Calculating the Product of a Complex Number and Its Conjugate

Oct 13, 2025 by miodeya.com 146 views

Introduction

In mathematics, the conjugate of a complex number is another complex number that has the same real part but an opposite imaginary part. The product of a complex number and its conjugate is a real number, which can be calculated using the formula: (a + bi)(a - bi) = a^2 + b^2. In this article, we will explore the product of ${$ -3+6i$}$ and its conjugate.

What is a Complex Number?

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. The real part of a complex number is the part that is not multiplied by i, and the imaginary part is the part that is multiplied by i.

What is the Conjugate of a Complex Number?

The conjugate of a complex number a + bi is a - bi. For example, the conjugate of 3 + 4i is 3 - 4i. The conjugate of a complex number has the same real part but an opposite imaginary part.

Calculating the Product of a Complex Number and Its Conjugate

To calculate the product of a complex number and its conjugate, we can use the formula: (a + bi)(a - bi) = a^2 + b^2. This formula is derived from the fact that (a + bi)(a - bi) = a^2 - abi + abi - b²ⁱ2 = a^2 + b^2.

Calculating the Product of ${$ -3+6i$}$ and Its Conjugate

To calculate the product of ${$ -3+6i$}$ and its conjugate, we need to find the conjugate of ${$ -3+6i$}$. The conjugate of ${$ -3+6i$}$ is ${$ -3-6i$}$. Now, we can calculate the product using the formula: (a + bi)(a - bi) = a^2 + b^2.

Step 1: Identify the Real and Imaginary Parts

The real part of ${$ -3+6i$}$ is -3, and the imaginary part is 6.

Step 2: Calculate the Product

Using the formula (a + bi)(a - bi) = a^2 + b^2, we can calculate the product as follows:

(-3)^2 + (6)^2 = 9 + 36 = 45

Step 3: Write the Product in the Form a + bi

Since the product is a real number, we can write it in the form a + bi, where a is the product and b is 0. Therefore, the product of ${$ -3+6i$}$ and its conjugate is 45 + 0i, which can be written as ${ $45\$}$ .

Conclusion

In this article, we calculated the product of ${$ -3+6i$}$ and its conjugate using the formula (a + bi)(a - bi) = a^2 + b^2. We found that the product is 45, which can be written as ${ $45\$}$ . This result is consistent with the answer choices provided.

Answer

The correct answer is C. ${ $45\$}$ .

Frequently Asked Questions

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

Q: How do I calculate the product of a complex number and its conjugate?

A: To calculate the product of a complex number and its conjugate, you can use the formula: (a + bi)(a - bi) = a^2 + b^2.

Q: What is the product of ${$ -3+6i$}$ and its conjugate?

A: The product of ${$ -3+6i$}$ and its conjugate is 45.

References

[1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
[2] "Conjugate of a Complex Number" by Math Is Fun. Retrieved from https://www.mathisfun.com/numbers/complex-conjugate.html
[3] "Product of a Complex Number and Its Conjugate" by Purplemath. Retrieved from https://www.purplemath.com/modules/complex.htm

Introduction

In our previous article, we explored the product of a complex number and its conjugate. In this article, we will answer some frequently asked questions about complex numbers and conjugates.

Q&A

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 conjugate of a complex number?

A: The conjugate of a complex number a + bi is a - bi.

Q: How do I calculate the product of a complex number and its conjugate?

A: To calculate the product of a complex number and its conjugate, you can use the formula: (a + bi)(a - bi) = a^2 + b^2.

Q: What is the product of ${$ -3+6i$}$ and its conjugate?

A: The product of ${$ -3+6i$}$ and its conjugate is 45.

Q: Can I use the conjugate of a complex number to simplify expressions?

A: Yes, you can use the conjugate of a complex number to simplify expressions. For example, if you have the expression (a + bi)(a - bi), you can simplify it to a^2 + b^2 using the formula.

Q: How do I find the conjugate of a complex number in the form a + bi?

A: To find the conjugate of a complex number in the form a + bi, you can simply change the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i.

Q: Can I use the conjugate of a complex number to solve equations?

A: Yes, you can use the conjugate of a complex number to solve equations. For example, if you have the equation (a + bi) = c, you can multiply both sides by the conjugate of (a + bi) to eliminate the imaginary part.

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 I add or subtract complex numbers?

A: Yes, you can add or subtract complex numbers. For example, if you have the complex numbers 3 + 4i and 2 - 3i, you can add them to get 5 + i.

Q: Can I multiply complex numbers?

A: Yes, you can multiply complex numbers. For example, if you have the complex numbers 3 + 4i and 2 - 3i, you can multiply them to get -9 + 5i.