What is a method for rationalizing the imaginary unit i in a fraction?

To understand why multiplying by ((a - bi)/(a - bi)) is a valid method for rationalizing a fraction that contains the imaginary unit (i), it is useful to recall the nature of complex numbers. Rationalizing typically refers to the process of eliminating roots or irrational numbers from the denominator. In this case, the goal is to eliminate the imaginary unit in the denominator.

When you multiply a complex fraction by ((a - bi)/(a - bi)), you utilize a fundamental property of complex numbers known as the conjugate. The conjugate of a complex number (a + bi) is (a - bi). When these two expressions are multiplied, they yield a real number:

[

(a + bi)(a - bi) = a^2 + b^2

]

Thus, multiplying the numerator and the denominator by ((a - bi)) effectively removes the imaginary part from the denominator because the denominator becomes (a^2 + b^2), a purely real number.

This transformation not only rationalizes the fraction but also maintains the value of the expression since you are essentially multiplying by 1, represented as ((a - bi)/(a - bi)), which does not change the