Complex Numbers Expressing Z = 1/w₁ + W₂ In Polar Form Modulus And Argument

In the realm of mathematics, complex numbers offer a fascinating extension to the familiar world of real numbers. They play a crucial role in various fields, including engineering, physics, and computer science. In this article, we will delve into a comprehensive exploration of complex numbers, focusing on the expression z = 1/w₁ + w₂, where w₁ = 1 + 3i and w₂ = -2 + i. Our journey will involve expressing z in the standard form a + bi, determining its modulus and argument, and finally, representing it in polar form. This exploration will not only solidify your understanding of complex numbers but also demonstrate their versatility in mathematical manipulations.

(a) Expressing z = 1/w₁ + w₂ in the Form a + bi

To express the complex number z = 1/w₁ + w₂ in the standard form a + bi, where a and b are real numbers, we need to perform algebraic manipulations to separate the real and imaginary parts. Given that w₁ = 1 + 3i and w₂ = -2 + i, the first step involves finding the reciprocal of w₁ and then adding it to w₂. This process requires a firm grasp of complex number arithmetic, particularly the concepts of complex conjugates and division. Let's break down the steps involved in this process to ensure a clear and thorough understanding.

Step 1: Finding the Reciprocal of w₁

The reciprocal of a complex number w₁ = 1 + 3i is given by 1/w₁ = 1/(1 + 3i). To express this in the a + bi form, we need to eliminate the imaginary part from the denominator. This is achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 1 + 3i is 1 - 3i. Therefore, we multiply 1/(1 + 3i) by (1 - 3i)/(1 - 3i):

1/w₁ = (1/(1 + 3i)) * ((1 - 3i)/(1 - 3i))

Step 2: Multiplying by the Complex Conjugate

When we multiply the numerator and the denominator by the complex conjugate, we get:

1/w₁ = (1 - 3i) / ((1 + 3i)(1 - 3i))

The denominator is now a product of the form (a + bi)(a - bi), which simplifies to a² + b² (since (bi)(-bi) = -b²i² = -b²(-1) = b²). In this case, a = 1 and b = 3, so the denominator becomes:

(1 + 3i)(1 - 3i) = 1² + 3² = 1 + 9 = 10

Thus, the reciprocal of w₁ simplifies to:

1/w₁ = (1 - 3i) / 10

Step 3: Separating the Real and Imaginary Parts of 1/w₁

Now, we can express 1/w₁ in the form a + bi by dividing both the real and imaginary parts of the numerator by the denominator:

1/w₁ = 1/10 - (3/10)i

So, the real part is 1/10 and the imaginary part is -3/10.

Step 4: Adding w₂ to 1/w₁

Now that we have 1/w₁ in the a + bi form, we can add it to w₂ = -2 + i. To add two complex numbers, we add their real parts and their imaginary parts separately:

z = 1/w₁ + w₂ = (1/10 - (3/10)i) + (-2 + i)

Step 5: Combining Real and Imaginary Parts

Combining the real parts, we have:

1/10 + (-2) = 1/10 - 2 = 1/10 - 20/10 = -19/10

Combining the imaginary parts, we have:

(-3/10)i + i = (-3/10)i + (10/10)i = (7/10)i

Step 6: Expressing z in the Form a + bi

Finally, we can express z in the form a + bi:

z = -19/10 + (7/10)i

Therefore, the complex number z expressed in the form a + bi is -19/10 + (7/10)i. This detailed breakdown illustrates the process of finding the reciprocal of a complex number, adding complex numbers, and expressing the result in the standard a + bi form. This is a fundamental concept in complex number theory and is essential for further analysis.

(b) Finding the Modulus and Argument of z

Having expressed z in the form a + bi, where z = -19/10 + (7/10)i, the next step is to determine its modulus and argument. The modulus of a complex number represents its distance from the origin in the complex plane, while the argument represents the angle it makes with the positive real axis. These two parameters provide an alternative way to represent complex numbers, which is particularly useful in various applications, including signal processing and electrical engineering. Let's explore how to calculate the modulus and argument of z in detail.

Step 1: Understanding the Modulus of a Complex Number

The modulus of a complex number z = a + bi, denoted as |z|, is the distance from the origin to the point (a, b) in the complex plane. It is calculated using the Pythagorean theorem:

|z| = √(a² + b²)

This formula arises from considering the real part a and the imaginary part b as the lengths of the sides of a right-angled triangle, with the modulus |z| being the length of the hypotenuse. This geometric interpretation provides a clear understanding of the modulus as a measure of the magnitude or absolute value of the complex number.

Step 2: Calculating the Modulus of z

For z = -19/10 + (7/10)i, the real part a is -19/10 and the imaginary part b is 7/10. Plugging these values into the formula for the modulus, we get:

|z| = √((-19/10)² + (7/10)²) = √((361/100) + (49/100)) = √(410/100) = √(41/10)

Simplifying further:

|z| = √(4.1) ≈ 2.0248

Thus, the modulus of z is approximately 2.0248. This value represents the distance of the complex number z from the origin in the complex plane.

Step 3: Understanding the Argument of a Complex Number

The argument of a complex number z = a + bi, denoted as arg(z), is the angle θ that the line connecting the origin to the point (a, b) makes with the positive real axis. The argument is typically measured in radians and can be found using the arctangent function:

θ = arctan(b/a)

However, the arctangent function only gives values in the range (-π/2, π/2), which corresponds to the first and fourth quadrants of the complex plane. To find the correct argument in all four quadrants, we need to consider the signs of a and b. This involves adjusting the angle based on the quadrant in which the complex number lies.

