Complex Numbers: Standard Form and Operations

Complex Numbers

Standard Form of a Complex Number

Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit defined by the property that $i^2 = -1$.

• Standard form: $a + bi$

• Real part: The value of a

• Imaginary part: The value of b

Example: $3 - 2i$ is a complex number with real part 3 and imaginary part -2.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, combine their real parts and their imaginary parts separately.

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Subtraction: $(a + bi) - (c + di) = (a - c) + (b - d)i$

Example:

• Given $(1 - 3i) + (8 + 7i)$

• Combine real parts: $1 + 8 = 9$

• Combine imaginary parts: $-3i + 7i = 4i$

• Result: $9 + 4i$

Properties of Complex Numbers

• Commutative Property: Addition and multiplication of complex numbers are commutative.

• Associative Property: Addition and multiplication are associative.

• Distributive Property: Multiplication distributes over addition.

Applications of Complex Numbers

• Complex numbers are used in engineering, physics, and applied mathematics, especially in the study of electrical circuits, signal processing, and quantum mechanics.

Summary Table: Operations with Complex Numbers

Operation	Formula	Example
Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	$(1 - 3i) + (8 + 7i) = 9 + 4i$
Subtraction	$(a + bi) - (c + di) = (a - c) + (b - d)i$	$(5 + 2i) - (3 + 6i) = 2 - 4i$