%5E2.jpeg "(-1-3i)^2")
Squaring a Complex Number: (-1 - 3i)^2
Let's explore how to square the complex number (-1 - 3i).
Understanding Complex Numbers
A complex number is a number 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.
Squaring (-1 - 3i)
To square a complex number, we simply multiply it by itself:
(-1 - 3i)^2 = (-1 - 3i) * (-1 - 3i)
Now, we can use the distributive property (also known as FOIL) to expand this expression:
(-1 - 3i) * (-1 - 3i) = (-1)(-1) + (-1)(-3i) + (-3i)(-1) + (-3i)(-3i)
Simplifying this further:
= 1 + 3i + 3i + 9i^2
Since i^2 = -1, we can substitute:
= 1 + 3i + 3i + 9(-1)
Combining like terms:
= -8 + 6i
Conclusion
Therefore, (-1 - 3i)^2 = -8 + 6i. This demonstrates that squaring a complex number results in another complex number, with both real and imaginary components.