Divide, Writing The Answer In Standard Form: $\frac{-5-2i}{12+13i}$.Provide Your Answer Below:

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Introduction

In mathematics, division of complex numbers is a fundamental operation that involves dividing one complex number by another. The process of dividing complex numbers is similar to dividing real numbers, but it requires a different approach due to the presence of imaginary units. In this article, we will explore the process of dividing complex numbers and provide a step-by-step guide on how to write the answer in standard form.

What are Complex Numbers?

Before we dive into the process of dividing complex numbers, let's first understand what complex numbers are. 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. The imaginary unit $i$ is defined as the square root of $-1$, denoted by $i = \sqrt{-1}$. Complex numbers can be represented graphically on a complex plane, with the real part $a$ on the x-axis and the imaginary part $b$ on the y-axis.

The Process of Dividing Complex Numbers

To divide two complex numbers, we can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator: The conjugate of a complex number $a+bi$ is defined as $a-bi$. By multiplying the numerator and denominator by the conjugate of the denominator, we can eliminate the imaginary part from the denominator.
2. Simplify the expression: After multiplying the numerator and denominator by the conjugate of the denominator, we can simplify the expression by combining like terms.
3. Write the answer in standard form: The final step is to write the answer in standard form, which is in the form $a+bi$.

Example: Dividing Complex Numbers

Let's consider the example of dividing the complex numbers $\frac{-5-2i}{12+13i}$. To divide these complex numbers, we can follow the steps outlined above.

Step 1: Multiply the numerator and denominator by the conjugate of the denominator

The conjugate of the denominator $12+13i$ is $12-13i$. We can multiply the numerator and denominator by the conjugate of the denominator as follows:

$$\frac{-5-2i}{12+13i} \cdot \frac{12-13i}{12-13i}$$

Step 2: Simplify the expression

After multiplying the numerator and denominator by the conjugate of the denominator, we can simplify the expression by combining like terms:

$$\frac{(-5-2i)(12-13i)}{(12+13i)(12-13i)}$$

Expanding the numerator and denominator, we get:

$$\frac{-60+65i-24i+26i^2}{144-169i^2}$$

Since $i^2 = -1$, we can substitute this value into the expression:

$$\frac{-60+65i-24i-26}{144+169}$$

Combining like terms, we get:

$$\frac{-86+41i}{313}$$

Step 3: Write the answer in standard form

The final step is to write the answer in standard form, which is in the form $a+bi$. In this case, the answer is:

$$\frac{-86}{313} + \frac{41}{313}i$$

Conclusion

In this article, we have explored the process of dividing complex numbers and provided a step-by-step guide on how to write the answer in standard form. By following the steps outlined above, we can divide complex numbers and express the answer in standard form. This is an essential skill in mathematics, particularly in algebra and calculus.

Frequently Asked Questions

• What is the conjugate of a complex number? The conjugate of a complex number $a+bi$ is defined as $a-bi$.
• How do I multiply complex numbers? To multiply complex numbers, we can use the distributive property and FOIL method.
• How do I divide complex numbers? To divide complex numbers, we can use the steps outlined above: multiply the numerator and denominator by the conjugate of the denominator, simplify the expression, and write the answer in standard form.

Final Thoughts

Dividing complex numbers is a fundamental operation in mathematics that requires a different approach than dividing real numbers. By following the steps outlined above, we can divide complex numbers and express the answer in standard form. This is an essential skill in mathematics, particularly in algebra and calculus. With practice and patience, you can master the art of dividing complex numbers and become proficient in mathematics.

Introduction

Dividing complex numbers can be a challenging task, especially for those who are new to the concept. However, with practice and patience, you can master the art of dividing complex numbers and become proficient in mathematics. In this article, we will answer some of the most frequently asked questions about dividing complex numbers.

Q&A

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a+bi$ is defined as $a-bi$. The conjugate of a complex number is used to eliminate the imaginary part from the denominator when dividing complex numbers.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and FOIL method. The FOIL method is a mnemonic device that helps you remember the order of operations when multiplying complex numbers:

• F: First, multiply the first terms in each parentheses
• O: Outer, multiply the outer terms in each parentheses
• I: Inner, multiply the inner terms in each parentheses
• L: Last, multiply the last terms in each parentheses

For example, to multiply the complex numbers $(3+4i)$ and $(5-2i)$, you can use the FOIL method as follows:

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

• F: $3 \cdot 5 = 15$
• O: $3 \cdot -2i = -6i$
• I: $4i \cdot 5 = 20i$
• L: $4i \cdot -2i = -8i^2$

Since $i^2 = -1$, we can substitute this value into the expression:

$15 - 6i + 20i - 8(-1)$

Combining like terms, we get:

$23 + 14i$

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the following steps:

1. Multiply the numerator and denominator by the conjugate of the denominator
2. Simplify the expression
3. Write the answer in standard form

For example, to divide the complex numbers $\frac{-5-2i}{12+13i}$, you can follow the steps outlined above.

Q: What is the standard form of a complex number?

A: The standard form of a complex number is in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use the following steps:

1. Combine like terms
2. Simplify the expression using the rules of arithmetic
3. Write the answer in standard form

For example, to simplify the complex expression $3(2+4i) + 2(1-3i)$, you can follow the steps outlined above.

Q: What is the difference between a complex number and a real 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. A real number is a number that can be expressed in the form $a$, where $a$ is a real number.

Q: Can complex numbers be used in real-world applications?

A: Yes, complex numbers can be used in real-world applications such as engineering, physics, and computer science. Complex numbers are used to represent quantities that have both magnitude and direction, such as electrical currents and voltages.

Conclusion

Dividing complex numbers can be a challenging task, but with practice and patience, you can master the art of dividing complex numbers and become proficient in mathematics. In this article, we have answered some of the most frequently asked questions about dividing complex numbers. We hope that this article has been helpful in clarifying any doubts you may have had about dividing complex numbers.

Final Thoughts

Dividing complex numbers is an essential skill in mathematics, particularly in algebra and calculus. With practice and patience, you can master the art of dividing complex numbers and become proficient in mathematics. Remember to always follow the steps outlined above and to simplify complex expressions using the rules of arithmetic. With time and practice, you will become proficient in dividing complex numbers and be able to apply this skill to real-world problems.