Multiplying Complex Numbers (9 - 4i)(2 + 5i) A Step-by-Step Guide

Understanding Complex Number Multiplication

In the realm of mathematics, especially when dealing with complex numbers, the operation of multiplication might seem a bit intricate at first. However, once you grasp the fundamental principles, it becomes a straightforward process. Complex numbers, which extend the real number system by including the imaginary unit i (where i² = -1), are expressed in the form a + bi, where a and b are real numbers. The term a represents the real part, and bi represents the imaginary part. Multiplying complex numbers involves a combination of the distributive property and the unique property of i. This detailed exploration will guide you through the multiplication of the complex numbers (9 - 4i) and (2 + 5i), providing a clear understanding of each step involved. We will delve into the mechanics of the operation, ensuring you can confidently tackle similar problems in the future. This foundation is crucial for more advanced topics in complex analysis and other related fields. This process requires meticulous attention to detail, as any small error in the application of the distributive property or the handling of the imaginary unit can lead to an incorrect result. Therefore, each step is explained thoroughly, reinforcing the underlying concepts. Mastering this skill is essential for anyone venturing into higher-level mathematics and engineering, where complex numbers play a pivotal role. As we proceed, you'll notice the similarities between complex number multiplication and the multiplication of binomials in algebra, highlighting the interconnectedness of mathematical concepts. This understanding not only enhances your computational abilities but also deepens your appreciation for the elegance and consistency of mathematical principles.

Step-by-Step Multiplication of (9 - 4i)(2 + 5i)

To effectively multiply the complex numbers (9 - 4i) and (2 + 5i), we will employ the distributive property, a fundamental concept in algebra. This property, often remembered by the acronym FOIL (First, Outer, Inner, Last), provides a systematic way to multiply two binomials. In this case, the complex numbers (9 - 4i) and (2 + 5i) can be treated as binomials, allowing us to apply the same principle. Let's break down the process step by step, ensuring a clear and thorough understanding. First, we multiply the first terms of each complex number: 9 multiplied by 2. Then, we multiply the outer terms: 9 multiplied by 5i. Next, we multiply the inner terms: -4i multiplied by 2. Finally, we multiply the last terms: -4i multiplied by 5i. Each of these individual multiplications contributes to the final product, and by following the FOIL method, we ensure that no term is overlooked. This method not only simplifies the multiplication process but also minimizes the chances of errors. By carefully tracking each term and its corresponding sign, we can confidently arrive at the correct result. Furthermore, this step-by-step approach helps in visualizing the distribution of terms, making it easier to understand the underlying algebraic principles. The result obtained from this initial application of the distributive property is a complex expression that needs further simplification, particularly concerning the i² term, which we will address in the subsequent steps. This methodical approach is crucial for maintaining accuracy and building a strong foundation in complex number arithmetic.

1. First: Multiply the first terms: 9 * 2 = 18
2. Outer: Multiply the outer terms: 9 * (5i) = 45i
3. Inner: Multiply the inner terms: (-4i) * 2 = -8i
4. Last: Multiply the last terms: (-4i) * (5i) = -20i²

Simplifying the Expression

After applying the distributive property, we now have the expression 18 + 45i - 8i - 20i². The next crucial step is to simplify this expression by combining like terms and addressing the i² term. This simplification process is essential because it transforms the expression into its standard complex number form, a + bi, where a represents the real part and b represents the imaginary part. To begin, let's focus on the imaginary terms, 45i and -8i. These terms can be combined because they both involve the imaginary unit i. Adding their coefficients (45 and -8) gives us 37i. Now, let's turn our attention to the i² term. Recall that i is defined as the square root of -1, which means i² is equal to -1. This fundamental property of the imaginary unit is key to simplifying expressions involving complex numbers. Replacing i² with -1 in the expression -20i² gives us -20 * (-1), which equals 20. This transformation converts the imaginary term -20i² into a real number, which can then be combined with the other real number in the expression. The ability to manipulate the i² term is a cornerstone of complex number arithmetic, and understanding its significance is vital for accurately simplifying complex expressions. This step not only reduces the complexity of the expression but also brings it closer to the standard form, making it easier to interpret and use in further calculations. The simplified expression now consists of a combination of real and imaginary terms, paving the way for the final step of expressing the result in the form a + bi.

