Simplifying Complex Number Expressions Step-by-Step Guide

Hey guys! Let's dive into simplifying complex number expressions, specifically the expression $(6 + 3i)(1 + i)$. This might seem daunting at first, but we'll break it down step by step, making it super easy to understand. This comprehensive guide will not only help you solve this particular problem but also equip you with the skills to tackle similar complex number problems with confidence. We'll go through the fundamental concepts, the step-by-step solution, common pitfalls to avoid, and some practice problems to solidify your understanding. So, let's get started!

Understanding Complex Numbers

Before we jump into the solution, let's quickly recap what complex numbers are. Complex numbers are numbers that can be 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 (i.e., $i = \sqrt{-1}$). In this form:

• a is the real part of the complex number.
• b is the imaginary part of the complex number.
• i is the imaginary unit, where $i^2 = -1$.

The expression we are going to simplify, $(6 + 3i)(1 + i)$, involves multiplying two complex numbers. To do this, we'll use the distributive property (also known as the FOIL method), just like we do with binomials in algebra. This involves multiplying each term in the first complex number by each term in the second complex number and then simplifying the result. Understanding this foundation is crucial, as complex numbers are not just abstract mathematical concepts but have real-world applications in fields like electrical engineering, quantum mechanics, and signal processing. So, grasping the basics ensures you're well-prepared for more advanced topics later on.

When dealing with complex numbers, it's also essential to remember the powers of i. We know that $i = \sqrt{-1}$, so:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 * i = -i$
• $i^4 = i^2 * i^2 = (-1) * (-1) = 1$

These powers of i cycle through these four values, which is critical for simplifying expressions involving complex numbers. In our problem, we will encounter $i^2$, which simplifies to -1, a key step in reaching the final answer. Understanding these cyclical properties helps in simplifying more complex expressions and lays the groundwork for more advanced manipulations of complex numbers. The ability to quickly recall these powers of i can significantly speed up your problem-solving process and reduce the likelihood of errors.

Key Operations with Complex Numbers

Working with complex numbers involves several key operations, such as addition, subtraction, multiplication, and division. Each of these operations has its own set of rules, but they all build upon the fundamental structure of complex numbers. For our problem, we'll focus on multiplication, but it's helpful to have a general understanding of other operations as well.

1. Addition and Subtraction: To add or subtract complex numbers, you simply add or subtract the real parts and the imaginary parts separately. For example:

   • $(a + bi) + (c + di) = (a + c) + (b + d)i$
   • $(a + bi) - (c + di) = (a - c) + (b - d)i

2. Multiplication: To multiply complex numbers, you use the distributive property (FOIL method), just like with binomials. For example:

   • $(a + bi)(c + di) = a(c + di) + bi(c + di) = ac + adi + bci + bdi^2$. Remember, $i^2 = -1$, so you can simplify further to $(ac - bd) + (ad + bc)i$.

3. Division: Dividing complex numbers involves multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of $a + bi$ is $a - bi$. This process eliminates the imaginary part from the denominator, making it a real number. Understanding these operations provides a comprehensive view of how complex numbers can be manipulated and used in various contexts. Knowing the rules for each operation ensures you can handle any complex number problem that comes your way.

Step-by-Step Solution

Now that we've covered the basics of complex numbers and their operations, let's tackle the given expression: $(6 + 3i)(1 + i)$. We'll use the distributive property (FOIL method) to multiply these two complex numbers.

Step 1: Apply the Distributive Property (FOIL Method)

The FOIL method stands for First, Outer, Inner, Last. It's a handy way to remember how to multiply two binomials (or in this case, two complex numbers).

• First: Multiply the first terms in each complex number: $6 * 1 = 6$
• Outer: Multiply the outer terms: $6 * i = 6i$
• Inner: Multiply the inner terms: $3i * 1 = 3i$
• Last: Multiply the last terms: $3i * i = 3i^2$

So, $(6 + 3i)(1 + i) = 6 + 6i + 3i + 3i^2$ is the result of applying the distributive property. This expansion is a crucial step, as it sets the stage for simplifying the expression further. Each term must be accounted for to ensure accuracy. The FOIL method provides a structured approach, reducing the chance of missing any terms during multiplication.

Step 2: Simplify the Expression

Now, let's simplify the expression we obtained in the previous step: $6 + 6i + 3i + 3i^2$. The first thing we can do is combine the like terms, which in this case are the terms with i.

Combine the imaginary terms: $6i + 3i = 9i$

So our expression now looks like: $6 + 9i + 3i^2$

Next, we need to deal with the $i^2$ term. Remember that $i^2 = -1$, so we can substitute -1 for $i^2$:

$3i^2 = 3(-1) = -3$

Now our expression is: $6 + 9i - 3$. Substituting $i^2$ with -1 is a critical step in simplifying complex number expressions. It transforms the imaginary unit into a real number, allowing us to combine it with other real numbers in the expression. This substitution is based on the fundamental definition of the imaginary unit, and mastering this step is essential for accurate simplification.

