%5E3.jpeg "(1+i)^3")
Exploring the Power of Complex Numbers: (1 + i)³
In the realm of complex numbers, understanding the behavior of powers can be intriguing. Let's delve into the expansion of (1 + i)³.
Understanding Complex Numbers
Before we embark on the calculation, let's quickly recap the fundamentals of complex numbers. A complex number is expressed as a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit defined as the square root of -1.
Expanding the Expression
We can expand (1 + i)³ using the distributive property or the binomial theorem. Let's use the distributive property:
(1 + i)³ = (1 + i)(1 + i)(1 + i)
First, let's multiply the first two factors:
(1 + i)(1 + i) = 1 + i + i + i²
Remember, i² = -1. Therefore:
(1 + i)(1 + i) = 1 + 2i - 1 = 2i
Now, we multiply this result by the remaining factor (1 + i):
2i (1 + i) = 2i + 2i² = 2i - 2
The Result
Therefore, (1 + i)³ simplifies to -2 + 2i.
Visualizing the Result
Complex numbers can be represented graphically on the complex plane. The real part of the number is plotted on the horizontal axis, and the imaginary part is plotted on the vertical axis. The result, -2 + 2i, would be located two units to the left of the origin and two units above the origin.
Conclusion
Expanding expressions like (1 + i)³ helps us understand the unique properties of complex numbers and their behavior when raised to powers. While the calculation might seem simple at first glance, it reveals the interplay between real and imaginary parts in a fascinating way.