We have: 18 + 45i - 8i - 20i²

Remember that i² = -1. Substitute -1 for i²: 18 + 45i - 8i - 20(-1)

This simplifies to: 18 + 45i - 8i + 20

Combining Like Terms

The expression 18 + 45i - 8i + 20 now consists of both real and imaginary terms that need to be combined to reach the final simplified form. The process of combining like terms is a fundamental algebraic operation that involves grouping terms with similar variables or, in this case, similar units (real or imaginary). To begin, let's identify the real terms in the expression: 18 and 20. These terms are both constants and can be directly added together. Adding 18 and 20 gives us 38, which will be the real part of our final complex number. Next, we focus on the imaginary terms: 45i and -8i. As we discussed earlier, these terms can be combined by adding their coefficients. Adding 45 and -8 gives us 37, resulting in the imaginary term 37i. Now that we have combined both the real and imaginary terms, we can express the simplified complex number in the standard form a + bi. This form clearly separates the real and imaginary components, making it easy to visualize and interpret the complex number. The real part, 38, represents the horizontal component on the complex plane, while the imaginary part, 37i, represents the vertical component. Understanding how to combine like terms is not only essential for simplifying complex expressions but also for various other mathematical operations, such as solving equations and manipulating algebraic expressions. This step solidifies the connection between algebraic principles and complex number arithmetic, reinforcing the broader mathematical context.

Combine the real parts: 18 + 20 = 38

Combine the imaginary parts: 45i - 8i = 37i

Final Result: Expressing in a + bi Form

After simplifying the expression and combining like terms, we arrive at the final result, which represents the product of the complex numbers (9 - 4i) and (2 + 5i). The simplified form is 38 + 37i, which is expressed in the standard complex number notation a + bi. In this notation, a represents the real part of the complex number, and b represents the imaginary part. In our result, 38 is the real part, and 37 is the coefficient of the imaginary unit i. This standard form is crucial because it allows for easy comparison and further manipulation of complex numbers. When a complex number is expressed in the a + bi form, it can be readily plotted on the complex plane, where the real part corresponds to the horizontal axis and the imaginary part corresponds to the vertical axis. This geometric representation provides a visual understanding of complex numbers and their operations. Furthermore, the a + bi form is essential for performing other complex number operations, such as addition, subtraction, division, and finding the magnitude and argument of a complex number. The ability to confidently express complex numbers in this form is a fundamental skill in complex analysis and various fields of engineering and physics, where complex numbers are widely used. This final step not only provides the answer to our initial multiplication problem but also reinforces the importance of standard notation in mathematics for clarity and consistency. The result, 38 + 37i, is a single complex number that encapsulates the combined effect of the original two complex numbers.

Therefore, (9 - 4i)(2 + 5i) = 38 + 37i

Conclusion

In conclusion, the multiplication of complex numbers, as demonstrated with the example (9 - 4i)(2 + 5i), involves a systematic application of the distributive property followed by careful simplification. The process begins with expanding the product using the FOIL method (First, Outer, Inner, Last), ensuring that each term in the first complex number is multiplied by each term in the second complex number. This results in an expression containing both real and imaginary terms, including the crucial i² term. The next critical step is simplifying the expression by substituting -1 for i², which transforms the imaginary term into a real number. This substitution is a fundamental aspect of complex number arithmetic and allows us to combine like terms effectively. By combining the real parts and the imaginary parts separately, we arrive at the simplified complex number in the standard form a + bi. This form clearly delineates the real and imaginary components, making the result easily interpretable and usable for further calculations. The final result, 38 + 37i, represents the product of the two original complex numbers and is a single complex number in its simplest form. This entire process highlights the importance of understanding the properties of complex numbers, particularly the definition of i and its square, as well as the application of algebraic principles like the distributive property and combining like terms. Mastering these skills is essential for anyone working with complex numbers in mathematics, engineering, or other scientific disciplines. The ability to confidently multiply complex numbers is a building block for more advanced concepts and applications in complex analysis.