Simplifying Complex Numbers: A Step-by-Step Guide

by Alex Johnson 50 views

Have you ever stumbled upon an equation filled with 'i's and wondered what to do with it? You're not alone! Complex numbers, with that intriguing imaginary unit 'i', can seem daunting at first. But fear not! This guide will break down the process of simplifying complex expressions, using the example i(−3+5i)(3−4i)i(-3+5i)(3-4i) as our roadmap. We'll walk through each step in a friendly, easy-to-understand way, so you can confidently tackle similar problems in the future. Understanding complex numbers is very important in mathematics, especially in algebra and calculus. Without understanding the imaginary unit i, it may be very difficult to understand complex mathematical concepts.

Understanding Complex Numbers

Before diving into the simplification, let's quickly recap what complex numbers are. A complex number is essentially a combination of a real number and an imaginary number. It's written in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit. The imaginary unit 'i' is defined as the square root of -1 (i=−1i = \sqrt{-1}). This seemingly simple definition unlocks a whole new realm of mathematical possibilities.

Complex numbers might sound abstract, but they have practical applications in various fields, including electrical engineering, quantum mechanics, and even computer graphics. They allow us to solve equations that have no solutions within the realm of real numbers alone. For example, the equation x2+1=0x^2 + 1 = 0 has no real solutions, but it has two complex solutions: i and -i. The beauty of complex numbers lies in their ability to provide solutions where real numbers fall short.

Working with complex numbers involves similar operations to working with real numbers, such as addition, subtraction, multiplication, and division. However, we need to keep in mind the special property of 'i': i2=−1i^2 = -1. This property is crucial when simplifying expressions involving complex numbers, as it allows us to eliminate 'i' from the denominator or to reduce higher powers of 'i'. Mastering these operations is essential for anyone delving deeper into mathematics and its applications. Complex numbers provide a powerful tool for solving a wide range of problems in various scientific and engineering disciplines, making their understanding indispensable.

Step 1: Distribute 'i' into the First Parenthesis

Our mission is to simplify i(−3+5i)(3−4i)i(-3+5i)(3-4i). The first logical step is to distribute the 'i' into the first parenthesis, which is (-3 + 5i). This means we'll multiply each term inside the parenthesis by 'i'. This is very similar to how we distribute a constant in a regular algebraic expression. The distributive property states that a(b + c) = ab + ac, and this applies equally well to complex numbers.

So, let's do the math:

  • i * (-3) = -3i
  • i * (5i) = 5i²

Remember that crucial rule: i² = -1. This is the key that unlocks the door to simplification. Substitute -1 for i²:

  • 5i² = 5 * (-1) = -5

Now, let’s rewrite our expression after this distribution and simplification. The expression i(-3 + 5i) becomes -3i - 5. It’s important to keep track of the signs and the order of operations to avoid any mistakes. This simplified form makes the next steps much easier to handle. By applying the distributive property and the rule for i², we've taken the first step towards simplifying the entire expression. This process highlights the importance of understanding the fundamental properties of complex numbers.

Step 2: Rewrite the Expression

After distributing 'i' and simplifying, our expression now looks like this: (-5 - 3i)(3 - 4i). Notice that we've rearranged the terms in the first parenthesis to have the real part (-5) first and the imaginary part (-3i) second. This is a standard practice when working with complex numbers and helps to keep things organized. It's similar to writing a complex number in its standard form, which is a + bi.

The order of terms might seem like a small detail, but it significantly enhances clarity and reduces the chance of errors in the subsequent steps. By convention, writing the real part first makes it easier to identify and combine like terms later on. Think of it as setting the stage for the main act – the multiplication of the two complex numbers. A well-organized expression is half the battle won in mathematical simplification. This step emphasizes the importance of methodical arrangement in mathematical problem-solving.

Step 3: Multiply the Two Complex Numbers

Now comes the core of the simplification process: multiplying the two complex numbers (-5 - 3i) and (3 - 4i). We can use the FOIL method (First, Outer, Inner, Last) to ensure we multiply each term in the first parenthesis by each term in the second parenthesis. This method provides a systematic approach to multiplying binomials, which is exactly what we have here with complex numbers.

Let's break it down:

  • First: (-5) * (3) = -15
  • Outer: (-5) * (-4i) = 20i
  • Inner: (-3i) * (3) = -9i
  • Last: (-3i) * (-4i) = 12i²

So, when we multiply the two complex numbers, we get: -15 + 20i - 9i + 12i². This is a crucial step where careful attention to signs and coefficients is paramount. Each term must be multiplied correctly to ensure the final result is accurate. The FOIL method provides a structured way to handle this, but it's essential to double-check each multiplication to avoid errors. This meticulous multiplication lays the groundwork for the subsequent simplification, where we'll combine like terms and use the property of i² to further reduce the expression.

Step 4: Simplify and Combine Like Terms

We're almost there! Our expression now reads: -15 + 20i - 9i + 12i². But we're not done until we've simplified it completely. The first thing to do is to deal with that i² term. Remember, i² = -1. So, we can substitute -1 for i²:

  • 12i² = 12 * (-1) = -12

Now our expression looks like this: -15 + 20i - 9i - 12. The next step is to combine like terms. This means grouping the real numbers together and the imaginary numbers together. Think of it as sorting your groceries: you put all the fruits together, all the vegetables together, and so on. Similarly, we'll combine the real parts and the imaginary parts of our complex number.

  • Real parts: -15 - 12 = -27
  • Imaginary parts: 20i - 9i = 11i

By combining like terms, we simplify the expression to its most concise form. This is a fundamental step in algebra and is essential for expressing mathematical results in their simplest terms. The ability to identify and combine like terms is a crucial skill for any mathematician or anyone working with mathematical expressions. This process ensures that the final answer is clear, easy to understand, and free from unnecessary complexity.

Step 5: Final Result

And there we have it! After all the distribution, multiplication, and simplification, we arrive at our final answer: -27 + 11i. This is the simplified form of the original expression, i(−3+5i)(3−4i)i(-3+5i)(3-4i). We've successfully transformed a seemingly complicated expression into a neat and tidy complex number in the standard form a + bi.

The final result represents the culmination of all the steps we've taken, from understanding complex numbers to applying the distributive property and simplifying like terms. It's a testament to the power of methodical problem-solving in mathematics. This process demonstrates how complex expressions can be broken down into manageable steps, leading to a clear and concise solution. The final answer, -27 + 11i, not only provides the solution to the problem but also showcases the elegance and precision of mathematical simplification.

Conclusion

Simplifying complex numbers might seem like a maze at first, but with a clear understanding of the basic principles and a step-by-step approach, it becomes a manageable and even enjoyable task. Remember the key concepts: the definition of 'i' (i=−1i = \sqrt{-1}), the property i² = -1, the distributive property, and the FOIL method. By mastering these tools, you can confidently navigate the world of complex numbers and tackle more advanced mathematical problems.

This example, i(−3+5i)(3−4i)i(-3+5i)(3-4i), serves as a great illustration of the process. By breaking it down into smaller steps, we were able to simplify the expression and arrive at the final answer: -27 + 11i. Practice is key to mastering any mathematical concept, so don't hesitate to try similar problems on your own. The more you practice, the more comfortable and confident you'll become in working with complex numbers. Keep exploring and keep simplifying!

For further learning on complex numbers, you can check out resources like Khan Academy's complex numbers section. This is an excellent resource for further expanding your knowledge and tackling even more complex problems.