Polar Form Of 7√3 - 7i A Step-by-Step Guide

To understand the polar form of a complex number, it is essential to first grasp the fundamentals of complex numbers and their representation in the complex plane. A complex number, typically denoted as z, is expressed in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1. The complex plane, also known as the Argand diagram, provides a visual representation of complex numbers, with the horizontal axis representing the real part (a) and the vertical axis representing the imaginary part (b). In this plane, a complex number z = a + bi can be plotted as a point (a, b).

The polar form of a complex number offers an alternative way to represent it, using its distance from the origin (also known as the modulus or magnitude) and the angle it makes with the positive real axis (also known as the argument). This representation is particularly useful in various mathematical operations, such as multiplication, division, and exponentiation of complex numbers. The polar form is expressed as z = r(cos θ + i sin θ), where r represents the modulus of z, denoted as |z|, and θ represents the argument of z, denoted as arg(z).

To convert a complex number from its rectangular form (a + bi) to polar form, we need to determine the values of r and θ. The modulus r can be calculated using the Pythagorean theorem: r = √(a² + b²). The argument θ can be found using the arctangent function: θ = arctan(b/a). However, it's important to consider the quadrant in which the complex number lies to ensure the correct angle is obtained. If the complex number lies in the first or fourth quadrant, the arctangent function directly gives the correct angle. If it lies in the second or third quadrant, we need to add π (or 180°) to the result of the arctangent function.

Now, let's focus on the complex number 7√3 - 7i. Our goal is to express this complex number in its polar form. We will follow the steps outlined above to find the modulus and argument of this complex number. First, we identify the real and imaginary parts: a = 7√3 and b = -7. Next, we calculate the modulus r using the formula r = √(a² + b²). Substituting the values, we get r = √((7√3)² + (-7)²) = √(147 + 49) = √196 = 14. Therefore, the modulus of the complex number 7√3 - 7i is 14. This value represents the distance of the complex number from the origin in the complex plane.

Next, we need to determine the argument θ. We start by calculating arctan(b/a) = arctan(-7 / 7√3) = arctan(-1/√3). The arctangent of -1/√3 is -π/6 or -30°. However, we need to consider the quadrant in which the complex number lies. Since the real part is positive (7√3) and the imaginary part is negative (-7), the complex number lies in the fourth quadrant. In the fourth quadrant, the angle can be expressed as either a negative angle or a positive angle greater than 270°. Therefore, -30° is a valid angle. To express it as a positive angle, we can add 360° to get 330°. Hence, the argument θ is 330°. Now that we have both the modulus r = 14 and the argument θ = 330°, we can express the complex number 7√3 - 7i in polar form as 14(cos(330°) + i sin(330°)). This representation provides a clear understanding of the complex number's magnitude and direction in the complex plane.

In conclusion, the polar form of the complex number 7√3 - 7i is 14(cos(330°) + i sin(330°)). This result is obtained by calculating the modulus and argument of the complex number and then substituting these values into the polar form equation. Understanding the polar form of complex numbers is crucial for various mathematical applications, including complex number operations and signal processing. The process of converting between rectangular and polar forms allows for a deeper understanding of complex number behavior and simplifies many calculations.

The process of converting a complex number from its rectangular form (a + bi) to its polar form (r(cos θ + i sin θ)) involves a series of well-defined steps. This conversion is a fundamental concept in complex number theory and has numerous applications in mathematics, physics, and engineering. To effectively convert a complex number, one must understand the relationship between the rectangular and polar coordinate systems and how to calculate the modulus and argument of the complex number. The steps outlined below provide a comprehensive guide to this conversion process.

The initial step in converting a complex number to polar form is identifying the real and imaginary parts. In the rectangular form a + bi, a represents the real part and b represents the imaginary part. For the complex number 7√3 - 7i, the real part (a) is 7√3, and the imaginary part (b) is -7. These values are crucial for calculating the modulus and argument, which are the components of the polar form. Correctly identifying these parts is essential for the subsequent steps in the conversion process. The real and imaginary parts provide the coordinates in the complex plane, which are used to determine the distance from the origin and the angle with the positive real axis.

