Multiplying Complex Numbers: A Step-by-Step Guide

Complex numbers might seem intimidating at first, but multiplying them is a straightforward process once you understand the basics. In this guide, we'll break down how to multiply the complex number $-5i$ by the complex number $(-3 + i)$ and express the result in standard form (a + bi), where 'a' represents the real part and 'b' represents the imaginary part. Let's dive in and simplify this expression together!

Understanding Complex Numbers

Before we jump into the multiplication, let's quickly recap what complex numbers are. A complex number is a number that can be expressed in the form a + bi, where:

• a is the real part.
• b is the imaginary part.
• i is the imaginary unit, defined as the square root of -1 ($i = \sqrt{-1}$), which means $i^2 = -1$.

The expression we're working with, $-5i(-3 + i)$, involves multiplying a pure imaginary number ($-5i$) by a complex number $(-3 + i)$. Remember, a pure imaginary number is a complex number where the real part is zero.

Why are Complex Numbers Important?

You might be wondering, where do complex numbers fit into the real world? Despite their name, complex numbers are incredibly useful in various fields, including:

• Electrical Engineering: Analyzing AC circuits.
• Quantum Mechanics: Describing the behavior of particles.
• Fluid Dynamics: Modeling fluid flow.
• Mathematics: Solving polynomial equations and exploring number theory.

Complex numbers provide a powerful tool for solving problems that cannot be solved using real numbers alone. Now that we have a grasp of what complex numbers are and their significance, let's proceed with multiplying them.

Step-by-Step Multiplication of $-5i(-3 + i)$

To multiply $-5i$ by $(-3 + i)$, we'll use the distributive property, just like we do with regular algebraic expressions. The distributive property states that a(b + c) = ab + ac. Applying this to our problem:

$-5i(-3 + i) = (-5i) * (-3) + (-5i) * (i)$

Let's break this down step by step:

Step 1: Multiply $-5i$ by $-3$

$(-5i) * (-3) = 15i$

Remember, multiplying two negative numbers results in a positive number. So, $-5$ multiplied by $-3$ is $15$, and we keep the imaginary unit i.

Step 2: Multiply $-5i$ by $i$

$(-5i) * (i) = -5i^2$

Here, we're multiplying $-5i$ by i. This gives us $-5$ times $i$ squared ($-5i^2$).

Step 3: Substitute $i^2$ with $-1$

Now comes a crucial step: we know that $i^2 = -1$. So, we can substitute $i^2$ in our expression:

$-5i^2 = -5 * (-1) = 5$

Step 4: Combine the Results

Now that we've multiplied both parts, let's combine the results from Step 1 and Step 3:

$15i + 5$

Step 5: Express in Standard Form

The standard form of a complex number is a + bi, where a is the real part and b is the imaginary part. Our current expression, $15i + 5$, has the imaginary part first and then the real part. Let's rearrange it to match the standard form:

$5 + 15i$

And there you have it! We've successfully multiplied $-5i$ by $(-3 + i)$ and expressed the answer in standard complex number form.

Key Concepts in Complex Number Multiplication

Mastering complex number multiplication requires a firm grasp of a few key concepts. Let's solidify our understanding:

1. The Distributive Property

The distributive property is your best friend when multiplying complex numbers. It allows you to break down the multiplication into simpler steps.

Example:

To multiply $(2 + 3i)(4 - i)$, distribute each term in the first complex number across the terms in the second complex number:

$(2 + 3i)(4 - i) = 2(4) + 2(-i) + 3i(4) + 3i(-i)$

2. The Imaginary Unit i

The imaginary unit i is the cornerstone of complex numbers. Remember that $i = \sqrt{-1}$ and, crucially, $i^2 = -1$. This relationship is essential for simplifying expressions.

Example:

When you encounter $i^2$ during multiplication, always replace it with $-1$:

$(3i)^2 = 3^2 * i^2 = 9 * (-1) = -9$

3. Standard Form a + bi

The standard form a + bi provides a consistent way to represent complex numbers, making it easier to compare and perform operations on them. Always aim to express your final answer in this form.

