(2-3i).jpeg "(2+3i)(2-3i)")
Exploring Complex Number Multiplication: (2+3i)(2-3i)
In the realm of mathematics, complex numbers introduce an intriguing dimension beyond real numbers. One of the fundamental operations with complex numbers is multiplication. Let's delve into the multiplication of (2+3i) and (2-3i).
Understanding Complex Numbers
Complex numbers are 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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials.
(2+3i)(2-3i) = 2(2-3i) + 3i(2-3i)
Expanding the terms, we get:
= 4 - 6i + 6i - 9i²
The Power of i²
The key to simplifying this expression lies in the fact that i² = -1. Substituting this value, we get:
= 4 - 6i + 6i - 9(-1)
Final Result
Simplifying further, we obtain:
= 4 + 9
= 13
Conclusion
The product of (2+3i) and (2-3i) is a real number, 13. This exemplifies a notable property of complex conjugates. Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. When multiplied, they always yield a real number.