 + b_1a_2)i$.
• Key for understanding rotation and scaling in the complex plane.

Division of Complex Numbers

Using the Conjugate:

• The conjugate of $z = a + bi$ is $z^* = a - bi$.
• Divide $\frac{z_1}{z_2}$ by multiplying numerator and denominator with $z_2^*$ : $\frac{z_1}{z_2}$ = $\frac{z_1 \cdot z_2^*}{z_2 \cdot z_2^*}$.
• Eliminates the imaginary part in the denominator.
• Essential in complex equations and electrical engineering.

Example Questions

Addition Example

Problem Statement:

Compute $(3 + 4i) + (5 - 2i)$.

Solution:

1. Separate Real and Imaginary Parts:

• Real: $3 + 5$.
• Imaginary: $4i - 2i$.

2. Combine and Simplify:

• Real: $3 + 5 = 8$.
• Imaginary: $4i - 2i = 2i$.

3. Result:

• Sum is $8 + 2i$.

Multiplication Example

Problem Statement:

Determine the product of $(2 + 3i)$ and $(4 - 5i)$.

Solution:

1. Apply Distributive Property:

• Perform $(2 + 3i)(4 - 5i)$.

2. Multiply Each Pair of Terms:

• Calculate $2 \cdot 4$, $2 \cdot -5i$, $3i \cdot 4$, and $3i \cdot -5i$.

3. Combine Results:

• Sum up $8 - 10i + 12i - 15i^2$.

4. Use $i^2 = -1$:

• Convert $-15i^2$ to $+15$.

5. Combine Like Terms:

• Add $8 - 10i + 12i + 15$ together.

6. Final Product:

• Result is $23 + 2i$.

Division Example

Problem Statement:

Divide $(1 + 2i)$ by $(3 - 4i)$.

Solution:

1. Multiply by Conjugate:

• Use $\frac{3 + 4i}{3 + 4i}$ to clear the imaginary part in the denominator.
• Thus, $\frac{1 + 2i}{3 - 4i} \cdot \frac{3 + 4i}{3 + 4i}$.

2. Simplify Numerator:

• Calculate $(1 + 2i)(3 + 4i)$.
• Expand to get $3 + 4i + 6i + 8$ (using $i^2 = -1$).

3. Simplify Numerator Further:

• Combine like terms: $11 + 10i$.

4. Simplify Denominator:

• Calculate $(3 - 4i)(3 + 4i)$.
• Expand to $9 + 16$ (imaginary terms cancel out).

5. Complete Division:

• Divide $\frac{11 + 10i}{25}$.

6. Final Answer:

• Separate into real and imaginary parts: $\frac{11}{25} + \frac{10}{25}i$