Example:

If you end up with an expression like $7i - 4$, rearrange it to the standard form:

$-4 + 7i$

4. Multiplying by a Real Number

Multiplying a complex number by a real number is straightforward – simply distribute the real number to both the real and imaginary parts of the complex number.

Example:

To multiply $3(2 + 5i)$, distribute the $3$:

$3(2 + 5i) = 3 * 2 + 3 * 5i = 6 + 15i$

5. Multiplying by a Pure Imaginary Number

When multiplying by a pure imaginary number (like $-5i$ in our original problem), the process is similar to distributing, but be mindful of the i terms and the fact that $i^2 = -1$.

Example:

To multiply $-2i(1 - 4i)$, distribute the $-2i$:

$-2i(1 - 4i) = -2i * 1 + (-2i) * (-4i) = -2i + 8i^2$

Then, replace $i^2$ with $-1$:

$-2i + 8(-1) = -2i - 8$

Finally, express in standard form:

$-8 - 2i$

By understanding these key concepts, you'll be well-equipped to tackle a wide range of complex number multiplication problems. Practice is key, so try working through various examples to solidify your skills.

Common Mistakes to Avoid

When multiplying complex numbers, it's easy to make a few common mistakes. Being aware of these pitfalls can help you avoid them and ensure accurate results.

1. Forgetting that $i^2 = -1$

This is the most critical point to remember! Whenever you encounter $i^2$, always replace it with $-1$. Forgetting this will lead to incorrect simplifications.

Example of a Mistake:

Incorrect: $(2i)(3i) = 6i^2 = 6$

Correct: $(2i)(3i) = 6i^2 = 6(-1) = -6$

2. Incorrectly Distributing

Just like with regular algebraic expressions, you need to distribute carefully. Make sure each term in the first complex number is multiplied by each term in the second complex number.

Example of a Mistake:

Incorrect: $2i(-3 + i) = -6i + 2i$

Correct: $2i(-3 + i) = -6i + 2i^2 = -6i + 2(-1) = -2 - 6i$

3. Not Expressing the Final Answer in Standard Form

Remember, the standard form is a + bi. Make sure your final answer is in this form, with the real part first and the imaginary part second.

Example of a Mistake:

Leaving the answer as $9i + 4$

Correct: Expressing it as $4 + 9i$

4. Sign Errors

Be extra careful with signs, especially when dealing with negative numbers and the imaginary unit i. A small sign error can change the entire result.

Example of a Mistake:

Incorrect: $-3i(-2 - i) = 6i - 3i^2 = 6i - 3(-1) = 6i + 3$

Correct: $-3i(-2 - i) = 6i + 3i^2 = 6i + 3(-1) = -3 + 6i$

5. Overcomplicating the Process

Complex number multiplication is methodical but not overly complicated. Break it down step by step, focus on the distributive property and the value of $i^2$, and you'll be fine.

By keeping these common mistakes in mind, you can improve your accuracy and confidence when multiplying complex numbers.

Practice Problems

To truly master complex number multiplication, practice is essential. Here are a few problems you can try on your own. Work through them step by step, and don't forget to express your answers in standard form (a + bi).

1. $(3 + 2i)(1 - i)$
2. $-4i(2 + 5i)$
3. $(4 - 3i)^2$ (Hint: This means $(4 - 3i)(4 - 3i)$)
4. $(1 + i)(1 - i)$
5. $(5 - 2i)(5 + 2i)$

Check your answers by working through them carefully, and if you encounter any difficulties, review the steps and concepts we've covered in this guide. The more you practice, the more comfortable you'll become with multiplying complex numbers.

Conclusion

Multiplying complex numbers might have seemed daunting at first, but by understanding the key concepts – the distributive property, the imaginary unit i, and standard form – you can confidently tackle these problems. Remember to break down the process into manageable steps, watch out for common mistakes, and practice regularly. With a solid foundation, you'll be well-prepared to work with complex numbers in various mathematical and real-world contexts.

For further exploration and practice, consider checking out resources like Khan Academy's complex number section. Happy multiplying!