Solved problems: Multiplying and dividing complex numbers in polar form

📚 Multiplying Complex Numbers in Polar Form

When multiplying complex numbers in polar form, we multiply their magnitudes and add their angles. If we have two complex numbers, $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, their product $z_1z_2$ is given by:

$z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$.

• 📏 Magnitudes: Multiply the magnitudes $r_1$ and $r_2$.
• 📐 Angles: Add the angles $\theta_1$ and $\theta_2$.

📚 Dividing Complex Numbers in Polar Form

When dividing complex numbers in polar form, we divide their magnitudes and subtract their angles. If we have two complex numbers, $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, their quotient $\frac{z_1}{z_2}$ is given by:

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$.

• ➗ Magnitudes: Divide the magnitude $r_1$ by $r_2$.
• ➖ Angles: Subtract the angle $\theta_2$ from $\theta_1$.

✨ Example: Multiplication

Let $z_1 = 2(\cos(30^\circ) + i\sin(30^\circ))$ and $z_2 = 3(\cos(60^\circ) + i\sin(60^\circ))$.

Then, $z_1z_2 = (2)(3)[\cos(30^\circ + 60^\circ) + i\sin(30^\circ + 60^\circ)] = 6(\cos(90^\circ) + i\sin(90^\circ)) = 6(0 + i) = 6i$.

🧮 Example: Division

Let $z_1 = 8(\cos(120^\circ) + i\sin(120^\circ))$ and $z_2 = 2(\cos(30^\circ) + i\sin(30^\circ))$.

Then, $\frac{z_1}{z_2} = \frac{8}{2}[\cos(120^\circ - 30^\circ) + i\sin(120^\circ - 30^\circ)] = 4(\cos(90^\circ) + i\sin(90^\circ)) = 4(0 + i) = 4i$.

📝 Practice Quiz

Here are some practice problems to test your understanding:

1. ❓ If $z_1 = 4(\cos(45^\circ) + i\sin(45^\circ))$ and $z_2 = 2(\cos(15^\circ) + i\sin(15^\circ))$, find $z_1z_2$.
2. ➗ If $z_1 = 10(\cos(135^\circ) + i\sin(135^\circ))$ and $z_2 = 5(\cos(45^\circ) + i\sin(45^\circ))$, find $\frac{z_1}{z_2}$.
3. ➕ If $z_1 = 3(\cos(60^\circ) + i\sin(60^\circ))$ and $z_2 = 2(\cos(30^\circ) + i\sin(30^\circ))$, find $z_1z_2$.

🔑 Solutions

1. $z_1z_2 = 8(\cos(60^\circ) + i\sin(60^\circ)) = 8(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = 4 + 4i\sqrt{3}$.
2. $\frac{z_1}{z_2} = 2(\cos(90^\circ) + i\sin(90^\circ)) = 2(0 + i) = 2i$.
3. $z_1z_2 = 6(\cos(90^\circ) + i\sin(90^\circ)) = 6(0 + i) = 6i$.

📚 Understanding Complex Numbers in Polar Form

Complex numbers can be represented in rectangular form ($a + bi$) or polar form ($r(\cos \theta + i \sin \theta)$), where $r$ is the magnitude and $\theta$ is the angle. Converting between these forms is crucial for understanding multiplication and division.

🧮 Multiplication of Complex Numbers in Polar Form

To multiply two complex numbers in polar form, multiply their magnitudes and add their angles. If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then:

$z_1 \cdot z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$.

• 📏 Magnitude: Multiply the magnitudes ($r_1$ and $r_2$).
• 📐 Angle: Add the angles ($\theta_1$ and $\theta_2$).

➗ Division of Complex Numbers in Polar Form

To divide two complex numbers in polar form, divide their magnitudes and subtract their angles. If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then:

$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$.

• 📉 Magnitude: Divide the magnitudes ($r_1$ by $r_2$).
• 🧭 Angle: Subtract the angles ($\theta_2$ from $\theta_1$).

