Embarking on the journey through complex numbers, the polar form offers a unique and insightful perspective, especially when dealing with multiplication and division of complex numbers. This form, which utilises the modulus and argument, and the application of De Moivre’s theorem, is pivotal in simplifying complex number arithmetic and understanding the geometric interpretation of complex numbers.
Modulus and Argument
Modulus
- The modulus of a complex number, denoted as |z|, is essentially its absolute value or magnitude.
- It is the distance from the origin (0,0) to the point representing the complex number in the complex plane.
- Mathematically, for a complex number z = a + bi, the modulus is calculated as |z| = sqrt(a2 + b2).
- The modulus is always non-negative and is particularly useful in representing the size of the complex number.
Argument
- The argument of a complex number, denoted as arg(z), is the angle formed by the line segment from the origin to z and the positive real axis.
- It is measured in radians or degrees and can be found using trigonometric and inverse trigonometric functions.
- The principal value of the argument, Arg(z), is the value of the argument such that -pi < theta <= pi.
Example 1
Consider z = 3 + 4i. The modulus is |z| = sqrt(32 + 42) = 5. The argument can be found using theta = arctan(b/a) = arctan(4/3).
Polar Form Representation
Conversion between Forms
- To convert from rectangular to polar form, use the modulus and argument as described above.
- To convert from polar to rectangular form, use: a = r cos theta b = r sin theta
Example 2
For z = 3 + 4i, the polar form is: z = 5(cos arctan(4/3) + i sin arctan(4/3))
De Moivre's Theorem
Theorem Statement
If z = r(cos theta + i sin theta), then for any integer n, zn = rn (cos ntheta + i sin ntheta)
Applications
- Calculating Powers: Directly find the power of a complex number without multiplying it by itself repeatedly.
- Roots of Unity: Determine the nth roots of 1, which are crucial in polynomial equations.
- Complex Number Division: Facilitates division by converting to polar form and using properties of exponents.
Example 3
Find z2 where z = 3 + 4i.
Using De Moivre's theorem and the polar form calculated above: z2 = 25(cos 2arctan(4/3) + i sin 2arctan(4/3))
Detailed Exploration of Concepts
Geometric Interpretation
- The modulus represents the distance of the complex number from the origin in the complex plane.
- The argument gives the direction (or angle) at which the complex number is located from the real axis.
Polar Form and Operations
- Addition and Subtraction: While the polar form is not typically used for addition and subtraction, it can still be done by converting back to rectangular form.
- Multiplication: The product of two complex numbers in polar form is a complex number with a modulus equal to the product of the moduli and an argument equal to the sum of the arguments.
- Division: The quotient of two complex numbers in polar form is a complex number with a modulus equal to the quotient of the moduli and an argument equal to the difference of the arguments.
De Moivre’s Theorem and Complex Powers
- Positive Integer Powers: The theorem allows for easy calculation of positive integer powers of complex numbers.
- Negative Powers: For negative powers, the reciprocal of the modulus is taken, and the sign of the argument is reversed.
- Non-Integer Powers: Non-integer powers involve finding roots and can be used to solve equations like zn = w.
Complex Roots and De Moivre’s Theorem
- Principal Root: The principal nth root is found by dividing the argument by n and taking the nth root of the modulus.
- Additional Roots: Additional roots are found by adding multiples of (2*pi/n) to the argument of the principal root.
Example 4
Find the square root of z = 3 + 4i.
Using De Moivre’s theorem and considering n = 1/2, we find the principal root and then determine additional roots by considering the periodicity of sine and cosine.
Applications in Mathematics
- Solving Polynomial Equations: De Moivre’s theorem aids in finding the roots of polynomial equations, especially when complex roots are involved.
- Trigonometry: The theorem provides relationships between the trigonometric functions of multiple angles, facilitating the derivation of double and triple angle formulas.
Example 5
Solve z3 = 1.
Using De Moivre’s theorem, we find the three cube roots of unity and express them in rectangular form.
