Dividing Complex Numbers | Square Root of minus 1 | i = √-1

This video demonstrates how to divide complex numbers and express the result in the standard form a + bi, where i = √-1. The key technique is to multiply both the numerator and denominator of the complex fraction by the complex conjugate of the denominator. This process eliminates the imaginary part from the denominator, simplifying the division. Here's why it works: Multiplying the numerator and denominator by the same value is equivalent to multiplying by 1, which doesn't change the value of the expression. By using the complex conjugate, we leverage the difference of squares pattern: (a + bi)(a - bi) = a² - (bi)² = a² + b². This results in a real number in the denominator. The complex conjugate of a complex number a + bi is a - bi. We simply change the sign of the imaginary part. This video uses four examples to illustrate the process, covering various scenarios: Basic Division: A straightforward example like (2 + 3i) / (1 - i) demonstrates the core method. Pure Imaginary Denominator: An example such as (5 + i) / (2i) shows how to handle cases where the denominator is purely imaginary. This often involves multiplying by i/i or -2i/-2i. Larger Coefficients: An example with larger numbers reinforces that the process remains the same, even with more complex arithmetic. Simplifying the Result: Emphasis is placed on expressing the final answer in the standard form a + bi, clearly separating the real and imaginary components. The video also highlights common student errors, including: Failure to distribute correctly when multiplying complex numbers. Incorrect simplification of i² = -1. Not expressing the final result in the standard a + bi form. Visual aids are used throughout the video to enhance understanding, such as clearly writing out complex conjugates, highlighting multiplication steps, and boxing the final answer. By covering these key points and addressing potential pitfalls, this video provides a clear and effective tutorial on dividing complex numbers.