✏️ Example 1: Multiplication

Let $z_1 = 2(\cos 30^\circ + i \sin 30^\circ)$ and $z_2 = 3(\cos 60^\circ + i \sin 60^\circ)$.

Then, $z_1 \cdot z_2 = (2 \cdot 3)[\cos(30^\circ + 60^\circ) + i \sin(30^\circ + 60^\circ)] = 6(\cos 90^\circ + i \sin 90^\circ) = 6(0 + i) = 6i$.

🖋️ Example 2: Division

Let $z_1 = 8(\cos 120^\circ + i \sin 120^\circ)$ and $z_2 = 4(\cos 30^\circ + i \sin 30^\circ)$.

Then, $\frac{z_1}{z_2} = \frac{8}{4}[\cos(120^\circ - 30^\circ) + i \sin(120^\circ - 30^\circ)] = 2(\cos 90^\circ + i \sin 90^\circ) = 2(0 + i) = 2i$.

💡 Key Principles

• ➕ Addition of Angles: When multiplying complex numbers, the angles are added.
• ➖ Subtraction of Angles: When dividing complex numbers, the angles are subtracted.
• 🔢 Magnitude Operations: Magnitudes are multiplied during multiplication and divided during division.

➗ Practice Quiz

1. Express $z_1 = 5(\cos 45^\circ + i \sin 45^\circ)$ and $z_2 = 2(\cos 15^\circ + i \sin 15^\circ)$. Find $z_1 \cdot z_2$.
2. Given $z_1 = 10(\cos 135^\circ + i \sin 135^\circ)$ and $z_2 = 5(\cos 45^\circ + i \sin 45^\circ)$. Calculate $\frac{z_1}{z_2}$.
3. If $z_1 = 4(\cos 60^\circ + i \sin 60^\circ)$ and $z_2 = 2(\cos 30^\circ + i \sin 30^\circ)$, determine $z_1 \cdot z_2$.
4. Calculate $\frac{z_1}{z_2}$ where $z_1 = 12(\cos 180^\circ + i \sin 180^\circ)$ and $z_2 = 3(\cos 60^\circ + i \sin 60^\circ)$.
5. Find the product of $z_1 = 3(\cos 90^\circ + i \sin 90^\circ)$ and $z_2 = 4(\cos 0^\circ + i \sin 0^\circ)$.

✅ Conclusion

Multiplying and dividing complex numbers in polar form simplifies calculations by converting them into straightforward magnitude and angle operations. Mastery of these concepts enhances problem-solving skills in various mathematical and engineering contexts.

📚 Understanding Complex Numbers in Polar Form

Complex numbers, often expressed in the form $a + bi$, can also be represented in polar form as $r(\cos \theta + i \sin \theta)$ or $re^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument. This representation simplifies multiplication and division.

📜 A Brief History

The use of polar coordinates to represent complex numbers became prominent in the 18th century, thanks to mathematicians like Abraham de Moivre and Leonhard Euler. De Moivre's theorem, a cornerstone in complex number theory, elegantly demonstrates the power of polar representation in simplifying complex number operations.

➗ Multiplication of Complex Numbers in Polar Form

If you have two complex numbers, $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, their product $z_1z_2$ is given by:

$z_1z_2 = r_1r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$.

In simpler terms, you multiply the magnitudes and add the angles.

• ➕ Multiply Magnitudes: 🔢 Multiply the magnitudes $r_1$ and $r_2$.
• 📐 Add Arguments: ➕ Add the arguments (angles) $\theta_1$ and $\theta_2$.

➗ Division of Complex Numbers in Polar Form

For division, if $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then $z_1/z_2$ is:

$\frac{z_1}{z_2} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$.

Here, you divide the magnitudes and subtract the angles.

• ➗ Divide Magnitudes: ➗ Divide the magnitude $r_1$ by $r_2$.
• ➖ Subtract Arguments: 📐 Subtract the argument (angle) $\theta_2$ from $\theta_1$.

💡 Practical Examples

Example 1: Multiplication

Let $z_1 = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$ and $z_2 = 3(\cos(\frac{\pi}{6}) + i \sin(\frac{\pi}{6}))$.