FAQ
The argument of a complex number is not unique due to the periodic nature of trigonometric functions. Specifically, sine and cosine functions are periodic with period 2π, meaning that adding any multiple of 2π to the argument of a complex number will not change its position in the complex plane. Mathematically, if θ is an argument of a complex number z, then θ + 2kπ, where k is an integer, is also an argument of z. This property is crucial in solving problems involving roots of complex numbers, as it allows us to find all possible values for the argument and, consequently, all distinct roots.
No, the modulus (or absolute value) of a complex number cannot be negative. By definition, the modulus of a complex number z = a + bi is |z| = sqrt(a2 + b2), which is the distance from the origin to the point representing the complex number in the complex plane. Distance is always non-negative, so the modulus of a complex number is always non-negative. This is crucial in understanding the geometric representation of complex numbers and ensures that calculations involving the modulus, such as converting between rectangular and polar forms, remain valid and consistent.
The polar form of complex numbers, z = r(cos(θ) + i sin(θ)), significantly simplifies the multiplication and division operations. When multiplying two complex numbers in polar form, z1 = r1(cos(θ1) + i sin(θ1)) and z2 = r2(cos(θ2) + i sin(θ2)), you simply multiply the moduli and add the arguments: z1 × z2 = r1r2(cos(θ1 + θ2) + i sin(θ1 + θ2)). Similarly, for division, you divide the moduli and subtract the arguments. This simplification is particularly useful in calculations involving powers and roots of complex numbers, making the operations more straightforward and less computationally intensive.
In electrical engineering, the argument of a complex number, especially in the context of impedance, is of paramount significance as it represents the phase difference between the voltage across and the current through an electrical component. When impedance is represented in polar form, the modulus indicates the magnitude of the impedance, while the argument provides the phase angle. This phase angle is crucial in analysing AC circuits, as it helps engineers to understand the phase relationship between different circuit elements, enabling them to design circuits that function optimally by considering the phase shifts introduced by various components.
Euler's Formula, e(iθ) = cos(θ) + i sin(θ), provides a profound link between exponential and trigonometric functions and is pivotal in expressing complex numbers in their polar form. When a complex number is represented as z = r(cos(θ) + i sin(θ)), using Euler's Formula, it can also be written as z = re(iθ). This form is particularly useful in simplifying operations like multiplication and finding powers of complex numbers, as it allows us to work with the modulus and argument separately. Moreover, it elegantly connects algebra, geometry, and calculus in the study of complex numbers, providing a versatile tool in various mathematical analyses.
Practice Questions
To find the polar form of z = 1 - i, we calculate the modulus and argument of z. The modulus r is given by r = sqrt(12 + (-1)2) = sqrt(2). The argument theta can be found as theta = arctan(-1/1) = -45 degrees or -pi/4 radians. Thus, the polar form of z is z = sqrt(2)(cos(-pi/4) + i sin(-pi/4)). To find the fourth power of z using De Moivre’s theorem, we raise the modulus to the fourth power and multiply the argument by 4: z4 = (sqrt(2))4 (cos(4(-pi/4)) + i sin(4(-pi/4))) = 2(cos(-pi) + i sin(-pi)) = -2, since cos(-pi) = -1 and sin(-pi) = 0.
First, we find the polar form of z = -1 + sqrt(3)i. The modulus r is r = sqrt((-1)2 + (sqrt(3))2) = 2 and the argument theta is theta = arctan(sqrt(3)/(-1)) = 2pi/3. So, z = 2(cos(2pi/3) + i sin(2pi/3)). To find the cube roots, we use De Moivre’s theorem with n = 1/3: z(1/3) = 2(1/3)(cos((2pi/3 + 2kpi)/3) + i sin((2pi/3 + 2kpi)/3)), where k = 0, 1, 2. For k = 0: z_1 = 2(1/3)(cos(2pi/9) + i sin(2pi/9)). For k = 1: z_2 = 2(1/3)(cos(8pi/9) + i sin(8pi/9)). For k = 2: z_3 = 2(1/3)(cos(14pi/9) + i sin(14pi/9)). To express these roots in rectangular form, we can use the known trigonometric values for these angles or use a calculator to approximate them.
