Multiplying Complex Numbers - Step By Step Solution

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This article delves into the fascinating world of complex numbers, focusing on the multiplication of complex expressions. We will specifically address the problem of simplifying the expression -4i(-7 + 6i) and present the solution in the standard form of a complex number, which is a + bi. This exploration will not only provide a step-by-step solution to the given problem but also offer a comprehensive understanding of the underlying principles of complex number multiplication.

Complex Numbers: A Foundation

Before diving into the multiplication process, it is crucial to establish a solid understanding of 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. The imaginary unit, denoted by i, is defined as the square root of -1 (i = √-1). This means that i² = -1, a fundamental property that plays a crucial role in complex number arithmetic.

In the complex number a + bi, a is referred to as the real part, and b is referred to as the imaginary part. For instance, in the complex number 3 + 4i, 3 is the real part, and 4 is the imaginary part. Complex numbers extend the realm of real numbers by incorporating this imaginary component, allowing us to solve equations and explore mathematical concepts that are not possible within the real number system alone.

Complex numbers find extensive applications in various fields, including electrical engineering, quantum mechanics, and signal processing. Their ability to represent oscillations and wave phenomena makes them indispensable tools in these domains.

Multiplying Complex Numbers: The Distributive Property

The multiplication of complex numbers follows the distributive property, which is a fundamental principle in algebra. The distributive property states that for any numbers a, b, and c, the following holds true:

a( b + c ) = a b + a c

This property extends seamlessly to complex numbers. When multiplying a complex number by another complex number or a real number, we distribute the multiplier across both the real and imaginary parts of the complex number. This process ensures that each term in the first complex number is multiplied by each term in the second complex number, leading to a correct result.

For example, to multiply (2 + 3i) by 4, we distribute the 4 to both the real part (2) and the imaginary part (3i):

4(2 + 3i) = 4 * 2 + 4 * 3i = 8 + 12i

This simple example illustrates the core concept of complex number multiplication using the distributive property. The same principle applies when multiplying two complex numbers, albeit with a slightly more elaborate process.

Step-by-Step Solution for -4i(-7 + 6i)

Now, let's tackle the given problem: -4i(-7 + 6i). We will apply the distributive property to multiply the complex numbers.

  1. Distribute -4i: -4i(-7 + 6i) = (-4i)(-7) + (-4i)(6i)

  2. Multiply the terms: (-4i)(-7) = 28i (-4i)(6i) = -24i²

  3. Substitute i² with -1: Recall that i² = -1. Substitute -1 for i² in the expression: -24i² = -24(-1) = 24

  4. Combine the terms: Now we have 28i + 24.

  5. Write in the standard form a + bi: Rearrange the terms to match the standard form of a complex number, a + bi: 24 + 28i

Therefore, the simplified form of -4i(-7 + 6i) is 24 + 28i. This is our final answer, expressed in the standard form of a complex number.

Common Mistakes to Avoid

When multiplying complex numbers, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  • Forgetting to distribute: Always ensure that you distribute the multiplier to both the real and imaginary parts of the complex number. Failing to do so will lead to an incorrect result.
  • Incorrectly handling i²: Remember that i² = -1. This substitution is crucial for simplifying the expression and obtaining the correct answer. Forgetting to substitute or miscalculating the value of i² is a frequent source of error.
  • Combining real and imaginary terms incorrectly: Real and imaginary terms are distinct and cannot be directly combined. You can only add or subtract real terms with real terms and imaginary terms with imaginary terms. Mixing them up will result in an incorrect simplification.

By being mindful of these common mistakes, you can improve your accuracy and confidence in multiplying complex numbers.

Practice Problems

To solidify your understanding of complex number multiplication, try these practice problems:

  1. Multiply (3 + 2i) by (1 - i).
  2. Simplify 5i(2 - 4i).
  3. Expand (-2 + i)².

Working through these problems will give you valuable experience and help you develop a strong grasp of the concepts involved.

Conclusion

Multiplying complex numbers involves applying the distributive property and simplifying the expression using the fundamental property i² = -1. By understanding these principles and avoiding common mistakes, you can confidently perform complex number multiplication and express the result in the standard form a + bi. Complex numbers are a fascinating and essential part of mathematics, with wide-ranging applications in various scientific and engineering fields. Mastering complex number arithmetic is a valuable skill that opens doors to a deeper understanding of these applications and the mathematical world as a whole. The problem -4i(-7 + 6i) simplifies to 24 + 28i, demonstrating the practical application of these principles.

By consistently practicing and applying these techniques, you can enhance your proficiency in complex number manipulation and tackle more complex mathematical challenges with ease and accuracy. This comprehensive guide has equipped you with the knowledge and skills necessary to confidently multiply complex numbers and appreciate their significance in mathematics and beyond.