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Multiplying Complex Numbers: (4 + 3i)(4 - 3i)
This article will explore the multiplication of two complex numbers, (4 + 3i) and (4 - 3i). We'll demonstrate how to perform the operation and explain the resulting simplification.
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
To multiply two complex numbers, we can use the distributive property (or FOIL method) just like with binomials in algebra:
(4 + 3i)(4 - 3i) = (4 * 4) + (4 * -3i) + (3i * 4) + (3i * -3i)
Simplifying the terms:
= 16 - 12i + 12i - 9i²
Since i² = -1, we can substitute:
= 16 - 9(-1)
= 16 + 9
= 25
The Result
The product of (4 + 3i) and (4 - 3i) is 25. This demonstrates a key concept in complex numbers:
The product of a complex number and its conjugate is always a real number.
The conjugate of a complex number (a + bi) is (a - bi). In this case, (4 - 3i) is the conjugate of (4 + 3i). This relationship is often used in simplifying expressions involving complex numbers.