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Multiplying Complex Numbers: (6 + 3i)(6 - 3i)
This article will explore how to multiply the complex numbers (6 + 3i) and (6 - 3i). We'll also discuss the significance of this particular multiplication and its relation to the concept of conjugates.
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:
(6 + 3i)(6 - 3i) = 6(6 - 3i) + 3i(6 - 3i)
Expanding the terms:
= 36 - 18i + 18i - 9i²
Remember that i² = -1. Substituting this value:
= 36 - 9(-1)
= 36 + 9
= 45
Conjugates and Their Significance
The complex numbers (6 + 3i) and (6 - 3i) are complex conjugates. Notice that the only difference between them is the sign of the imaginary term. Multiplying a complex number by its conjugate always results in a real number. This is because the imaginary terms cancel each other out, leaving only the real terms.
This property is crucial in various mathematical operations involving complex numbers, particularly when dealing with division and finding the modulus of a complex number.
Summary
We have successfully multiplied the complex numbers (6 + 3i) and (6 - 3i), obtaining the result 45. This demonstrates the concept of complex conjugates and how their multiplication always yields a real number. This understanding is fundamental for working with complex numbers in various mathematical contexts.