Complex Numbers Express (1/z₁) + (1/(z₁ - 2z₂)) In X + Yi Form

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In the realm of mathematics, complex numbers play a crucial role, extending the real number system and enabling solutions to equations that are unsolvable within real numbers alone. Understanding complex number arithmetic, including operations like addition, subtraction, multiplication, and division, is fundamental in various fields such as electrical engineering, quantum mechanics, and signal processing. This article delves into a specific problem involving complex numbers, providing a step-by-step solution and detailed explanations to enhance understanding. This exploration will guide you through the process of expressing complex numbers in the standard form of x + yi, where x and y are real numbers, and i represents the imaginary unit (−1{\sqrt{-1}}). Mastering these concepts is essential for anyone venturing into advanced mathematics and its applications. Let's embark on this journey to unravel the intricacies of complex numbers and their manipulations.

Given two complex numbers, z1=1+2i{ z_1 = 1 + 2i } and z2=3−4i{ z_2 = 3 - 4i }, our task is to express the expression 1z1+1z1−2z2{ \frac{1}{z_1} + \frac{1}{z_1 - 2z_2} } in the standard form x + yi, where x and y are real numbers. This problem requires us to perform several operations involving complex numbers, including division and subtraction, before finally combining the results into the desired format. The process involves finding the reciprocals of complex numbers, subtracting complex numbers, and rationalizing the denominators of complex fractions. By meticulously following each step, we can arrive at the solution and gain a deeper understanding of complex number arithmetic. This exercise serves as a practical application of the fundamental principles governing complex number operations. Let's break down the problem into manageable steps and solve it systematically.

1. Finding the Reciprocal of z1{ z_1 }

The first step in solving the problem is to find the reciprocal of z1{ z_1 }, which is 1z1{ \frac{1}{z_1} }. Given that z1=1+2i{ z_1 = 1 + 2i }, we have:

1z1=11+2i{ \frac{1}{z_1} = \frac{1}{1 + 2i} }

To express this fraction in the form x + yi, we need to rationalize the denominator. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of 1+2i{ 1 + 2i } is 1−2i{ 1 - 2i }. Therefore, we multiply the fraction as follows:

11+2i×1−2i1−2i=1−2i(1+2i)(1−2i){ \frac{1}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} = \frac{1 - 2i}{(1 + 2i)(1 - 2i)} }

Now, we multiply out the denominator:

(1+2i)(1−2i)=1−2i+2i−4i2{ (1 + 2i)(1 - 2i) = 1 - 2i + 2i - 4i^2 }

Since i2=−1{ i^2 = -1 }, the expression simplifies to:

1−4(−1)=1+4=5{ 1 - 4(-1) = 1 + 4 = 5 }

Thus, we have:

1z1=1−2i5=15−25i{ \frac{1}{z_1} = \frac{1 - 2i}{5} = \frac{1}{5} - \frac{2}{5}i }

So, the reciprocal of z1{ z_1 } in the form x + yi is 15−25i{ \frac{1}{5} - \frac{2}{5}i }.

2. Calculating z1−2z2{ z_1 - 2z_2 }

The next step is to calculate z1−2z2{ z_1 - 2z_2 }. We are given that z1=1+2i{ z_1 = 1 + 2i } and z2=3−4i{ z_2 = 3 - 4i }. First, we multiply z2{ z_2 } by 2:

2z2=2(3−4i)=6−8i{ 2z_2 = 2(3 - 4i) = 6 - 8i }

Now, we subtract 2z2{ 2z_2 } from z1{ z_1 }:

z1−2z2=(1+2i)−(6−8i){ z_1 - 2z_2 = (1 + 2i) - (6 - 8i) }

Combine the real and imaginary parts:

(1−6)+(2i−(−8i))=−5+10i{ (1 - 6) + (2i - (-8i)) = -5 + 10i }

Therefore, z1−2z2=−5+10i{ z_1 - 2z_2 = -5 + 10i }.

