Simplifying Complex Number Expressions Converting (6-4i)(6-4i)-3(7-11i) To A + Bi Form
In the fascinating realm of complex numbers, mathematical expressions can sometimes appear daunting at first glance. However, with a systematic approach and a solid understanding of the underlying principles, we can unravel even the most intricate expressions. In this article, we will embark on a journey to simplify the expression (6-4i)(6-4i)-3(7-11i) and express it in the standard form a + bi, where a and b are rational numbers. This exploration will not only enhance our comprehension of complex number arithmetic but also showcase the power of algebraic manipulation in simplifying mathematical expressions.
Understanding Complex Numbers: The Foundation of Our Exploration
Before we delve into the simplification process, it is crucial to establish a firm understanding of complex numbers. At its core, a complex number is an extension of the real number system, incorporating the imaginary unit 'i'. This imaginary unit is defined as the square root of -1, a concept that expands the realm of numbers beyond the familiar real number line. A complex number is typically expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. These real and imaginary components interact in unique ways, governed by the rules of complex number arithmetic. To effectively manipulate complex numbers, we must grasp these rules and how they govern operations such as addition, subtraction, multiplication, and division.
The introduction of complex numbers opens up a whole new dimension in mathematics, allowing us to solve equations that were previously considered unsolvable within the realm of real numbers. For instance, the equation x^2 + 1 = 0 has no real solutions, but it has two complex solutions: x = i and x = -i. This ability to solve a wider range of equations makes complex numbers indispensable in various fields, including engineering, physics, and computer science. In electrical engineering, complex numbers are used to analyze alternating current circuits. In quantum mechanics, they are used to describe the wave functions of particles. And in computer science, they are used in signal processing and image analysis. Understanding complex numbers is therefore not just an academic exercise but a gateway to understanding and solving real-world problems.
Step-by-Step Simplification: A Journey Through Algebraic Manipulation
Now that we have a solid foundation in complex numbers, let's embark on the step-by-step simplification of the expression (6-4i)(6-4i)-3(7-11i). This process will involve a combination of algebraic techniques, including the distributive property and the rules of complex number arithmetic. By carefully applying each step, we will gradually transform the expression into its simplified form a + bi.
1. Expanding the First Term: Unleashing the Distributive Property
The first step in our simplification journey involves expanding the term (6-4i)(6-4i). This requires us to apply the distributive property, a fundamental principle of algebra that allows us to multiply expressions containing multiple terms. In this case, we will multiply each term in the first set of parentheses by each term in the second set of parentheses. This process will generate a series of terms that we can then combine and simplify.
When we apply the distributive property, we get:
(6-4i)(6-4i) = 6(6-4i) - 4i(6-4i)
Now, we distribute again:
= 36 - 24i - 24i + 16i^2
This expansion yields four terms: 36, -24i, -24i, and 16i^2. Notice the presence of the term i^2, which represents the square of the imaginary unit. As we know, i is defined as the square root of -1, so i^2 is equal to -1. This substitution is crucial for simplifying the expression further.
2. Simplifying i^2: Unveiling the Real Component
The presence of i^2 in our expanded expression provides an opportunity to simplify the expression further. Since i^2 is equal to -1, we can substitute -1 for i^2 in our expression. This substitution will transform the imaginary term 16i^2 into a real term, allowing us to combine it with the other real terms in the expression.
Substituting i^2 = -1, we get:
36 - 24i - 24i + 16i^2 = 36 - 24i - 24i + 16(-1)
Simplifying further:
= 36 - 24i - 24i - 16
Now, we can combine the real terms (36 and -16) to get:
= 20 - 24i - 24i
This simplification step highlights the unique nature of complex numbers. The imaginary unit 'i' allows us to introduce a term that, when squared, becomes a real number. This interplay between real and imaginary components is a key characteristic of complex number arithmetic.
3. Combining Imaginary Terms: Consolidating the Imaginary Component
Our expression now contains two imaginary terms: -24i and -24i. To further simplify the expression, we need to combine these terms. Just as we can combine like terms in algebraic expressions with real numbers, we can combine imaginary terms by adding their coefficients. In this case, we will add the coefficients of the imaginary unit 'i' to consolidate the imaginary component of the expression.
Combining the imaginary terms, we get:
20 - 24i - 24i = 20 - 48i
This step demonstrates the linearity of complex number arithmetic. We can treat the imaginary unit 'i' as a variable and combine like terms, just as we would with any other algebraic variable. This linearity simplifies the manipulation of complex expressions.
4. Distributing the Scalar: Expanding the Second Term
Now, let's turn our attention to the second term in the original expression: -3(7-11i). This term involves the multiplication of a scalar (-3) with a complex number (7-11i). To simplify this term, we will again apply the distributive property, multiplying the scalar by each term within the parentheses.
Applying the distributive property, we get:
-3(7-11i) = -3 * 7 - 3 * (-11i)
Simplifying further:
= -21 + 33i
This step highlights the importance of careful attention to signs when applying the distributive property. Multiplying a negative scalar by a negative term results in a positive term, and vice versa. Maintaining accuracy with signs is crucial for obtaining the correct result.
5. Combining Like Terms: The Final Synthesis
We have now simplified both parts of the original expression. The first part, (6-4i)(6-4i), has been simplified to 20 - 48i. The second part, -3(7-11i), has been simplified to -21 + 33i. To complete the simplification process, we need to combine these two simplified expressions.
Combining the simplified expressions, we get:
(6-4i)(6-4i) - 3(7-11i) = (20 - 48i) + (-21 + 33i)
To combine these expressions, we add the real parts together and the imaginary parts together:
= (20 - 21) + (-48i + 33i)
Simplifying further:
= -1 - 15i
This final step brings together all the individual simplifications we have performed. By combining like terms, we arrive at the expression in the desired form a + bi, where a = -1 and b = -15.
The Result: Expressing the Complex Number in Standard Form
After meticulously performing the simplification steps, we have successfully expressed the original expression (6-4i)(6-4i)-3(7-11i) in the standard form a + bi. The result is:
-1 - 15i
This final form clearly identifies the real part as -1 and the imaginary part as -15. This result showcases the power of algebraic manipulation and complex number arithmetic in simplifying seemingly complex expressions. By breaking down the expression into smaller, manageable steps, we were able to systematically simplify it and arrive at the desired form.
Conclusion: Mastering Complex Number Arithmetic
In this article, we have embarked on a journey to simplify the expression (6-4i)(6-4i)-3(7-11i) and express it in the standard form a + bi. Through a series of step-by-step simplifications, we have demonstrated the application of algebraic techniques and the rules of complex number arithmetic. This exploration has not only enhanced our understanding of complex numbers but also showcased the power of systematic problem-solving in mathematics. Mastering complex number arithmetic is an invaluable skill for anyone pursuing further studies in mathematics, science, or engineering. It opens doors to a deeper understanding of mathematical concepts and their applications in the real world. The ability to manipulate complex numbers with confidence and accuracy is a testament to one's mathematical proficiency.
The journey through complex number simplification is a testament to the beauty and elegance of mathematics. It demonstrates how seemingly complex problems can be solved through a combination of fundamental principles and meticulous application. As we continue our mathematical pursuits, we will encounter more intricate challenges, but the lessons learned from simplifying complex numbers will serve as a valuable foundation for tackling those challenges with confidence and precision.
