(4-5i)(4+5i)

2 min read Jun 16, 2024
(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.

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