Write Z1 And Z2 In Polar Form

Here's how to express complex numbers z1 and z2 in their polar forms, along with explanations and examples to guide you through the process.

Understanding Polar Form

The polar form of a complex number offers a different perspective compared to its rectangular (or Cartesian) form. Instead of representing a complex number z as z = a + bi (where a is the real part and b is the imaginary part), the polar form expresses z in terms of its magnitude (or modulus) r and its argument (or angle) θ.

Think of it this way: the rectangular form tells you how far to move horizontally (a) and vertically (b) in the complex plane to reach the point representing z. The polar form, on the other hand, tells you how far to move along a straight line from the origin (r) and the angle θ that this line makes with the positive real axis.

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

z = r(cos θ + i sin θ)

Often, this is abbreviated as:

z = r cis θ

Where cis θ = cos θ + i sin θ. Euler's formula provides an even more compact representation:

z = re^iθ

Understanding these different notations is crucial for working with complex numbers in polar form.

Finding r (the Modulus)

The modulus r represents the distance of the complex number z from the origin in the complex plane. Given a complex number z = a + bi, the modulus r is calculated as:

r = |z| = √(a² + b²)

This formula is derived from the Pythagorean theorem. The real part a and the imaginary part b form the two legs of a right triangle, and the modulus r is the length of the hypotenuse.

Example:

Let's say z1 = 3 + 4i. To find the modulus r1 of z1:

r1 = |z1| = √(3² + 4²) = √(9 + 16) = √25 = 5

Therefore, the modulus of z1 is 5.

Finding θ (the Argument)

The argument θ is the angle that the line connecting the origin to the complex number z makes with the positive real axis. It's measured in radians or degrees (radians are preferred in many mathematical contexts).

The argument θ is found using the arctangent function:

θ = arctan(b/a)

However, it's extremely important to consider the quadrant in which the complex number z lies in the complex plane. The arctangent function has a range of (-π/2, π/2) or (-90°, 90°), meaning it only gives correct angles in the first and fourth quadrants. We need to adjust the angle accordingly for complex numbers in the second and third quadrants.

Here's a breakdown of how to find θ based on the quadrant:

• Quadrant I (a > 0, b > 0): θ = arctan(b/a)
• Quadrant II (a < 0, b > 0): θ = arctan(b/a) + π (or arctan(b/a) + 180°)
• Quadrant III (a < 0, b < 0): θ = arctan(b/a) - π (or arctan(b/a) - 180°)
• Quadrant IV (a > 0, b < 0): θ = arctan(b/a)

Important Considerations:

• Principal Argument: The principal argument, denoted as Arg(z), is the value of θ that lies within the interval (-π, π] or (-180°, 180°]. When finding the argument, ensure you're providing the principal argument unless otherwise specified.
• a = 0: If a = 0, then the complex number lies on the imaginary axis. In this case:
  • If b > 0, θ = π/2 (or 90°)
  • If b < 0, θ = -π/2 (or -90°)

Examples:

Let's continue with z1 = 3 + 4i and introduce z2 = -1 - i.

• Finding θ1 for z1 = 3 + 4i:

  • a = 3, b = 4
  • Since a > 0 and b > 0, z1 is in Quadrant I.
  • θ1 = arctan(4/3) ≈ 0.927 radians (or 53.13°)

• Finding θ2 for z2 = -1 - i:

  • a = -1, b = -1
  • Since a < 0 and b < 0, z2 is in Quadrant III.
  • θ2 = arctan(-1/-1) - π = arctan(1) - π = π/4 - π = -3π/4 radians (or -135°)

Expressing z1 and z2 in Polar Form

Now that we've calculated the modulus and argument for z1 and z2, we can write them in polar form.

• z1 = 3 + 4i:

  • r1 = 5
  • θ1 ≈ 0.927 radians
  • Therefore, z1 = 5(cos(0.927) + i sin(0.927)) or z1 = 5 cis(0.927) or z1 = 5e^i0.927

• z2 = -1 - i:

  • r2 = √((-1)² + (-1)²) = √2
  • θ2 = -3π/4 radians
  • Therefore, z2 = √2(cos(-3π/4) + i sin(-3π/4)) or z2 = √2 cis(-3π/4) or z2 = √2e^-i3π/4

Additional Examples and Scenarios

Let's explore some more examples to solidify your understanding.

