Multiplying Complex Numbers: (6 + 2i)(6  2i)
This article explores the multiplication of complex numbers, specifically focusing on the product of (6 + 2i) and (6  2i).
Understanding Complex Numbers
Complex numbers are numbers that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as the square root of 1 (i² = 1).
Multiplication of Complex Numbers
To multiply complex numbers, we can use the distributive property, similar to how we multiply binomials.
Calculating (6 + 2i)(6  2i)

Expand the product: (6 + 2i)(6  2i) = 6(6  2i) + 2i(6  2i)

Apply the distributive property: = 36  12i + 12i  4i²

Simplify by substituting i² = 1: = 36  4(1)

Combine terms: = 36 + 4

Final Result: = 40
Observation
Notice that the product of (6 + 2i) and (6  2i) results in a real number (40). This is a special case because (6 + 2i) and (6  2i) are complex conjugates of each other.
Complex Conjugates
The complex conjugate of a complex number a + bi is a  bi. When multiplying a complex number by its conjugate, the imaginary terms cancel out, leaving only a real number.
Conclusion
We have successfully multiplied the complex numbers (6 + 2i) and (6  2i), resulting in the real number 40. This calculation highlights the concept of complex conjugates and their significance in simplifying complex number operations.