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Multiplying Complex Numbers: (6 + 7i)(6 - 7i)
This article explores the multiplication of complex numbers, specifically the product of (6 + 7i) and (6 - 7i).
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).
Multiplying Complex Numbers
Multiplying complex numbers is similar to multiplying binomials, using the distributive property (or FOIL method).
Step 1: Expand the product:
(6 + 7i)(6 - 7i) = 6(6 - 7i) + 7i(6 - 7i)
Step 2: Distribute:
= 36 - 42i + 42i - 49i²
Step 3: Simplify using i² = -1:
= 36 - 49(-1)
Step 4: Combine real terms:
= 36 + 49
Step 5: Final result:
= 85
Key Observation
Notice that the result of multiplying (6 + 7i) and (6 - 7i) is a real number (85). This is because (6 - 7i) is the complex conjugate of (6 + 7i).
The complex conjugate of a complex number a + bi is a - bi. Multiplying a complex number by its conjugate always results in a real number. This property is often used to simplify expressions involving complex numbers.
Conclusion
The product of (6 + 7i) and (6 - 7i) is 85. This demonstrates the concept of multiplying complex numbers and the special case of multiplying a complex number by its conjugate, which always yields a real number.