(4%2B5i).jpeg "(4-5i)(4+5i)")
Multiplying Complex Numbers: (4 - 5i)(4 + 5i)
This article will explore the multiplication of the complex numbers (4 - 5i) and (4 + 5i).
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 complex numbers, we can use the distributive property (also known as FOIL method):
(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
Solving (4 - 5i)(4 + 5i)
Let's apply this to our problem:
(4 - 5i)(4 + 5i) = (4 * 4) + (4 * 5i) - (5i * 4) - (5i * 5i)
Simplifying:
(4 - 5i)(4 + 5i) = 16 + 20i - 20i - 25i²
Since i² = -1, we can substitute:
(4 - 5i)(4 + 5i) = 16 + 20i - 20i + 25
Combining like terms:
(4 - 5i)(4 + 5i) = 41
Conclusion
The product of (4 - 5i) and (4 + 5i) is 41. This result highlights an important property of complex numbers. When multiplying a complex number by its conjugate (obtained by changing the sign of the imaginary part), the result is always a real number. In this case, the product is a real number, 41, which is the sum of the squares of the real and imaginary parts of the original complex numbers.