Step 3: Combine Real Terms

The final step is to combine the real terms in our expression: $6 + 9i - 3$.

Combine the real terms: $6 - 3 = 3$

So, our simplified expression is: $3 + 9i$. This is the final simplified form of the complex number expression. By combining the real and imaginary parts, we express the result in the standard form of a complex number, a + bi. The process of combining like terms is a fundamental algebraic skill that applies to complex numbers as well, ensuring the final answer is in its simplest form.

Therefore, $(6 + 3i)(1 + i) = 3 + 9i$.

Common Mistakes to Avoid

When working with complex numbers, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results. Let's discuss some of these common errors:

1. Forgetting the Distributive Property: One of the most common mistakes is not correctly applying the distributive property (FOIL method) when multiplying complex numbers. Make sure to multiply each term in the first complex number by each term in the second complex number. Forgetting even one term can lead to an incorrect answer. To avoid this, always write out each step explicitly and double-check your work.

2. Incorrectly Simplifying $i^2$: Another frequent error is misinterpreting $i^2$. Remember that $i^2 = -1$, not 1. This substitution is crucial for simplifying complex number expressions. Failing to substitute correctly can lead to significant errors in your final answer. Always remember the fundamental definition of $i$ and its powers.

3. Combining Real and Imaginary Terms Incorrectly: Real and imaginary terms cannot be combined directly. For example, you cannot add 3 and 9i to get 12i. The real part and the imaginary part must be kept separate. The final answer should always be in the form a + bi. Mixing these terms is a common mistake that can be easily avoided by careful attention to the structure of complex numbers.

4. Arithmetic Errors: Simple arithmetic mistakes, such as incorrect addition or subtraction, can also lead to wrong answers. Double-check your calculations at each step to minimize these errors. Even small arithmetic errors can propagate through the problem and result in an incorrect final answer. Using a calculator for complex arithmetic can also help reduce these mistakes.

By being mindful of these common mistakes, you can significantly improve your accuracy when working with complex numbers. Always take your time, show your work, and double-check each step to ensure you arrive at the correct solution.

Practice Problems

To solidify your understanding of simplifying complex number expressions, let's work through a few practice problems. These examples will help you apply the concepts we've discussed and build your confidence in tackling similar problems.

Practice Problem 1: Simplify $(2 - i)(3 + 2i)$

Solution:

1. Apply the distributive property (FOIL method):
   • $(2 - i)(3 + 2i) = 2(3) + 2(2i) - i(3) - i(2i) = 6 + 4i - 3i - 2i^2$
2. Simplify the expression:
   • Combine like terms: $6 + 4i - 3i - 2i^2 = 6 + i - 2i^2$
   • Substitute $i^2$ with -1: $6 + i - 2(-1) = 6 + i + 2$
3. Combine real terms:
   • $6 + i + 2 = 8 + i$

So, $(2 - i)(3 + 2i) = 8 + i$

Practice Problem 2: Simplify $(-1 + 4i)(2 - 3i)$

Solution:

1. Apply the distributive property (FOIL method):
   • $(-1 + 4i)(2 - 3i) = -1(2) - 1(-3i) + 4i(2) + 4i(-3i) = -2 + 3i + 8i - 12i^2$
2. Simplify the expression:
   • Combine like terms: $-2 + 3i + 8i - 12i^2 = -2 + 11i - 12i^2$
   • Substitute $i^2$ with -1: $-2 + 11i - 12(-1) = -2 + 11i + 12$
3. Combine real terms:
   • $-2 + 11i + 12 = 10 + 11i$

So, $(-1 + 4i)(2 - 3i) = 10 + 11i$

Practice Problem 3: Simplify $(5 + 2i)(5 - 2i)$

Solution:

1. Apply the distributive property (FOIL method):
   • $(5 + 2i)(5 - 2i) = 5(5) + 5(-2i) + 2i(5) + 2i(-2i) = 25 - 10i + 10i - 4i^2$
2. Simplify the expression:
   • Combine like terms: $25 - 10i + 10i - 4i^2 = 25 - 4i^2$
   • Substitute $i^2$ with -1: $25 - 4(-1) = 25 + 4$
3. Combine real terms:
   • $25 + 4 = 29$

So, $(5 + 2i)(5 - 2i) = 29$

These practice problems demonstrate the step-by-step process of simplifying complex number expressions. By working through these examples, you can reinforce your understanding and improve your problem-solving skills. Remember to apply the distributive property, simplify terms, and substitute $i^2$ with -1 to arrive at the correct answer.

Conclusion

In this guide, we've walked through the process of simplifying the complex number expression $(6 + 3i)(1 + i)$. We started with a quick review of complex numbers and their properties, then broke down the problem into manageable steps. We applied the distributive property (FOIL method), simplified the expression by combining like terms and substituting $i^2$ with -1, and arrived at the final answer: $3 + 9i$. Hopefully, after this extensive breakdown and practice problems, you guys can solve any complex numbers problem! Remember to practice regularly and apply what you've learned to various problems. With consistent effort, you'll become more confident and proficient in working with complex numbers.