%5E2.jpeg "(1-2i)^2")
Understanding the Square of a Complex Number: (1-2i)^2
This article explores the squaring of a complex number, specifically (1-2i)^2. We'll delve into the process of calculating this and understand the result in the context of complex numbers.
Understanding Complex Numbers
Complex numbers are numbers that extend the real number system by including the imaginary unit 'i', where i² = -1. They are expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.
Squaring the Complex Number
To square (1-2i)², we simply multiply it by itself:
(1-2i)² = (1-2i)(1-2i)
Now, we can use the distributive property (or FOIL method) to expand the expression:
(1-2i)(1-2i) = 1(1) + 1(-2i) - 2i(1) - 2i(-2i)
Simplifying the terms:
= 1 - 2i - 2i + 4i²
Remembering that i² = -1, we can substitute:
= 1 - 2i - 2i + 4(-1)
Combining the real and imaginary terms:
= (1 - 4) + (-2 - 2)i
Finally, we get:
** (1-2i)² = -3 - 4i**
Visualizing the Result
Complex numbers can be represented graphically on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Squaring a complex number often results in a rotation and scaling of the original number on the complex plane.
In this case, squaring (1-2i) results in the complex number -3 - 4i. This visually means that the complex number is rotated and scaled on the complex plane.
Conclusion
By understanding the basic operations with complex numbers, we can successfully square a complex number like (1-2i)², arriving at the result -3 - 4i. This process involves utilizing the distributive property, recognizing the value of i² and simplifying the expression. The result can be visualized on the complex plane, offering a deeper understanding of the effect of squaring complex numbers.