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Multiplying Complex Numbers: (4 - 3i)(4 + 3i)
This article explores the multiplication of complex numbers, specifically the product of (4 - 3i) and (4 + 3i). We will walk through the process and explain the significance of this result.
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 (also known as FOIL for first, outer, inner, last). Let's break down the multiplication:
(4 - 3i)(4 + 3i) =
- First: 4 * 4 = 16
- Outer: 4 * 3i = 12i
- Inner: -3i * 4 = -12i
- Last: -3i * 3i = -9i²
Remember that i² = -1. Substituting this into the equation:
16 + 12i - 12i - 9(-1) = 16 + 9 = 25
The Result and Significance
Therefore, (4 - 3i)(4 + 3i) = 25.
This result highlights an important property of 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) and (4 + 3i) are conjugates of each other. This property is often used to simplify expressions involving complex numbers.