Step 4: Determining the Quadrant of z

For z = -19/10 + (7/10)i, the real part a is -19/10 (negative) and the imaginary part b is 7/10 (positive). This means that z lies in the second quadrant of the complex plane, where the real part is negative and the imaginary part is positive.

Step 5: Calculating the Argument of z

First, we calculate the reference angle using the arctangent function:

θ' = arctan(|b/a|) = arctan(|(7/10) / (-19/10)|) = arctan(7/19)

Using a calculator, we find:

θ' ≈ arctan(0.3684) ≈ 0.3564 radians

Since z lies in the second quadrant, the argument θ is given by:

θ = π - θ' = π - 0.3564 ≈ 2.7852 radians

Alternatively, we can use the atan2 function, which is a variation of the arctangent function that takes into account the signs of both a and b to determine the correct quadrant:

θ = atan2(b, a) = atan2(7/10, -19/10) ≈ 2.7852 radians

Thus, the argument of z is approximately 2.7852 radians. This angle represents the direction of the complex number z from the origin in the complex plane.

Step 6: Summarizing the Modulus and Argument

In summary, the modulus of z = -19/10 + (7/10)i is approximately 2.0248, and its argument is approximately 2.7852 radians. These two values provide a complete description of the complex number z in terms of its magnitude and direction.

(c) Expressing z in Polar Form

Now that we have determined the modulus and argument of z, we can express it in polar form. The polar form of a complex number provides an alternative representation that is particularly useful in situations involving rotations and oscillations. It is defined in terms of the modulus |z| and the argument θ and offers a concise way to represent complex numbers.

Step 1: Understanding Polar Form

The polar form of a complex number z is given by:

z = |z|(cos(θ) + i sin(θ))

where |z| is the modulus of z and θ is its argument. This representation arises from the geometric interpretation of complex numbers in the complex plane. The modulus |z| represents the distance from the origin, and the argument θ represents the angle with the positive real axis. The terms cos(θ) and sin(θ) correspond to the real and imaginary components of a point on the unit circle, respectively, scaled by the modulus |z|.

Step 2: Euler's Formula and Polar Form

An alternative way to write the polar form is using Euler's formula:

e^(iθ) = cos(θ) + i sin(θ)

Using Euler's formula, we can express the polar form of z as:

z = |z|e^(iθ)

This compact notation is widely used in mathematics and engineering due to its convenience and the insights it provides into the properties of complex numbers. The exponential form highlights the rotational aspect of complex numbers and simplifies many calculations.

Step 3: Substituting Modulus and Argument

We have already calculated the modulus |z| ≈ 2.0248 and the argument θ ≈ 2.7852 radians for z = -19/10 + (7/10)i. Substituting these values into the polar form, we get:

z ≈ 2.0248(cos(2.7852) + i sin(2.7852))

This is the polar form of z using trigonometric functions.

Step 4: Expressing z in Exponential Polar Form

Using Euler's formula, we can also express z in the exponential polar form:

z ≈ 2.0248e^(2.7852i)

This form is particularly useful in various applications, such as electrical engineering and signal processing, where complex exponentials simplify calculations involving oscillations and rotations.

Step 5: Verifying the Polar Form

To verify that our polar form is correct, we can convert it back to the rectangular form a + bi using the trigonometric form:

z ≈ 2.0248(cos(2.7852) + i sin(2.7852))

Using a calculator, we find:

cos(2.7852) ≈ -0.9387

sin(2.7852) ≈ 0.3441

So,

z ≈ 2.0248(-0.9387 + 0.3441i)

Distributing the modulus:

z ≈ -1.9007 + 0.6966i

These values are close to the original rectangular form -19/10 + (7/10)i = -1.9 + 0.7i, which confirms that our polar form is accurate.

Step 6: Conclusion

Therefore, the complex number z = -19/10 + (7/10)i can be expressed in polar form as z ≈ 2.0248(cos(2.7852) + i sin(2.7852)) or in exponential polar form as z ≈ 2.0248e^(2.7852i). This transformation from rectangular to polar form demonstrates the versatility of complex number representation and its utility in various mathematical and engineering contexts.

In this comprehensive exploration, we have successfully expressed the complex number z = 1/w₁ + w₂, where w₁ = 1 + 3i and w₂ = -2 + i, in the standard form a + bi. We meticulously calculated the reciprocal of w₁, added it to w₂, and arrived at z = -19/10 + (7/10)i. Furthermore, we determined the modulus and argument of z, which are approximately 2.0248 and 2.7852 radians, respectively. Finally, we expressed z in both trigonometric and exponential polar forms, providing a complete understanding of its representation in the complex plane.

This journey through complex number manipulation has highlighted the importance of understanding complex conjugates, the Pythagorean theorem in the context of complex numbers, and the use of the arctangent function to find arguments. The polar form representation, facilitated by Euler's formula, offers a powerful tool for simplifying complex number operations, particularly in fields such as electrical engineering and physics.

By mastering these concepts, you are well-equipped to tackle more advanced topics in complex analysis and appreciate the profound role complex numbers play in various scientific and engineering disciplines. This detailed exploration serves as a solid foundation for further studies and applications in the fascinating world of complex numbers.