Example 3: z3 = -2 + 2√3i

1. Find the modulus r3:

   • r3 = √((-2)² + (2√3)²) = √(4 + 12) = √16 = 4

2. Find the argument θ3:

   • a = -2, b = 2√3
   • Since a < 0 and b > 0, z3 is in Quadrant II.
   • θ3 = arctan((2√3)/-2) + π = arctan(-√3) + π = -π/3 + π = 2π/3 radians (or 120°)

3. Polar Form of z3:

   • z3 = 4(cos(2π/3) + i sin(2π/3)) or z3 = 4 cis(2π/3) or z3 = 4e^i2π/3

Example 4: z4 = 5i

1. Find the modulus r4:

   • z4 = 0 + 5i
   • r4 = √(0² + 5²) = √25 = 5

2. Find the argument θ4:

   • a = 0, b = 5
   • Since a = 0 and b > 0, z4 lies on the positive imaginary axis.
   • θ4 = π/2 radians (or 90°)

3. Polar Form of z4:

   • z4 = 5(cos(π/2) + i sin(π/2)) or z4 = 5 cis(π/2) or z4 = 5e^iπ/2

Example 5: z5 = -7

1. Find the modulus r5:

   • z5 = -7 + 0i
   • r5 = √((-7)² + 0²) = √49 = 7

2. Find the argument θ5:

   • a = -7, b = 0
   • Since a < 0 and b = 0, z5 lies on the negative real axis.
   • θ5 = π radians (or 180°)

3. Polar Form of z5:

   • z5 = 7(cos(π) + i sin(π)) or z5 = 7 cis(π) or z5 = 7e^iπ

Working with Degrees:

While radians are often preferred in theoretical mathematics, you might encounter situations where degrees are used. The process remains the same, but you'll use the degree mode on your calculator and express the argument in degrees. Remember to use the appropriate quadrant adjustments for degrees as well.

Why Use Polar Form?

While the rectangular form is convenient for addition and subtraction of complex numbers, the polar form shines when it comes to multiplication, division, and raising complex numbers to powers.

• Multiplication: If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then:

  • z1 * z2 = r1r2(cos(θ1 + θ2) + i sin(θ1 + θ2))

  In other words, to multiply complex numbers in polar form, you multiply their moduli and add their arguments.

• Division: If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then:

  • z1 / z2 = (r1/r2)(cos(θ1 - θ2) + i sin(θ1 - θ2))

  To divide complex numbers in polar form, you divide their moduli and subtract their arguments.

• De Moivre's Theorem: This theorem provides a powerful tool for raising a complex number to a power:

  • [r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))

  To raise a complex number to the power n, you raise the modulus to the power n and multiply the argument by n.

These operations are significantly simpler in polar form than in rectangular form, especially for higher powers.

Converting Back from Polar Form to Rectangular Form

To convert a complex number from polar form z = r(cos θ + i sin θ) back to rectangular form z = a + bi, you simply use the following relationships:

• a = r cos θ
• b = r sin θ

Example:

Let's convert z = 2(cos(π/3) + i sin(π/3)) back to rectangular form.

• a = 2 cos(π/3) = 2 * (1/2) = 1
• b = 2 sin(π/3) = 2 * (√3/2) = √3

Therefore, z = 1 + √3i.

Practical Applications

The polar form of complex numbers isn't just a mathematical curiosity; it has important applications in various fields:

• Electrical Engineering: Analyzing alternating current (AC) circuits is greatly simplified using complex numbers in polar form. The impedance, voltage, and current can be represented as complex numbers, and the phase relationships between them are easily handled using polar form operations.

• Physics: Wave phenomena, such as light and sound, can be described using complex numbers in polar form. The amplitude and phase of the wave are naturally represented by the modulus and argument, respectively.

• Signal Processing: The Fourier transform, a fundamental tool in signal processing, relies heavily on complex numbers and their polar representation. Analyzing the frequency components of a signal is made easier using the polar form.

• Navigation: Representing directions and distances in navigation systems can be done efficiently using complex numbers.

Common Mistakes to Avoid

• Forgetting Quadrant Adjustments: This is the most common mistake. Always visualize the complex number in the complex plane to determine the correct quadrant before calculating the argument.

• Using the Wrong Mode on Your Calculator: Ensure your calculator is in radian mode when working with radians and degree mode when working with degrees.

• Confusing Modulus and Absolute Value: While the modulus is often referred to as the absolute value of a complex number, remember that the absolute value of a real number is simply its distance from zero, while the modulus of a complex number is its distance from the origin in the complex plane.

• Not Simplifying: Always simplify your answers as much as possible. For example, if you find arctan(1), simplify it to π/4 or 45°.

Conclusion

Expressing complex numbers in polar form provides a powerful alternative to the rectangular form, particularly when dealing with multiplication, division, powers, and roots. Understanding how to find the modulus and argument, and remembering to account for quadrant adjustments, are key to mastering this concept. By practicing with various examples and understanding the applications of polar form, you can unlock a deeper understanding of complex numbers and their role in various scientific and engineering disciplines. Remember to always consider the context of the problem and choose the form (rectangular or polar) that best suits the task at hand.