Next, the modulus r, which represents the distance from the origin to the point representing the complex number in the complex plane, must be calculated. The modulus is a non-negative real number and is calculated using the Pythagorean theorem. The formula for the modulus is r = √(a² + b²). Substituting the real and imaginary parts of 7√3 - 7i, we get r = √((7√3)² + (-7)²) = √(147 + 49) = √196 = 14. Therefore, the modulus of the complex number 7√3 - 7i is 14. The modulus gives the magnitude of the complex number and is a key component in the polar representation. A larger modulus indicates a greater distance from the origin.

Following the calculation of the modulus, the next step is to determine the argument θ, which represents the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. The argument is typically measured in radians or degrees and can be found using the arctangent function. The initial calculation for the argument is θ = arctan(b/a). For the complex number 7√3 - 7i, this gives us θ = arctan(-7 / 7√3) = arctan(-1/√3). The arctangent of -1/√3 is -π/6 radians or -30°. However, it is essential to consider the quadrant in which the complex number lies to ensure the correct angle is obtained. This is because the arctangent function has a period of π, and the result might need adjustment depending on the signs of a and b.

Considering the quadrant is a critical part of finding the correct argument. The complex number 7√3 - 7i lies in the fourth quadrant because the real part (7√3) is positive, and the imaginary part (-7) is negative. In the fourth quadrant, the angle can be expressed as a negative angle or a positive angle greater than 270°. The angle -30° is a valid representation of the argument. To express it as a positive angle, we can add 360° to get 330°. Therefore, the argument θ for the complex number 7√3 - 7i is 330°. Understanding the quadrant allows for the correct interpretation of the angle and ensures the polar form accurately represents the complex number.

Finally, with both the modulus (r = 14) and the argument (θ = 330°) determined, we can express the complex number in its polar form. The polar form of a complex number is given by the equation z = r(cos θ + i sin θ). Substituting the values for r and θ, we get 7√3 - 7i = 14(cos(330°) + i sin(330°)). This is the polar form of the complex number 7√3 - 7i. The polar form provides a clear representation of the magnitude and direction of the complex number in the complex plane. It is particularly useful in operations involving complex numbers, such as multiplication, division, and exponentiation.

In summary, the step-by-step conversion of 7√3 - 7i to polar form involves identifying the real and imaginary parts, calculating the modulus, determining the argument while considering the quadrant, and then expressing the complex number in the form r(cos θ + i sin θ). This process ensures an accurate representation of the complex number in polar coordinates, facilitating various mathematical manipulations and applications. The ability to convert between rectangular and polar forms is a fundamental skill in complex number theory, providing a comprehensive understanding of complex number behavior.

To fully grasp the polar form of a complex number, it is essential to delve deeper into the concepts of modulus and argument. These two components are fundamental in representing complex numbers in polar coordinates and provide valuable insights into their geometric properties. The modulus and argument collectively define the position of a complex number in the complex plane, offering an alternative perspective to the rectangular form. Understanding these concepts is crucial for various applications, including complex number operations, signal processing, and electrical engineering.

The modulus of a complex number, often denoted as |z|, represents the distance from the origin (0, 0) to the point representing the complex number in the complex plane. In the rectangular form z = a + bi, the modulus is calculated using the Pythagorean theorem: |z| = √(a² + b²). The modulus is always a non-negative real number, providing a measure of the magnitude or absolute value of the complex number. For example, if we consider the complex number 7√3 - 7i, the modulus is calculated as √((7√3)² + (-7)²) = √(147 + 49) = √196 = 14. This means the complex number 7√3 - 7i is 14 units away from the origin in the complex plane. The modulus is a key parameter in understanding the size or intensity of the complex number.

The modulus has several important properties. It is always a non-negative real number, as it represents a distance. The modulus of a product of two complex numbers is the product of their moduli: |z₁z₂| = |z₁||z₂|. Similarly, the modulus of a quotient of two complex numbers is the quotient of their moduli: |z₁/z₂| = |z₁|/|z₂|, provided that z₂ ≠ 0. These properties make the modulus a powerful tool in simplifying complex number operations. The modulus also provides a way to compare the magnitudes of different complex numbers, making it useful in applications where relative sizes are important.

On the other hand, the argument of a complex number, denoted as arg(z), represents the angle between the positive real axis and the line connecting the origin to the complex number in the complex plane. The argument is typically measured in radians or degrees and is denoted by θ. The principal argument is usually defined to be in the interval (-π, π] or (-180°, 180°]. The argument can be found using the arctangent function: θ = arctan(b/a). However, it is crucial to consider the quadrant in which the complex number lies to ensure the correct angle is obtained. For the complex number 7√3 - 7i, the initial arctangent calculation gives arctan(-7 / 7√3) = arctan(-1/√3) = -π/6 or -30°. Since 7√3 - 7i lies in the fourth quadrant, the argument can be expressed as -30° or 330°.

