%5E2.jpeg "(1-i)^2")
Exploring the Square of (1 - i)
In the realm of complex numbers, the expression (1 - i)² often arises in various mathematical contexts. Let's delve into understanding its value and the significance it holds.
Understanding Complex Numbers
Before we embark on the calculation, let's refresh our memory on complex numbers. Complex numbers are 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).
Calculating (1 - i)²
To square (1 - i), we can apply the distributive property or simply use the formula:
(a - b)² = a² - 2ab + b²
Substituting (1 - i) for (a - b):
(1 - i)² = 1² - 2(1)(i) + i²
Simplifying further:
(1 - i)² = 1 - 2i - 1
Therefore, (1 - i)² = -2i
Significance of the Result
The result, -2i, is a purely imaginary number. This demonstrates that squaring a complex number can result in a complex number with a different form, potentially eliminating the real component.
Visual Representation
Complex numbers can be visualized on the complex plane, where the real part is represented on the horizontal axis and the imaginary part on the vertical axis.
The number (1 - i) lies in the fourth quadrant of the complex plane, with coordinates (1, -1). Squaring this number effectively rotates it by 90 degrees counterclockwise and doubles its magnitude. This results in the purely imaginary number -2i, which lies on the negative imaginary axis.
Conclusion
The calculation of (1 - i)² provides a valuable insight into the behavior of complex numbers under squaring. It showcases how complex numbers can transform in the complex plane, transitioning from a combination of real and imaginary components to a purely imaginary number in this case. Understanding such manipulations is crucial for solving complex equations and analyzing various mathematical models that employ complex numbers.