Complex numbers are an essential concept in mathematics, extending the idea of numbers beyond the real number line. A complex number is usually expressed in the form:

z = a + bi

where:
– a is the real part of z (Re(z))
– b is the imaginary part of z (Im(z))
– i is the imaginary unit, defined as i^2 = -1

Understanding the algebraic properties of complex numbers is crucial for solving equations, analyzing functions, and exploring advanced mathematics.

Addition of Complex Numbers

If z1 = a + bi and z2 = c + di are two complex numbers, their sum is:

z1 + z2 = (a + c) + (b + d)i

Example:
(3 + 2i) + (1 + 4i) = (3+1) + (2+4)i = 4 + 6i

Property: Addition of complex numbers is commutative and associative:
– Commutative: z1 + z2 = z2 + z1
– Associative: (z1 + z2) + z3 = z1 + (z2 + z3)

Subtraction of Complex Numbers

The difference of two complex numbers is:

z1 – z2 = (a – c) + (b – d)i

Example:
(5 + 3i) – (2 + i) = (5-2) + (3-1)i = 3 + 2i

 Multiplication of Complex Numbers

For z1 = a + bi and z2 = c + di, the product is:

z1 * z2 = (ac – bd) + (ad + bc)i

Example:
(1 + 2i)(3 + 4i) = (1*3 – 2*4) + (1*4 + 2*3)i = -5 + 10i

Property: Multiplication is commutative and associative, and distributes over addition:
z1 * z2 = z2 * z1, z1(z2 + z3) = z1 * z2 + z1 * z3

Conjugate of a Complex Number

The conjugate of z = a + bi is:

z̅ = a – bi

Properties of Conjugates:
1. z + z̅ = 2a (real number)
2. z * z̅ = a^2 + b^2 (non-negative real number)
3. z̅1 + z̅2 = z̅1 + z̅2
4. z̅1 * z̅2 = z̅1 * z̅2

Division of Complex Numbers

To divide z1 by z2 ≠ 0, multiply numerator and denominator by the conjugate of the denominator:

z1 / z2 = (a+bi)/(c+di) * (c-di)/(c-di) = ((ac+bd) + (bc-ad)i)/(c^2 + d^2)

Example:
(1 + 2i)/(3 + 4i) = ((1*3 + 2*4) + (2*3 – 1*4)i)/(3^2 + 4^2) = (11 + 2i)/25 = 11/25 + 2/25i

 Important Algebraic Properties Summary

Conclusion

Complex numbers form a complete algebraic system with well-defined addition, subtraction, multiplication, and division. Mastering these basic algebraic properties helps in solving quadratic equations, analyzing signals in engineering, and exploring higher mathematics in a structured way.