Solving Complex Number Equations: Unveiling the Steps and Understanding the Logic

In the complex number realm, equations can sometimes appear cryptic when presented out of context. This article aims to demystify how to solve equations involving complex numbers by breaking down the process step-by-step. Specifically, we will explore the equation:

1 - 2i/12 - 2i2 1 - 2i/14

To comprehend the solution to this equation and the logic behind it, it is essential to revisit some fundamental concepts of complex numbers and algebraic manipulations. By understanding these foundational elements, readers will be better equipped to tackle similar problems with ease.

Understanding the Basics

To delve into the intricacies of the given equation, let's first review some important basic concepts:

1. Imaginary Numbers

Imaginary numbers are a subset of complex numbers, where the square root of a negative number is involved. The most common imaginary number is i, which is defined as the square root of -1. In other words, i2 -1.

2. Complex Numbers

Complex numbers consist of a real part and an imaginary part. They are expressed in the form a bi, where a is the real part, and b is the imaginary part. In the equation at hand, -2i is the imaginary part.

3. Algebraic Manipulation and Simplification

Algebraic manipulation involves operations such as addition, subtraction, multiplication, and division applied to expressions. Simplification involves reducing an expression to its simplest form.

Solving the Equation Step-by-Step

Let's break down the given equation and solve it step-by-step:

1 - 2i/12 - 2i2 1 - 2i/(12 - 2i2) i2 -1

Step 1: Simplify the Denominator

The main challenge in this equation is the denominator. Let's start by simplifying it:

12 - 2i2

Since i2 -1, we substitute -1 for i2:

12 - 2(-1) 1 2 3

Step 2: Rewrite the Equation

Now that we've simplified the denominator, the equation becomes:

1 - 2i/3

Let's simplify this further:

1 - 2i/3 1 - (2/3)i

Step 3: Equate to Given Result

This simplified equation, 1 - 2i/3, does not equate to 1 - 2i/14 as stated in the initial problem. Let's investigate why this might be the case:

1 - 2i/12 - 2i2 ≠ 1 - 2i/14

The final step in the equation states:

1 - 2i/(12 - 2i2) i2 -1

This is because when we simplify the denominator (12 - 2i2), we get 3, not 14. Therefore, the final simplified form is indeed -1, which matches the right-hand side of the equation.

Conclusion

Through this detailed step-by-step breakdown, we have unveiled the logic and methodical approach required to solve complex number equations. Understanding the basics of imaginary numbers, complex numbers, and algebraic manipulation is crucial for tackling more intricate problems. If you encounter similar equations, remember to simplify the denominator correctly, substitute i2 with -1, and perform the operations carefully.

Related Keywords

Complex number Imaginary number Algebraic manipulation