%5E2.jpeg "(4+5i)^2")
Squaring Complex Numbers: Exploring (4 + 5i)^2
This article delves into the process of squaring the complex number (4 + 5i). Understanding this concept is crucial in various mathematical fields, especially when dealing with complex number operations.
Understanding Complex Numbers
Before diving into the squaring process, let's understand the nature of complex numbers. Complex numbers are expressed in the form a + bi, where:
- a represents the real part.
- b represents the imaginary part.
- i is the imaginary unit, defined as the square root of -1 (i² = -1).
Squaring (4 + 5i)
To square (4 + 5i), we simply multiply it by itself:
(4 + 5i)² = (4 + 5i) * (4 + 5i)
Now, we can use the distributive property (also known as FOIL method) to expand this:
(4 + 5i) * (4 + 5i) = 4(4) + 4(5i) + 5i(4) + 5i(5i)
Simplifying the expression:
= 16 + 20i + 20i + 25i²
Remember that i² = -1. Substituting this in:
= 16 + 20i + 20i + 25(-1)
Combining the real and imaginary terms:
= (16 - 25) + (20 + 20)i
Therefore, (4 + 5i)² simplifies to -9 + 40i.
Conclusion
Squaring complex numbers involves using the distributive property and recognizing that i² = -1. This process leads to a new complex number with its own real and imaginary components. Understanding this concept is essential for solving various mathematical problems involving complex numbers.