3. Finding the Reciprocal of z1−2z2{ z_1 - 2z_2 }

Now, we need to find the reciprocal of z1−2z2{ z_1 - 2z_2 }, which is 1z1−2z2{ \frac{1}{z_1 - 2z_2} }. We have already calculated that z1−2z2=−5+10i{ z_1 - 2z_2 = -5 + 10i }, so we need to find:

1−5+10i{ \frac{1}{-5 + 10i} }

Similar to step 1, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of −5+10i{ -5 + 10i } is −5−10i{ -5 - 10i }. Therefore, we multiply the fraction as follows:

1−5+10i×−5−10i−5−10i=−5−10i(−5+10i)(−5−10i){ \frac{1}{-5 + 10i} \times \frac{-5 - 10i}{-5 - 10i} = \frac{-5 - 10i}{(-5 + 10i)(-5 - 10i)} }

Now, we multiply out the denominator:

(−5+10i)(−5−10i)=(−5)2−(10i)2=25−100i2{ (-5 + 10i)(-5 - 10i) = (-5)^2 - (10i)^2 = 25 - 100i^2 }

Since i2=−1{ i^2 = -1 }, the expression simplifies to:

25−100(−1)=25+100=125{ 25 - 100(-1) = 25 + 100 = 125 }

Thus, we have:

1z1−2z2=−5−10i125{ \frac{1}{z_1 - 2z_2} = \frac{-5 - 10i}{125} }

Divide both the real and imaginary parts by 125:

−5125−10125i=−125−225i{ \frac{-5}{125} - \frac{10}{125}i = -\frac{1}{25} - \frac{2}{25}i }

So, the reciprocal of z1−2z2{ z_1 - 2z_2 } in the form x + yi is −125−225i{ -\frac{1}{25} - \frac{2}{25}i }.

4. Adding the Reciprocals

The final step is to add the two reciprocals we found in steps 1 and 3:

1z1+1z1−2z2=(15−25i)+(−125−225i){ \frac{1}{z_1} + \frac{1}{z_1 - 2z_2} = \left(\frac{1}{5} - \frac{2}{5}i\right) + \left(-\frac{1}{25} - \frac{2}{25}i\right) }

Combine the real and imaginary parts:

(15−125)+(−25i−225i){ \left(\frac{1}{5} - \frac{1}{25}\right) + \left(-\frac{2}{5}i - \frac{2}{25}i\right) }

To add the fractions, we need a common denominator. The least common denominator for 5 and 25 is 25. So, we rewrite the fractions:

(525−125)+(−1025i−225i){ \left(\frac{5}{25} - \frac{1}{25}\right) + \left(-\frac{10}{25}i - \frac{2}{25}i\right) }

Now, we can add the fractions:

425−1225i{ \frac{4}{25} - \frac{12}{25}i }

Therefore, the final expression in the form x + yi is 425−1225i{ \frac{4}{25} - \frac{12}{25}i }.

The expression 1z1+1z1−2z2{ \frac{1}{z_1} + \frac{1}{z_1 - 2z_2} }, where z1=1+2i{ z_1 = 1 + 2i } and z2=3−4i{ z_2 = 3 - 4i }, simplifies to 425−1225i{ \frac{4}{25} - \frac{12}{25}i } in the form x + yi. This solution involves several steps, including finding reciprocals of complex numbers, subtracting complex numbers, and rationalizing denominators. Each step is crucial in arriving at the correct answer. By understanding these operations, one can effectively manipulate complex numbers and express them in the desired form. This exercise not only reinforces the basic principles of complex number arithmetic but also provides a practical application of these concepts. Mastering these skills is essential for further studies in mathematics and related fields.

In conclusion, we have successfully expressed the given complex number expression in the form x + yi. The process involved a series of operations, each requiring careful attention to detail. We began by finding the reciprocal of z1{ z_1 }, then calculated z1−2z2{ z_1 - 2z_2 }, and subsequently found the reciprocal of that result. Finally, we added the two reciprocals to obtain the final expression. This problem highlights the importance of understanding complex number arithmetic and the techniques involved in manipulating complex numbers. The ability to perform these operations is fundamental in various branches of mathematics, physics, and engineering. By working through this problem, we have gained a deeper appreciation for the elegance and utility of complex numbers. The solution demonstrates how complex numbers, though seemingly abstract, can be handled with precision and lead to concrete results. This exercise serves as a valuable learning experience for anyone seeking to enhance their understanding of complex numbers and their applications.