Simplifying Complex Numbers Finding The Product Of I(6 + 7i)(6 - 7i)
In the realm of mathematics, complex numbers hold a special place, extending the familiar number system to encompass the imaginary unit, denoted by i, where i² = -1. These numbers, expressed in the form a + bi, where a and b are real numbers, open up a fascinating world of mathematical possibilities. Among the intriguing concepts within complex numbers is the notion of complex conjugates. The complex conjugate of a number a + bi is simply a - bi, and vice versa. The product of a complex number and its conjugate possesses a remarkable property: it always results in a real number. This property stems from the cancellation of the imaginary terms during multiplication, leaving only the sum of the squares of the real and imaginary parts. In this article, we will embark on a journey to explore this fascinating property by delving into the product of a specific complex number and its conjugate, unraveling the steps involved in simplifying the expression i(6 + 7i)(6 - 7i) and presenting the final result in standard form.
The expression i(6 + 7i)(6 - 7i) presents an opportunity to witness the interplay between complex numbers and their conjugates. To simplify this expression, we will first focus on the product of the two complex conjugates, (6 + 7i) and (6 - 7i). As mentioned earlier, the multiplication of complex conjugates yields a real number, and we will see this principle in action. The distributive property, often referred to as the FOIL method (First, Outer, Inner, Last), will guide us through the multiplication process. We will multiply the first terms, then the outer terms, followed by the inner terms, and finally the last terms. By carefully combining like terms and remembering that i² = -1, we will arrive at a simplified expression for the product of the conjugates. This simplified expression will then be multiplied by the imaginary unit, i, to complete the simplification of the original expression. The final result will be expressed in standard form, a + bi, where a and b are real numbers, showcasing the real and imaginary components of the complex number.
Step-by-Step Simplification: Unraveling the Expression
Let's embark on a step-by-step journey to simplify the expression i(6 + 7i)(6 - 7i). Our first task is to tackle the multiplication of the complex conjugates, (6 + 7i) and (6 - 7i). This is where the distributive property, or FOIL method, comes into play. We will carefully multiply each term in the first complex number by each term in the second complex number, ensuring that we account for all possible combinations. This meticulous approach will lay the foundation for simplifying the expression and revealing its underlying structure. As we proceed with the multiplication, we will pay close attention to the imaginary unit, i, and its properties. Specifically, we will remember that i² = -1, a crucial piece of information that will help us eliminate the imaginary component when multiplying the conjugates. By keeping this property in mind, we will be able to navigate the simplification process smoothly and accurately.
Multiplying the Conjugates: A Detailed Walkthrough
To begin, let's multiply the first terms of the complex conjugates: 6 multiplied by 6 equals 36. This forms the first part of our product. Next, we move on to the outer terms: 6 multiplied by -7i gives us -42i. This is the second term in our product. Now, we consider the inner terms: 7i multiplied by 6 yields 42i. Notice that this term is the opposite of the outer term, a characteristic feature of complex conjugate multiplication. Finally, we multiply the last terms: 7i multiplied by -7i gives us -49i². Here, the i² term appears, which we know is equal to -1. This crucial substitution will play a significant role in simplifying the expression. By combining these individual products, we obtain the expression 36 - 42i + 42i - 49i². The next step involves simplifying this expression by combining like terms and substituting -1 for i².
Simplifying the Product: Unveiling the Real Number
As we examine the expression 36 - 42i + 42i - 49i², we notice that the imaginary terms, -42i and 42i, cancel each other out. This cancellation is a direct consequence of the complex conjugate property, where the imaginary components have opposite signs and therefore eliminate each other upon multiplication. This leaves us with 36 - 49i². Now, we substitute -1 for i², which gives us 36 - 49(-1). This simplifies to 36 + 49. Adding these two real numbers, we arrive at the result 85. This elegant outcome demonstrates the fundamental property of complex conjugates: their product is always a real number. The imaginary components vanish, leaving behind a purely real value. With the product of the conjugates simplified to 85, we are now ready to incorporate the imaginary unit, i, and complete the simplification of the original expression.
Incorporating the Imaginary Unit: The Final Touch
Having simplified the product of the complex conjugates to 85, we now turn our attention back to the original expression, i(6 + 7i)(6 - 7i). We have effectively reduced the expression to i(85). This final step involves multiplying the real number, 85, by the imaginary unit, i. The multiplication is straightforward: 85 multiplied by i simply gives us 85i. This result represents a purely imaginary number, where the real component is zero. To express this result in standard form, we write it as 0 + 85i. This form clearly highlights the real and imaginary components of the complex number. The real component is 0, and the imaginary component is 85. This completes the simplification process, providing us with the final answer in standard form.
