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Multiplying Complex Numbers: (5 + 2i)(5 - 2i)
This article explores the multiplication of complex numbers, specifically focusing on the product of (5 + 2i) and (5 - 2i).
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:
(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 Formula
Let's apply this formula to our problem:
(5 + 2i)(5 - 2i) = (5 * 5 - 2 * -2) + (5 * -2 + 2 * 5)i
Simplifying the expression:
(5 + 2i)(5 - 2i) = (25 + 4) + ( -10 + 10)i
Therefore, the product of (5 + 2i) and (5 - 2i) is:
(5 + 2i)(5 - 2i) = 29
Key Observation:
Notice that the product of (5 + 2i) and (5 - 2i) is a real number. This is because (5 - 2i) is the complex conjugate of (5 + 2i). The product of a complex number and its conjugate always results in a real number.
Conclusion
Multiplying complex numbers requires careful application of the distributive property and recognizing the special case of multiplying a number by its conjugate. In the case of (5 + 2i)(5 - 2i), the product simplifies to a real number, highlighting the relationship between complex numbers and their conjugates.