(2%2B2i)%3D8i.jpeg "(2+2i)(2+2i)=8i")
A Complex Calculation: Unveiling the Error in (2 + 2i)(2 + 2i) = 8i
The equation (2 + 2i)(2 + 2i) = 8i is incorrect. Let's break down why and explore the correct solution.
Understanding Complex Numbers
Complex numbers are numbers of 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 (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
Correct Calculation
Let's apply this to our problem:
(2 + 2i)(2 + 2i) = (2 * 2 - 2 * 2) + (2 * 2 + 2 * 2)i
Simplifying:
(2 + 2i)(2 + 2i) = 0 + 8i
Therefore, the correct answer is 8i, not just 8i. The real part of the product is zero.
Key Takeaway
It's crucial to remember the rules for multiplying complex numbers. The imaginary unit 'i' plays a significant role, and its square (i²) is always -1. This leads to the unique properties of complex number multiplication.