Products And Quotients Of Complex NumbersFind The Product Z 1 Z 2 Z_1 Z_2 Z 1 ​ Z 2 ​ And The Quotient Z 1 Z 2 \frac{z_1}{z_2} Z 2 ​ Z 1 ​ ​ . Express Your Answers In Polar Form. Then Convert Each Answer To Rectangular Form A + B I A + B I A + Bi .Given:$z_1 = 3\left(\cos

Introduction

Complex numbers are an extension of the real numbers, and they have numerous applications in mathematics, physics, and engineering. In this article, we will explore the product and quotient of complex numbers, and we will express our answers in polar form. We will also convert each answer to rectangular form.

Polar Form of Complex Numbers

A complex number $z$ can be expressed in polar form as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of the complex number and $\theta$ is the argument. The magnitude of a complex number is given by $r = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number, respectively. The argument of a complex number is given by $\theta = \tan^{-1} \left(\frac{b}{a}\right)$.

Product of Complex Numbers

The product of two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$ is given by $z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$. This can be verified by multiplying the two complex numbers using the distributive property and the fact that $i^2 = -1$.

Example 1

Find the product of $z_1 = 3(\cos \pi/4 + i \sin \pi/4)$ and $z_2 = 2(\cos \pi/3 + i \sin \pi/3)$.

Solution

Using the formula for the product of complex numbers, we have:

$$z_1 z_2 = 3 \cdot 2 (\cos (\pi/4 + \pi/3) + i \sin (\pi/4 + \pi/3))$$

$$= 6 (\cos (7\pi/12) + i \sin (7\pi/12))$$

To convert this to rectangular form, we can use the fact that $\cos (7\pi/12) = \cos (2\pi - 5\pi/12) = \cos (5\pi/12)$ and $\sin (7\pi/12) = \sin (2\pi - 5\pi/12) = -\sin (5\pi/12)$. Therefore, we have:

$$z_1 z_2 = 6 (\cos (5\pi/12) - i \sin (5\pi/12))$$

Using a calculator, we can find that $\cos (5\pi/12) = 0.2588$ and $\sin (5\pi/12) = 0.9659$. Therefore, we have:

$$z_1 z_2 = 6 (0.2588 - 0.9659i)$$

$$= 1.5532 - 5.7954i$$

Quotient of Complex Numbers

The quotient of two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$ is given by $z_1/z_2 = r_1/r_2 (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$. This can be verified by dividing the two complex numbers using the distributive property and the fact that $i^2 = -1$.

Example 2

Find the quotient of $z_1 = 3(\cos \pi/4 + i \sin \pi/4)$ and $z_2 = 2(\cos \pi/3 + i \sin \pi/3)$.

Solution

Using the formula for the quotient of complex numbers, we have:

$$z_1/z_2 = 3/2 (\cos (\pi/4 - \pi/3) + i \sin (\pi/4 - \pi/3))$$

$$= 1.5 (\cos (-\pi/12) + i \sin (-\pi/12))$$

To convert this to rectangular form, we can use the fact that $\cos (-\pi/12) = \cos (\pi/12)$ and $\sin (-\pi/12) = -\sin (\pi/12)$. Therefore, we have:

$$z_1/z_2 = 1.5 (\cos (\pi/12) - i \sin (\pi/12))$$

Using a calculator, we can find that $\cos (\pi/12) = 0.9659$ and $\sin (\pi/12) = 0.2588$. Therefore, we have:

$$z_1/z_2 = 1.5 (0.9659 - 0.2588i)$$

$$= 1.4464 - 0.3892i$$

Conclusion

Introduction

In our previous article, we explored the product and quotient of complex numbers, and we expressed our answers in polar form. We also converted each answer to rectangular form. In this article, we will answer some frequently asked questions about the product and quotient of complex numbers.

Q: What is the product of two complex numbers?

A: The product of two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$ is given by $z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$.

Q: How do I find the product of two complex numbers?

A: To find the product of two complex numbers, you can use the formula $z_1 z_2 = r_1 r_2 (\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$. You can also use the distributive property and the fact that $i^2 = -1$ to multiply the two complex numbers.

Q: What is the quotient of two complex numbers?

A: The quotient of two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$ is given by $z_1/z_2 = r_1/r_2 (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$.

Q: How do I find the quotient of two complex numbers?

A: To find the quotient of two complex numbers, you can use the formula $z_1/z_2 = r_1/r_2 (\cos (\theta_1 - \theta_2) + i \sin (\theta_1 - \theta_2))$. You can also use the distributive property and the fact that $i^2 = -1$ to divide the two complex numbers.

Q: Can I use the product and quotient formulas to simplify complex number operations?

A: Yes, you can use the product and quotient formulas to simplify complex number operations. For example, you can use the product formula to simplify the expression $(a + bi)(c + di)$, and you can use the quotient formula to simplify the expression $(a + bi)/(c + di)$.

Q: Can I use the product and quotient formulas to convert complex numbers from polar form to rectangular form?

A: Yes, you can use the product and quotient formulas to convert complex numbers from polar form to rectangular form. For example, you can use the product formula to convert the polar form $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ to the rectangular form $z_1 = a + bi$, and you can use the quotient formula to convert the polar form $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$ to the rectangular form $z_2 = c + di$.

Q: Are there any other formulas for the product and quotient of complex numbers?

A: Yes, there are other formulas for the product and quotient of complex numbers. For example, you can use the formula $z_1 z_2 = (a_1 + b_1 i)(a_2 + b_2 i) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1)i$ to find the product of two complex numbers, and you can use the formula $z_1/z_2 = (a_1 + b_1 i)/(a_2 + b_2 i) = ((a_1 a_2 + b_1 b_2)/(a_2^2 + b_2^2)) + ((a_2 b_1 - a_1 b_2)/(a_2^2 + b_2^2))i$ to find the quotient of two complex numbers.

Conclusion

In this article, we have answered some frequently asked questions about the product and quotient of complex numbers. We have also provided formulas for the product and quotient of complex numbers, and we have explained how to use these formulas to simplify complex number operations and to convert complex numbers from polar form to rectangular form.