The argument provides directional information about the complex number. It indicates the orientation of the complex number in the complex plane relative to the positive real axis. The argument is periodic, meaning that adding integer multiples of 2π (or 360°) to the argument does not change the complex number. For example, 330° and -30° represent the same direction in the complex plane. The argument is essential in various applications, such as representing alternating currents in electrical engineering and analyzing wave phenomena in physics.

The argument also has several important properties. The argument of a product of two complex numbers is the sum of their arguments: arg(z₁z₂) = arg(z₁) + arg(z₂). Similarly, the argument of a quotient of two complex numbers is the difference of their arguments: arg(z₁/z₂) = arg(z₁) - arg(z₂). These properties are particularly useful in complex number multiplication and division, where the polar form simplifies the operations. The argument helps in understanding how complex numbers combine and interact in various mathematical and physical contexts.

In conclusion, the modulus and argument are fundamental concepts in representing complex numbers in polar form. The modulus provides the magnitude or distance from the origin, while the argument provides the directional information. Together, they define the position of a complex number in the complex plane and facilitate various complex number operations. Understanding the modulus and argument is crucial for effectively working with complex numbers and applying them in diverse fields such as mathematics, physics, and engineering. The polar form, using the modulus and argument, offers a powerful tool for analyzing and manipulating complex numbers.

The polar form of complex numbers is not just a mathematical curiosity; it is a powerful tool that simplifies various operations and provides valuable insights in numerous fields. Understanding why the polar form matters involves appreciating its advantages in simplifying complex number arithmetic, its role in solving equations, and its applications in science and engineering. The polar form offers a unique perspective on complex numbers, making it an indispensable concept for anyone working with these mathematical entities.

One of the primary reasons the polar form matters is its ability to simplify complex number multiplication and division. In rectangular form (a + bi), multiplying two complex numbers involves applying the distributive property and combining like terms, which can be cumbersome. Similarly, dividing complex numbers in rectangular form requires multiplying the numerator and denominator by the conjugate of the denominator, which can also be quite involved. However, in polar form (r(cos θ + i sin θ)), multiplication and division become much simpler. When multiplying two complex numbers in polar form, you simply multiply their moduli and add their arguments. When dividing two complex numbers, you divide their moduli and subtract their arguments. These operations are far more straightforward than their rectangular form counterparts.

For instance, consider multiplying two complex numbers z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂). In polar form, the product z₁z₂ is given by r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)). This simple formula highlights the ease with which complex number multiplication can be performed in polar form. Similarly, for division, z₁/z₂ = (r₁/r₂)(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂)), where z₂ ≠ 0. These operations demonstrate the efficiency and elegance of the polar form in handling complex number arithmetic. The simplification of multiplication and division is a key advantage of using polar form, making it a preferred method in many calculations.

The polar form is also instrumental in raising complex numbers to powers and finding roots, a process known as De Moivre's Theorem. De Moivre's Theorem states that for any complex number z = r(cos θ + i sin θ) and any integer n, zⁿ = rⁿ(cos(nθ) + i sin(nθ)). This theorem provides a straightforward method for calculating powers of complex numbers. In rectangular form, raising a complex number to a power involves repeated multiplication, which can be tedious. De Moivre's Theorem transforms this task into a simple calculation involving the modulus and argument. Similarly, finding the nth roots of a complex number is greatly simplified using the polar form and De Moivre's Theorem.

De Moivre's Theorem not only simplifies calculations but also provides a deeper understanding of the geometric behavior of complex numbers. It shows how repeated multiplication of a complex number affects its position in the complex plane, with the modulus raised to the power and the argument multiplied by the power. This geometric interpretation is invaluable in various applications, including signal processing and control systems. The ability to easily calculate powers and roots is a significant advantage of the polar form, making it a powerful tool in complex analysis.

Beyond mathematical operations, the polar form has significant applications in various fields of science and engineering. In electrical engineering, complex numbers are used to represent alternating currents (AC) and voltages. The polar form is particularly useful in AC circuit analysis, where the modulus represents the amplitude of the current or voltage, and the argument represents the phase angle. Using the polar form, engineers can easily analyze the behavior of AC circuits, calculate impedances, and determine power factors. The ability to represent sinusoidal signals using complex numbers in polar form greatly simplifies circuit analysis and design.

