%5E2.jpeg "(5+2i)^2")
Squaring Complex Numbers: Exploring (5 + 2i)^2
This article delves into the process of squaring the complex number (5 + 2i). We'll explore the concept of complex numbers, the rules for squaring them, and arrive at the final result.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by incorporating the imaginary unit, i, where i² = -1. They are expressed in the form a + bi, where a and b are real numbers.
Squaring Complex Numbers
Squaring a complex number involves multiplying it by itself. The process is similar to squaring real numbers, but we need to keep in mind the properties of i.
Calculating (5 + 2i)²
Let's break down the calculation step-by-step:
(5 + 2i)² = (5 + 2i) * (5 + 2i)
Expanding the product using the distributive property, we get:
(5 + 2i)² = 5 * (5 + 2i) + 2i * (5 + 2i)
Simplifying further:
(5 + 2i)² = 25 + 10i + 10i + 4i²
Since i² = -1, we can substitute:
(5 + 2i)² = 25 + 10i + 10i + 4(-1)
Combining real and imaginary terms:
(5 + 2i)² = (25 - 4) + (10 + 10)i
Therefore, the final result is:
(5 + 2i)² = 21 + 20i
Conclusion
Squaring a complex number involves careful application of the distributive property and the fundamental property of the imaginary unit, i² = -1. By following these steps, we can efficiently calculate the square of any complex number. In this case, (5 + 2i)² simplifies to 21 + 20i.