$z_1z_2 = (2)(3) [\cos(\frac{\pi}{3} + \frac{\pi}{6}) + i \sin(\frac{\pi}{3} + \frac{\pi}{6})] = 6(\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})) = 6i$.

Example 2: Division

Let $z_1 = 4(\cos(\frac{2\pi}{3}) + i \sin(\frac{2\pi}{3}))$ and $z_2 = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$.

$\frac{z_1}{z_2} = \frac{4}{2} [\cos(\frac{2\pi}{3} - \frac{\pi}{3}) + i \sin(\frac{2\pi}{3} - \frac{\pi}{3})] = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})) = 1 + i\sqrt{3}$.

🔑 Key Principles

• 🧭 Magnitude: 📏 The magnitude $r$ represents the distance from the origin to the point in the complex plane.
• 🧮 Argument: 📐 The argument $\theta$ is the angle formed with the positive real axis.
• 🔄 Euler's Formula: 💡 $e^{i\theta} = \cos \theta + i \sin \theta$ provides a compact notation for complex numbers in polar form.

🌍 Real-World Applications

Polar representation is crucial in various fields:

• 🧲 Electrical Engineering: ⚡ Analyzing AC circuits.
• 📡 Signal Processing: 🔊 Representing and manipulating signals.
• 🛰️ Physics: ⚛️ Quantum mechanics and wave phenomena.

📝 Conclusion

Multiplying and dividing complex numbers in polar form simplifies calculations by converting them into straightforward operations on magnitudes and angles. Understanding this concept is essential for advanced mathematical and engineering applications.

📚 Understanding Complex Numbers in Polar Form

Complex numbers, usually written in the form $a + bi$, can also be represented in polar form. This form uses the magnitude (or modulus) $r$ and the angle (or argument) $\theta$ to define the complex number.

📜 Historical Context

The development of complex numbers dates back to the 16th century, but their representation in polar form became significant with advancements in complex analysis. Mathematicians like Carl Friedrich Gauss contributed to formalizing these concepts.

📌 Key Principles

• 📏 Polar Representation: A complex number $z = a + bi$ can be written as $z = r(\cos \theta + i \sin \theta)$, where $r = \sqrt{a^2 + b^2}$ and $\theta = \arctan(\frac{b}{a})$.
• ➕ Multiplication: If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then $z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$. In essence, multiply the magnitudes and add the angles.
• ➗ Division: If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$. Divide the magnitudes and subtract the angles.
• 💡 Euler's Formula: This provides a compact notation: $z = re^{i\theta}$, where $e^{i\theta} = \cos \theta + i \sin \theta$.

🧮 Practical Examples

Example 1: Multiplication

Let $z_1 = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$ and $z_2 = 3(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})$.

Then, $z_1z_2 = (2)(3)[\cos(\frac{\pi}{3} + \frac{\pi}{6}) + i \sin(\frac{\pi}{3} + \frac{\pi}{6})] = 6(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 6(0 + i) = 6i$.

Example 2: Division

Let $z_1 = 8(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})$ and $z_2 = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$.

Then, $\frac{z_1}{z_2} = \frac{8}{2}[\cos(\frac{5\pi}{6} - \frac{\pi}{3}) + i \sin(\frac{5\pi}{6} - \frac{\pi}{3})] = 4(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2}) = 4(0 + i) = 4i$.

Example 3: Using Euler's Formula

Let $z_1 = 4e^{i\frac{2\pi}{3}}$ and $z_2 = 2e^{i\frac{\pi}{4}}$.

For multiplication: $z_1z_2 = (4)(2)e^{i(\frac{2\pi}{3} + \frac{\pi}{4})} = 8e^{i\frac{11\pi}{12}}$.

For division: $\frac{z_1}{z_2} = \frac{4}{2}e^{i(\frac{2\pi}{3} - \frac{\pi}{4})} = 2e^{i\frac{5\pi}{12}}$.

