%5E2.jpeg "(2-8i)^2")
Squaring Complex Numbers: (2 - 8i)²
In mathematics, complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying the equation i² = -1. Squaring a complex number involves multiplying it by itself.
Let's explore the process of squaring the complex number (2 - 8i)²:
Expanding the Expression
We can expand the expression using the distributive property (FOIL method):
(2 - 8i)² = (2 - 8i)(2 - 8i) = 2(2 - 8i) - 8i(2 - 8i) = 4 - 16i - 16i + 64i²
Simplifying the Expression
Remember that i² = -1. Substituting this value into our expression:
4 - 16i - 16i + 64i² = 4 - 16i - 16i + 64(-1) = 4 - 16i - 16i - 64 = -60 - 32i
Result
Therefore, (2 - 8i)² simplifies to -60 - 32i.
Visualizing the Result
It's helpful to visualize complex numbers as points on a complex plane. The real part (a) is represented on the horizontal axis, and the imaginary part (b) is represented on the vertical axis.
Squaring a complex number can be viewed as a geometric transformation. In the case of (2 - 8i)², squaring results in a complex number with a significantly larger magnitude and a different angle from the origin on the complex plane.