The Result in Standard Form: 0 + 85i
After meticulously simplifying the expression i(6 + 7i)(6 - 7i), we arrive at the result 0 + 85i. This is the standard form representation of the complex number, where the real component is 0 and the imaginary component is 85. This final answer encapsulates the journey we have undertaken, from multiplying complex conjugates to incorporating the imaginary unit. The process has showcased the fundamental properties of complex numbers and their conjugates, demonstrating how their interplay leads to elegant simplifications. The standard form representation provides a clear and concise way to express the complex number, highlighting its real and imaginary constituents. This result serves as a testament to the power and beauty of complex number arithmetic.
In conclusion, we have successfully simplified the expression i(6 + 7i)(6 - 7i) and expressed the result in standard form as 0 + 85i. This journey has not only provided us with a concrete answer but has also deepened our understanding of complex numbers, complex conjugates, and their properties. The process of multiplying complex conjugates, simplifying expressions, and expressing results in standard form is a cornerstone of complex number arithmetic. By mastering these techniques, we unlock a deeper appreciation for the intricate world of mathematics and its ability to reveal elegant solutions to seemingly complex problems.
Key Concepts Revisited: A Recap of the Journey
Throughout our exploration of the expression i(6 + 7i)(6 - 7i), we have encountered several key concepts that warrant revisiting. These concepts form the foundation of complex number arithmetic and are crucial for understanding the simplification process. First and foremost, we encountered the definition of a complex number as a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i² = -1. This definition extends the familiar number system to include imaginary components, opening up a new realm of mathematical possibilities. We also encountered the concept of complex conjugates, which are pairs of complex numbers that differ only in the sign of their imaginary component. The complex conjugate of a + bi is a - bi, and vice versa. The product of complex conjugates holds a special significance, as it always results in a real number. This property stems from the cancellation of the imaginary terms during multiplication, leaving only the sum of the squares of the real and imaginary parts.
The distributive property, also known as the FOIL method, played a pivotal role in simplifying the product of the complex conjugates. This property allows us to multiply each term in one complex number by each term in the other complex number, ensuring that we account for all possible combinations. By applying the distributive property systematically, we can expand the product of complex numbers and identify terms that can be combined or simplified. The property i² = -1 is another crucial element in complex number arithmetic. This property allows us to eliminate the imaginary unit when it appears squared, transforming imaginary terms into real terms. This substitution is essential for simplifying expressions and expressing complex numbers in standard form. The standard form of a complex number, a + bi, provides a clear and concise way to represent the number, highlighting its real and imaginary components. By expressing complex numbers in standard form, we can easily compare and manipulate them.
By revisiting these key concepts, we reinforce our understanding of complex number arithmetic and the techniques involved in simplifying expressions. The journey through i(6 + 7i)(6 - 7i) has served as a practical application of these concepts, demonstrating their power and elegance in solving mathematical problems. As we continue to explore the world of mathematics, these fundamental concepts will serve as valuable tools in our quest for knowledge and understanding.
Applications and Beyond: The Significance of Complex Numbers
The simplification of i(6 + 7i)(6 - 7i) serves as a microcosm of the broader applications and significance of complex numbers in mathematics, physics, and engineering. Complex numbers, initially conceived as an abstract mathematical concept, have proven to be remarkably versatile tools for solving a wide range of problems in various fields. Their ability to represent both magnitude and phase makes them indispensable in areas such as electrical engineering, signal processing, and quantum mechanics. In electrical engineering, complex numbers are used extensively to analyze alternating current (AC) circuits. The impedance of a circuit, which is the opposition to the flow of current, can be represented as a complex number, with the real part representing resistance and the imaginary part representing reactance. This complex representation allows engineers to analyze the behavior of AC circuits in a concise and elegant manner.
In signal processing, complex numbers are used to represent signals in the frequency domain. The Fourier transform, a fundamental tool in signal processing, decomposes a signal into its constituent frequencies, which can be represented as complex numbers. This allows engineers to analyze and manipulate signals in the frequency domain, enabling tasks such as filtering, compression, and modulation. In quantum mechanics, complex numbers are essential for describing the wave functions of particles. The wave function, which describes the probability amplitude of a particle's position and momentum, is a complex-valued function. Complex numbers allow physicists to capture the wave-like nature of particles and their probabilistic behavior. Beyond these specific applications, complex numbers play a fundamental role in various mathematical concepts, such as fractals, complex analysis, and number theory. Their ability to extend the number system and provide solutions to equations that have no real solutions makes them an indispensable tool in mathematical research.
As we delve deeper into the world of complex numbers, we uncover their profound connections to various scientific and engineering disciplines. Their versatility and elegance make them a valuable asset in solving real-world problems and pushing the boundaries of human knowledge. The simplification of i(6 + 7i)(6 - 7i) is but a glimpse into the vast landscape of complex number applications, inspiring us to explore further and appreciate the beauty and power of mathematics.
Repair Input Keyword
Find the product of i(6 + 7i)(6 - 7i) and express the result in standard form.
