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Multiplying Complex Numbers: (1/3 + 3i)(1/3 - 3i)
This article explores the multiplication of complex numbers, specifically focusing on the expression (1/3 + 3i)(1/3 - 3i). We will demonstrate how to perform the multiplication and analyze the result.
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.
Multiplying Complex Numbers
To multiply complex numbers, we use the distributive property, similar to multiplying binomials in algebra:
(a + bi)(c + di) = ac + adi + bci + bdi²
Since i² = -1, we can simplify this to:
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Applying the Multiplication to (1/3 + 3i)(1/3 - 3i)
Let's identify the values of 'a', 'b', 'c', and 'd' in our expression:
- a = 1/3
- b = 3
- c = 1/3
- d = -3
Now, we can apply the formula:
(1/3 + 3i)(1/3 - 3i) = (1/3 * 1/3 - 3 * -3) + (1/3 * -3 + 3 * 1/3)i
Simplifying the expression:
(1/3 + 3i)(1/3 - 3i) = (1/9 + 9) + (-1 + 1)i
Finally, we get:
(1/3 + 3i)(1/3 - 3i) = 82/9
Conclusion
The multiplication of (1/3 + 3i)(1/3 - 3i) results in a real number, 82/9. This is because the imaginary terms cancel out due to the opposite signs of 'b' and 'd'. This example illustrates how multiplying complex conjugates (numbers of the form a + bi and a - bi) always leads to a real number.