%5E2.jpeg "(1+4i)^2")
Squaring Complex Numbers: (1 + 4i)²
This article will explore the process of squaring the complex number (1 + 4i).
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 (i² = -1).
Squaring (1 + 4i)
To square (1 + 4i), we simply multiply it by itself:
(1 + 4i)² = (1 + 4i)(1 + 4i)
Now, we expand the expression using the distributive property (FOIL method):
(1 + 4i)(1 + 4i) = 1(1) + 1(4i) + 4i(1) + 4i(4i)
Simplifying the terms:
= 1 + 4i + 4i + 16i²
Since i² = -1, we substitute:
= 1 + 4i + 4i + 16(-1)
Combining like terms:
= 1 + 8i - 16
= -15 + 8i
Conclusion
Therefore, the square of the complex number (1 + 4i) is -15 + 8i. This demonstrates that squaring a complex number results in another complex number, where the real and imaginary components are calculated based on the original number's components and the property of the imaginary unit, i².