In physics, the polar form is used in wave mechanics, quantum mechanics, and signal processing. Wave phenomena, such as light and sound, can be described using complex numbers in polar form, where the modulus represents the amplitude of the wave, and the argument represents the phase. In quantum mechanics, wave functions are often represented using complex numbers, and the polar form helps in understanding the probabilistic nature of quantum systems. Signal processing utilizes complex numbers and the polar form to analyze and manipulate signals, such as audio and video signals. The polar form provides a convenient way to represent and process signals in the frequency domain, making it an essential tool in signal analysis and communication systems.

In conclusion, the polar form of complex numbers matters because it simplifies complex number arithmetic, facilitates the calculation of powers and roots, and has widespread applications in science and engineering. Its ability to streamline multiplication, division, exponentiation, and root extraction makes it an invaluable tool in mathematical analysis. Furthermore, its applications in electrical engineering, physics, and signal processing highlight its practical significance in solving real-world problems. The polar form provides a powerful and elegant way to represent and manipulate complex numbers, making it a fundamental concept in mathematics and its applications.

In summary, the polar form of the complex number 7√3 - 7i is 14(cos(330°) + i sin(330°)). This representation is obtained by first calculating the modulus, which is the distance from the origin to the complex number in the complex plane, and then determining the argument, which is the angle formed with the positive real axis. The modulus was found to be 14, and the argument was determined to be 330°, taking into account the quadrant in which the complex number lies. The polar form provides a clear and concise way to represent complex numbers, highlighting their magnitude and direction in the complex plane. This representation is particularly useful in various mathematical and practical applications.

Throughout this discussion, we have explored the fundamental concepts underlying the polar form of complex numbers. We began by defining complex numbers and their representation in the complex plane, emphasizing the importance of understanding the real and imaginary components. We then delved into the definition of the polar form, which expresses a complex number in terms of its modulus and argument. The modulus represents the magnitude or absolute value of the complex number, while the argument represents its direction or phase angle. We outlined the step-by-step process of converting a complex number from rectangular form (a + bi) to polar form (r(cos θ + i sin θ)), which involves calculating the modulus using the Pythagorean theorem and determining the argument using the arctangent function, while carefully considering the quadrant of the complex number.

We also examined the significance of the modulus and argument in representing complex numbers. The modulus provides information about the distance of the complex number from the origin, while the argument provides directional information. Together, they completely describe the position of the complex number in the complex plane. We discussed the properties of the modulus and argument, such as how they behave under multiplication and division of complex numbers. These properties are essential for simplifying complex number operations and understanding their geometric interpretations. Understanding the modulus and argument is crucial for effectively working with complex numbers in polar form.

Furthermore, we highlighted the importance of the polar form in simplifying complex number arithmetic. Multiplication and division, which can be cumbersome in rectangular form, become straightforward operations in polar form. Multiplication involves multiplying the moduli and adding the arguments, while division involves dividing the moduli and subtracting the arguments. These simplifications make the polar form a valuable tool in complex number calculations. We also discussed De Moivre's Theorem, which provides a simple method for raising complex numbers to powers and finding their roots. De Moivre's Theorem is a powerful application of the polar form, enabling efficient calculations of powers and roots.

Beyond mathematical operations, we explored the practical applications of the polar form in various fields. In electrical engineering, complex numbers in polar form are used to represent alternating currents and voltages, simplifying circuit analysis and design. In physics, the polar form is used in wave mechanics, quantum mechanics, and signal processing, providing a convenient way to represent and manipulate wave phenomena and signals. These applications underscore the versatility and importance of the polar form in solving real-world problems. The polar form is not just a theoretical concept; it is a practical tool with wide-ranging applications.

In conclusion, the polar form of a complex number, particularly the expression 14(cos(330°) + i sin(330°)) for 7√3 - 7i, offers a powerful and elegant way to represent complex numbers. It simplifies complex number operations, provides valuable geometric insights, and has numerous applications in science and engineering. Understanding the polar form is essential for anyone working with complex numbers, whether in mathematics, physics, engineering, or other related fields. The polar form not only facilitates calculations but also deepens our understanding of complex number behavior and their role in various phenomena. The ability to convert between rectangular and polar forms is a fundamental skill that empowers us to tackle complex problems with greater efficiency and insight.