Complex Number Z Cartesian Form Modulus And Argument Explained

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In the realm of mathematics, complex numbers hold a significant position, extending the familiar number system to include imaginary units. These numbers, expressed in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (āˆš-1), open up a world of fascinating properties and applications. This article delves into a specific complex number, z, defined by a trigonometric expression, and explores its representation in Cartesian form, along with its modulus and argument.

Understanding the Complex Number z

The complex number z is given by the following expression:

z = \frac{\sqrt{2} \left[ \cos{\frac{\pi}{4}} + i \sin{\frac{\pi}{4}} \right]}{\cos{\frac{\pi}{6}} - i \sin{\frac{\pi}{6}}}

This expression involves trigonometric functions and the imaginary unit 'i'. To fully grasp the nature of this complex number, we will embark on a step-by-step analysis, starting with its conversion into Cartesian form.

Expressing z in Cartesian Form (a + bi)

The Cartesian form of a complex number, a + bi, provides a clear representation of its real and imaginary components. To transform z into this form, we need to simplify the given expression by evaluating the trigonometric functions and performing the division. Let's break down the process:

First, evaluate the trigonometric functions:

  • cos(Ļ€/4) = āˆš2 / 2
  • sin(Ļ€/4) = āˆš2 / 2
  • cos(Ļ€/6) = āˆš3 / 2
  • sin(Ļ€/6) = 1 / 2

Substitute these values into the expression for z:

z = \frac{\sqrt{2} \left[ \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right]}{\frac{\sqrt{3}}{2} - i \frac{1}{2}}

Simplify the numerator:

z = \frac{1 + i}{\frac{\sqrt{3}}{2} - i \frac{1}{2}}

To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of (āˆš3/2 - i/2) is (āˆš3/2 + i/2). This crucial step eliminates the imaginary part from the denominator, paving the way for expressing z in the desired a + bi format. The conjugate of a complex number is found by simply changing the sign of the imaginary part. This process is essential for rationalizing the denominator and isolating the real and imaginary components of the complex number.

Multiply the numerator and denominator by the conjugate:

z = \frac{(1 + i) \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right)}{\left( \frac{\sqrt{3}}{2} - i \frac{1}{2} \right) \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right)}

Expand the numerator and denominator:

z = \frac{\frac{\sqrt{3}}{2} + i \frac{1}{2} + i \frac{\sqrt{3}}{2} + i^2 \frac{1}{2}}{\frac{3}{4} - i^2 \frac{1}{4}}

Since iĀ² = -1, substitute and simplify:

z = \frac{\frac{\sqrt{3}}{2} + i \frac{1}{2} + i \frac{\sqrt{3}}{2} - \frac{1}{2}}{\frac{3}{4} + \frac{1}{4}}

z = \frac{\left( \frac{\sqrt{3}}{2} - \frac{1}{2} \right) + i \left( \frac{1}{2} + \frac{\sqrt{3}}{2} \right)}{1}

Therefore, the Cartesian form of z is:

z = \left( \frac{\sqrt{3} - 1}{2} \right) + i \left( \frac{1 + \sqrt{3}}{2} \right)

This representation clearly shows the real part, (āˆš3 - 1)/2, and the imaginary part, (1 + āˆš3)/2, of the complex number z. This form is fundamental for various operations and interpretations related to complex numbers, including geometric representation and algebraic manipulations.

Finding the Modulus and Argument of z

The modulus and argument are two crucial properties that provide an alternative way to represent a complex number. The modulus, denoted as |z|, represents the distance of the complex number from the origin in the complex plane. It's essentially the magnitude or absolute value of the complex number. The argument, denoted as arg(z), represents the angle formed by the line connecting the complex number to the origin and the positive real axis, measured in radians. Together, the modulus and argument provide a polar coordinate representation of the complex number.

Calculating the Modulus

The modulus of a complex number z = a + bi is calculated using the following formula:

|z| = \sqrt{a^2 + b^2}

Using the Cartesian form of z that we derived earlier, a = (āˆš3 - 1)/2 and b = (1 + āˆš3)/2, we can calculate the modulus:

|z| = \sqrt{\left( \frac{\sqrt{3} - 1}{2} \right)^2 + \left( \frac{1 + \sqrt{3}}{2} \right)^2}

Expand the squares:

|z| = \sqrt{\frac{3 - 2\sqrt{3} + 1}{4} + \frac{1 + 2\sqrt{3} + 3}{4}}

Simplify:

|z| = \sqrt{\frac{4 - 2\sqrt{3} + 4 + 2\sqrt{3}}{4}}

|z| = \sqrt{\frac{8}{4}}

|z| = \sqrt{2}

Therefore, the modulus of z is āˆš2. This value indicates the distance of the complex number z from the origin in the complex plane, giving us a sense of its magnitude.

Determining the Argument

The argument of a complex number z = a + bi is the angle Īø such that:

cos(Īø) = \frac{a}{|z|}

sin(Īø) = \frac{b}{|z|}

and

Īø = arctan(\frac{b}{a})

However, it's crucial to consider the quadrant in which the complex number lies to determine the correct argument. The arctangent function only gives values in the range (-Ļ€/2, Ļ€/2), so we might need to add or subtract Ļ€ to get the correct angle.

Using the values a = (āˆš3 - 1)/2, b = (1 + āˆš3)/2, and |z| = āˆš2, we can calculate the cosine and sine of the argument:

cos(Īø) = \frac{\frac{\sqrt{3} - 1}{2}}{\sqrt{2}} = \frac{\sqrt{3} - 1}{2\sqrt{2}}

sin(Īø) = \frac{\frac{1 + \sqrt{3}}{2}}{\sqrt{2}} = \frac{1 + \sqrt{3}}{2\sqrt{2}}

Now, let's calculate the arctangent:

Īø = arctan(\frac{\frac{1 + \sqrt{3}}{2}}{\frac{\sqrt{3} - 1}{2}}) = arctan(\frac{1 + \sqrt{3}}{\sqrt{3} - 1})

Rationalize the denominator of the fraction inside the arctangent:

Īø = arctan(\frac{(1 + \sqrt{3})(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)}) = arctan(\frac{1 + 2\sqrt{3} + 3}{3 - 1})

Īø = arctan(\frac{4 + 2\sqrt{3}}{2}) = arctan(2 + \sqrt{3})

We know that tan(5Ļ€/12) = 2 + āˆš3, so:

Īø = \frac{5\pi}{12}

Since both the real and imaginary parts of z are positive, the complex number lies in the first quadrant, and the calculated argument is correct. Therefore, the argument of z is 5Ļ€/12.

Conclusion

In this exploration, we've successfully expressed the complex number z in Cartesian form, revealing its real and imaginary components. Furthermore, we've determined its modulus and argument, providing an alternative representation in polar coordinates. These calculations demonstrate the fundamental principles of complex number manipulation and highlight the interconnectedness of different representations. Understanding the Cartesian form, modulus, and argument is crucial for various applications of complex numbers in mathematics, physics, and engineering.

By converting the given trigonometric expression into the Cartesian form, (āˆš3 - 1)/2 + i(1 + āˆš3)/2, we gained a clear understanding of the real and imaginary components of z. The modulus, āˆš2, tells us the magnitude of the complex number, while the argument, 5Ļ€/12, specifies its direction in the complex plane. This comprehensive analysis provides a solid foundation for further exploration of complex number theory and its diverse applications.