Which Point Represents The Quotient Of 4 + 3 I − I \frac{4+3i}{-i} − I 4 + 3 I ​ ?A. B. C. D.

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on understanding the quotient of a complex number, specifically $\frac{4+3i}{-i}$. We will explore the concept of complex numbers, the rules for dividing complex numbers, and how to simplify the given expression.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$. The real part of a complex number is the part that is not multiplied by $i$, while the imaginary part is the part that is multiplied by $i$. For example, the complex number $3+4i$ has a real part of $3$ and an imaginary part of $4$.

Rules for Dividing Complex Numbers

When dividing complex numbers, we need to follow the rules for dividing fractions. The quotient of two complex numbers is defined as the product of the numerator and the reciprocal of the denominator. In other words, if we want to divide two complex numbers, we need to multiply the numerator by the reciprocal of the denominator.

Simplifying the Expression $\frac{4+3i}{-i}$

To simplify the expression $\frac{4+3i}{-i}$, we need to follow the rules for dividing complex numbers. We can start by multiplying the numerator by the reciprocal of the denominator:

$$\frac{4+3i}{-i} = \frac{4+3i}{-i} \cdot \frac{-i}{-i}$$

Multiplying the Numerator and the Reciprocal of the Denominator

When multiplying the numerator and the reciprocal of the denominator, we need to follow the rules for multiplying complex numbers. We can start by multiplying the real parts and the imaginary parts separately:

$$(4+3i) \cdot (-i) = -4i - 3i^2$$

Simplifying the Expression

Now that we have multiplied the numerator and the reciprocal of the denominator, we can simplify the expression by combining like terms:

$$-4i - 3i^2 = -4i + 3$$

Conclusion

In this article, we have explored the concept of complex numbers and the rules for dividing complex numbers. We have also simplified the expression $\frac{4+3i}{-i}$ by following the rules for dividing fractions and multiplying complex numbers. The final answer is $\boxed{3-4i}$.

Final Answer

The final answer is $\boxed{3-4i}$.

Discussion

The discussion category for this article is mathematics. If you have any questions or comments about this article, please feel free to ask.

Related Topics

• Complex numbers
• Dividing complex numbers
• Simplifying complex expressions

References

• "Complex Numbers" by Math Open Reference
• "Dividing Complex Numbers" by Purplemath
• "Simplifying Complex Expressions" by Mathway

Keywords

• Complex numbers
• Dividing complex numbers
• Simplifying complex expressions
• Quotient of complex numbers
• Imaginary unit
• Real part
• Imaginary part

Meta Description

This article provides an in-depth explanation of complex numbers and the rules for dividing complex numbers. We will explore the concept of complex numbers, the rules for dividing complex numbers, and how to simplify the given expression $\frac{4+3i}{-i}$. The final answer is $\boxed{3-4i}$.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In our previous article, we explored the concept of complex numbers and the rules for dividing complex numbers. In this article, we will provide a Q&A guide to help you better understand complex numbers and how to work with them.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies the equation $i^2 = -1$.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$, is a mathematical concept that satisfies the equation $i^2 = -1$. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

A: To add complex numbers, you need to add the real parts and the imaginary parts separately. For example, if you want to add $3+4i$ and $2+5i$, you would add the real parts (3 and 2) and the imaginary parts (4 and 5) separately:

$(3+4i) + (2+5i) = (3+2) + (4+5)i = 5+9i$

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately. For example, if you want to subtract $3+4i$ from $2+5i$, you would subtract the real parts (2 and 3) and the imaginary parts (5 and 4) separately:

$(2+5i) - (3+4i) = (2-3) + (5-4)i = -1+i$

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to follow the rules for multiplying binomials. For example, if you want to multiply $3+4i$ and $2+5i$, you would multiply the real parts and the imaginary parts separately:

$(3+4i)(2+5i) = (3)(2) + (3)(5i) + (4i)(2) + (4i)(5i)$

$= 6 + 15i + 8i + 20i^2$

$= 6 + 23i - 20$

$= -14 + 23i$

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to follow the rules for dividing fractions. For example, if you want to divide $3+4i$ by $2+5i$, you would multiply the numerator by the reciprocal of the denominator:

$\frac{3+4i}{2+5i} = \frac{3+4i}{2+5i} \cdot \frac{2-5i}{2-5i}$

$= \frac{(3+4i)(2-5i)}{(2+5i)(2-5i)}$

$= \frac{6-15i+8i-20i^2}{4-25i^2}$

$= \frac{6-7i+20}{4+25}$

$= \frac{26-7i}{29}$

$= \frac{26}{29} - \frac{7}{29}i$

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is a complex number with the same real part and the opposite imaginary part. For example, the conjugate of $3+4i$ is $3-4i$.

Q: How do I find the conjugate of a complex number?

A: To find the conjugate of a complex number, you need to change the sign of the imaginary part. For example, the conjugate of $3+4i$ is $3-4i$.

Conclusion

In this article, we have provided a Q&A guide to help you better understand complex numbers and how to work with them. We have covered topics such as adding, subtracting, multiplying, and dividing complex numbers, as well as finding the conjugate of a complex number. We hope that this guide has been helpful in your understanding of complex numbers.

Final Answer

The final answer is $\boxed{3-4i}$.

Discussion

The discussion category for this article is mathematics. If you have any questions or comments about this article, please feel free to ask.

Related Topics

• Complex numbers
• Adding complex numbers
• Subtracting complex numbers
• Multiplying complex numbers
• Dividing complex numbers
• Conjugate of a complex number

References

• "Complex Numbers" by Math Open Reference
• "Adding, Subtracting, Multiplying, and Dividing Complex Numbers" by Purplemath
• "Conjugate of a Complex Number" by Mathway

Keywords

• Complex numbers
• Adding complex numbers
• Subtracting complex numbers
• Multiplying complex numbers
• Dividing complex numbers
• Conjugate of a complex number
• Imaginary unit
• Real part
• Imaginary part