Simplifying Complex Number Expressions A Step-by-Step Guide
Complex numbers, an extension of the familiar real number system, involve the imaginary unit i, defined as the square root of -1. These numbers, expressed in the form a + bi, where a and b are real numbers, play a crucial role in various fields, including mathematics, physics, and engineering. One common task in complex number manipulation is simplifying expressions involving complex fractions. This article delves into a step-by-step approach to simplifying such expressions, focusing on the technique of multiplying the numerator and denominator by the conjugate of the denominator. Understanding complex numbers and their operations is fundamental for anyone delving into advanced mathematical concepts or engineering applications. Complex numbers are not just abstract mathematical constructs; they have tangible applications in areas like electrical engineering, quantum mechanics, and signal processing. Therefore, mastering the simplification of complex expressions is a valuable skill.
The expression we aim to simplify is a fraction where both the numerator and the denominator are complex numbers: $\frac{6-5 i}{7+6 i}$. The key to simplifying this expression lies in eliminating the imaginary component from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. In our case, the conjugate of 7 + 6i is 7 - 6i. This process is analogous to rationalizing the denominator when dealing with fractions involving radicals. By multiplying by the conjugate, we are essentially using the difference of squares identity, which will result in a real number in the denominator. This makes the expression easier to handle and allows us to express the complex number in its standard form, a + bi, where the real and imaginary parts are clearly separated. The method of using conjugates is a powerful tool in complex number algebra, and it is essential for simplifying complex fractions and performing other operations such as division and finding reciprocals.
To simplify a complex fraction like the one given, $ rac6-5 i}{7+6 i}$, we need to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Therefore, the conjugate of 7 + 6i is 7 - 6i. Now, we multiply both the numerator and the denominator by this conjugate{7+6 i} * \frac{7-6 i}{7-6 i}$. This step is crucial because it allows us to utilize the difference of squares identity in the denominator, which will result in a real number. Multiplying by the conjugate is not just a trick; it is a mathematically sound method based on the properties of complex numbers and their conjugates. This process ensures that we are not changing the value of the expression, as we are essentially multiplying by 1. The resulting expression will be equivalent to the original but will have a real denominator, making it easier to simplify further and express in the standard complex number form. The conjugate multiplication technique is a cornerstone of complex number arithmetic and is frequently used in various mathematical and engineering applications.
Step-by-Step Simplification
Multiplying by the Conjugate
The first step in simplifying the expression $\frac6-5 i}{7+6 i}$ is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 7 + 6i is 7 - 6i. This gives us{7+6 i} * \frac{7-6 i}{7-6 i}$. This step is crucial because it sets the stage for eliminating the imaginary part from the denominator. Multiplying by the conjugate is a strategic move that leverages the properties of complex numbers and the difference of squares identity. It is not just about getting rid of the i in the denominator; it's about transforming the expression into a form that is easier to work with and interpret. The act of multiplying by the conjugate is a fundamental technique in complex number simplification and is analogous to rationalizing the denominator in expressions involving radicals. It ensures that we maintain the value of the expression while changing its form to a more manageable one. This step is the foundation for the subsequent steps and ultimately leads to the simplified form of the complex number.
Expanding the Numerator and Denominator
Next, we expand both the numerator and the denominator:$\frac{(6-5 i)(7-6 i)}{(7+6 i)(7-6 i)}$. Expanding the numerator, we get: 6 * 7 + 6 * (-6i) + (-5i) * 7 + (-5i) * (-6i) = 42 - 36i - 35i + 30i2. Expanding the denominator, we use the difference of squares formula, (a + b)(a - b) = a2 - b2, which gives us: 72 - (6i)2 = 49 - 36i2. This expansion is a straightforward application of the distributive property and the difference of squares identity. It is a crucial step in breaking down the complex fraction into simpler terms. The expansion of the numerator and denominator allows us to see the individual terms and how they interact, particularly the i2 terms, which will simplify further since i2 = -1. This step is not just about performing the multiplication; it's about revealing the structure of the expression and setting the stage for simplification by combining like terms and substituting the value of i2. The careful expansion ensures that we do not lose any terms and that we can accurately simplify the expression in the following steps.
Simplifying Using i^2 = -1
Now, we simplify the expression by substituting i2 with -1. In the numerator, we have 42 - 36i - 35i + 30i2, which becomes 42 - 36i - 35i + 30(-1) = 42 - 36i - 35i - 30. In the denominator, we have 49 - 36i2, which becomes 49 - 36(-1) = 49 + 36. The substitution of i2 with -1 is a fundamental step in simplifying complex number expressions. It is based on the very definition of the imaginary unit i. This substitution allows us to transform the imaginary terms into real numbers, which can then be combined with the other real terms in the expression. The replacement of i2 with -1 is not just a mathematical trick; it's a direct application of the definition of i and is essential for reducing the expression to its simplest form. This step is the key to bridging the gap between the expanded form and the simplified form of the complex number, where the real and imaginary parts are clearly separated.
Combining Like Terms
Combining like terms in the numerator, we have (42 - 30) + (-36i - 35i) = 12 - 71i. In the denominator, we have 49 + 36 = 85. So, the expression becomes: $\frac{12-71 i}{85}$. This step involves grouping the real terms and the imaginary terms separately. It is a standard algebraic technique that helps to consolidate the expression and make it more readable. The process of combining like terms is not just about simplifying the expression; it's about revealing the structure of the complex number by separating its real and imaginary components. This separation is crucial for expressing the complex number in its standard form, a + bi, which is essential for various mathematical operations and applications. The result of this step is a simplified fraction where the numerator is a complex number and the denominator is a real number, paving the way for the final step of expressing the complex number in its standard form.
Expressing in Standard Form
Finally, we express the complex number in the standard form a + bi: $\frac{12-71 i}{85} = \frac{12}{85} - \frac{71}{85}i$. This step involves separating the real and imaginary parts of the complex number. We divide both the real and imaginary parts of the numerator by the denominator. This final step is crucial for expressing the complex number in its standard form, which is essential for comparing and performing operations with other complex numbers. The transformation into standard form is not just a cosmetic change; it's a fundamental requirement for many complex number operations and applications. The standard form a + bi clearly shows the real part (a) and the imaginary part (b) of the complex number, which is essential for tasks like plotting complex numbers on the complex plane, finding their magnitude and argument, and performing arithmetic operations. The final result, $\frac{12}{85} - \frac{71}{85}i$, is the simplified form of the original complex fraction, where the real and imaginary parts are clearly identified.
Conclusion
In conclusion, simplifying complex number expressions involves several key steps: multiplying by the conjugate of the denominator, expanding the numerator and denominator, simplifying using i2 = -1, combining like terms, and expressing the result in standard form. By following these steps, we can effectively simplify complex fractions and work with complex numbers in various mathematical and engineering contexts. The process of simplifying complex expressions is not just a mathematical exercise; it's a fundamental skill for anyone working with complex numbers in various fields. Mastering this skill allows for easier manipulation and interpretation of complex numbers, which are essential in areas like electrical engineering, quantum mechanics, and signal processing. The ability to simplify complex expressions is a testament to a strong understanding of complex number theory and its applications.
