%5E2.jpeg "(2+2i)^2")
Exploring the Square of a Complex Number: (2 + 2i)²
In the realm of complex numbers, squaring a complex number like (2 + 2i)² might seem daunting at first glance. However, with the right approach, it becomes a straightforward process. Let's delve into the calculation and uncover the intricacies.
Understanding Complex Numbers
Before diving into the square, let's quickly recap the fundamental concept of complex numbers. A complex number is represented 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.
Calculating (2 + 2i)²
To square (2 + 2i)², we can apply the standard algebraic approach of expanding the expression. Let's break down the steps:
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Expansion: (2 + 2i)² = (2 + 2i) * (2 + 2i)
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FOIL Method: We can use the FOIL (First, Outer, Inner, Last) method to multiply the two binomials:
- First: 2 * 2 = 4
- Outer: 2 * 2i = 4i
- Inner: 2i * 2 = 4i
- Last: 2i * 2i = 4i²
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Simplifying: Combining the terms and remembering that i² = -1, we get:
4 + 4i + 4i + 4(-1) = 4 + 8i - 4
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Final Result: The simplified expression for (2 + 2i)² is 8i.
Visualizing the Result
The result, 8i, represents a pure imaginary number on the complex plane. It lies on the imaginary axis, 8 units above the origin.
Key Takeaways
- Squaring a complex number involves multiplying it by itself.
- The FOIL method is a helpful tool for expanding binomial expressions.
- The imaginary unit 'i' plays a crucial role in complex number calculations.
By understanding the fundamentals of complex numbers and employing basic algebraic techniques, we can effortlessly square complex numbers like (2 + 2i)². This knowledge is essential for a deeper understanding of complex number arithmetic and its applications in various fields.