Multiplying Complex Numbers A Step By Step Solution For (2 + 4i)(7 - 5i)

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Introduction to Complex Number Multiplication

In the realm of mathematics, complex numbers extend the familiar number system by incorporating an imaginary unit, denoted as 'i', which is defined as the square root of -1. These numbers, expressed in the form a + bi (where 'a' and 'b' are real numbers), find applications in diverse fields such as electrical engineering, quantum mechanics, and signal processing. Multiplying complex numbers involves a process similar to multiplying binomials, with the crucial difference of simplifying any occurrences of i² to -1. This article will provide a comprehensive, step-by-step guide to multiplying the complex numbers (2 + 4i) and (7 - 5i), ensuring clarity and understanding for readers of all backgrounds. Understanding complex number multiplication is essential for anyone delving into advanced mathematical concepts and their real-world applications. The ability to accurately perform these calculations opens doors to solving complex problems in various scientific and engineering disciplines. Therefore, a thorough grasp of this fundamental operation is paramount for success in these fields.

Step-by-Step Multiplication of (2 + 4i)(7 - 5i)

The problem at hand is to multiply two complex numbers: (2 + 4i) and (7 - 5i). To accomplish this, we will employ the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first complex number is multiplied by each term in the second complex number. Let's break down the process step by step:

  1. First: Multiply the first terms of each complex number. In this case, it's 2 multiplied by 7, which equals 14. This is the initial step in expanding the product of the two complex numbers. Accurately multiplying the first terms sets the foundation for the subsequent steps.
  2. Outer: Multiply the outer terms of the complex numbers. This involves multiplying 2 by -5i, resulting in -10i. The term -10i represents the imaginary component derived from this particular multiplication. Correctly identifying and multiplying the outer terms is crucial for arriving at the correct final answer.
  3. Inner: Multiply the inner terms of the complex numbers. This means multiplying 4i by 7, which gives us 28i. The term 28i is another imaginary component that needs to be accounted for. Ensuring the accurate multiplication of the inner terms is vital for the overall correctness of the solution.
  4. Last: Multiply the last terms of each complex number. Here, we multiply 4i by -5i, yielding -20i². This term is particularly important because it involves i², which will need to be simplified in the next step. The product of the last terms, -20i², is a key component in the final calculation.

By meticulously following these steps, we expand the product of the complex numbers, setting the stage for simplification and arriving at the final answer. The application of the distributive property, or FOIL method, ensures that all terms are accounted for and accurately multiplied.

Simplifying the Expression

After applying the distributive property (FOIL) to (2 + 4i)(7 - 5i), we arrive at the expression: 14 - 10i + 28i - 20i². The next crucial step is to simplify this expression. Simplification in complex number arithmetic involves two primary actions: combining like terms and dealing with the imaginary unit 'i'.

  1. Combining Like Terms: We identify and combine the real and imaginary terms separately. The real term is 14, which stands alone for now. The imaginary terms are -10i and +28i. Combining these, we get -10i + 28i = 18i. This step reduces the number of terms and makes the expression more manageable.
  2. Simplifying i²: The term -20i² contains the imaginary unit squared. Recall that i is defined as the square root of -1, which means i² is equal to -1. Therefore, we substitute i² with -1 in the expression. This gives us -20 * (-1) = 20. This substitution is a fundamental step in simplifying complex number expressions.

By performing these simplifications, we transform the expression 14 - 10i + 28i - 20i² into a more concise and understandable form. The combination of like terms and the substitution of i² with -1 are essential techniques in complex number arithmetic. These steps bring us closer to the final answer in the standard form of a complex number, which is a + bi.

Final Calculation and the Standard Form

Following the simplification process, we now have the terms 14, 18i, and 20. To arrive at the final answer, we need to combine these terms and express the result in the standard form of a complex number, which is a + bi, where 'a' represents the real part and 'b' represents the imaginary part.

  1. Combining Real Terms: We add the real terms together. In this case, we have 14 and 20. Adding these gives us 14 + 20 = 34. This result represents the real component of our final complex number.
  2. Identifying the Imaginary Term: The imaginary term we have is 18i. This term represents the imaginary component of our final complex number. It remains as 18i since there are no other imaginary terms to combine it with.
  3. Expressing in Standard Form: Now, we combine the real and imaginary parts to express the answer in the standard form a + bi. The real part is 34, and the imaginary part is 18i. Therefore, the final answer is 34 + 18i. This is the complex number in its standard form, with the real and imaginary components clearly delineated.

By carefully combining the real and imaginary terms, we arrive at the solution 34 + 18i. This final step is crucial in presenting the answer in a clear and universally understood format. The standard form a + bi allows for easy comparison and manipulation of complex numbers in various mathematical and scientific contexts.

Conclusion: The Result of Multiplying (2 + 4i)(7 - 5i)

In conclusion, after meticulously performing the multiplication and simplification steps, the result of multiplying the complex numbers (2 + 4i) and (7 - 5i) is 34 + 18i. This outcome is achieved by applying the distributive property (FOIL method), combining like terms, and simplifying the imaginary unit i². The process underscores the importance of understanding the properties of complex numbers and their arithmetic operations.

The journey from the initial expression to the final answer involves several key steps. First, the distributive property is applied to expand the product. Then, like terms are combined to simplify the expression. Crucially, the term i² is replaced with -1, which is a fundamental rule in complex number arithmetic. Finally, the real and imaginary parts are combined to present the answer in the standard form a + bi.

The ability to accurately multiply complex numbers is a fundamental skill in various fields, including mathematics, physics, and engineering. Complex numbers provide a powerful tool for solving problems that cannot be addressed using real numbers alone. Understanding their properties and operations is essential for anyone working in these disciplines. Therefore, mastering the multiplication of complex numbers, as demonstrated in this step-by-step guide, is a valuable asset for students and professionals alike.

By following the detailed explanation and steps provided, readers can confidently approach similar problems involving complex number multiplication. The clear, methodical approach ensures that the underlying concepts are understood, leading to a deeper appreciation of complex number arithmetic and its applications.