📝 Conclusion

Multiplying and dividing complex numbers in polar form simplifies complex calculations to basic arithmetic operations on magnitudes and angles. Understanding these principles allows for efficient problem-solving in various fields, including electrical engineering and physics.

📚 Understanding Complex Numbers in Polar Form

Complex numbers, often written in the form $a + bi$, can also be represented in polar form. This representation simplifies multiplication and division. The polar form of a complex number $z = a + bi$ is given by $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude (or modulus) of $z$, and $\theta$ is the argument (or angle) of $z$.

📜 A Brief History

The development of complex numbers dates back to the 16th century, with mathematicians like Gerolamo Cardano grappling with solutions to cubic equations. However, it was in the 18th and 19th centuries that mathematicians such as Carl Friedrich Gauss and Jean-Robert Argand formalized the concept of complex numbers and their geometric representation. The polar form emerged as a natural way to express complex numbers, offering advantages in certain operations, particularly multiplication and division.

📌 Key Principles for Multiplication and Division

• 📏 Multiplication:
• ➕ If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then $z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$.
• 🔑 Multiply the magnitudes and add the angles.
• ➗ Division:
• ➖ If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then $\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)]$.
• 🔑 Divide the magnitudes and subtract the angles.

🧮 Step-by-Step Examples

Example 1: Multiplication

Let $z_1 = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$ and $z_2 = 3(\cos(\frac{\pi}{6}) + i \sin(\frac{\pi}{6}))$. Find $z_1z_2$.

Solution:

$z_1z_2 = (2)(3)[\cos(\frac{\pi}{3} + \frac{\pi}{6}) + i \sin(\frac{\pi}{3} + \frac{\pi}{6})] = 6[\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})] = 6(0 + i) = 6i$.

Example 2: Division

Let $z_1 = 8(\cos(\frac{5\pi}{6}) + i \sin(\frac{5\pi}{6}))$ and $z_2 = 2(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$. Find $\frac{z_1}{z_2}$.

Solution:

$\frac{z_1}{z_2} = \frac{8}{2}[\cos(\frac{5\pi}{6} - \frac{\pi}{3}) + i \sin(\frac{5\pi}{6} - \frac{\pi}{3})] = 4[\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})] = 4(0 + i) = 4i$.

✍️ Practice Quiz

Solve the following problems:

1. ❓Let $z_1 = 4(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))$ and $z_2 = 2(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))$. Find $z_1z_2$.
2. ➗ Let $z_1 = 10(\cos(\frac{2\pi}{3}) + i \sin(\frac{2\pi}{3}))$ and $z_2 = 5(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$. Find $\frac{z_1}{z_2}$.
3. ➕ Let $z_1 = 3(\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}))$ and $z_2 = 2(\cos(\pi) + i \sin(\pi))$. Find $z_1z_2$.

💡 Solutions to Practice Quiz

1. ✅ $z_1z_2 = 4 * 2 [\cos(\frac{\pi}{4} + \frac{\pi}{4}) + i \sin(\frac{\pi}{4} + \frac{\pi}{4})] = 8[\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2})] = 8i$
2. ✅ $\frac{z_1}{z_2} = \frac{10}{5} [\cos(\frac{2\pi}{3} - \frac{\pi}{3}) + i \sin(\frac{2\pi}{3} - \frac{\pi}{3})] = 2[\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})] = 2(\frac{1}{2} + i\frac{\sqrt{3}}{2}) = 1 + i\sqrt{3}$
3. ✅ $z_1z_2 = 3 * 2 [\cos(\frac{\pi}{2} + \pi) + i \sin(\frac{\pi}{2} + \pi)] = 6[\cos(\frac{3\pi}{2}) + i \sin(\frac{3\pi}{2})] = 6[0 - i] = -6i$

🔑 Conclusion

Multiplying and dividing complex numbers in polar form simplifies calculations by converting them into operations on magnitudes and angles. This method is especially useful in fields like electrical engineering and physics, where complex numbers frequently appear. Understanding these principles allows for efficient problem-solving and a deeper appreciation of complex number theory.