Properties of the Additive Inverse of Complex Numbers

Oct 13, 2025 by miodeya.com 136 views

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The additive inverse of a complex number is a crucial concept in understanding the properties of complex numbers. In this article, we will explore the concept of the additive inverse of complex numbers and provide a step-by-step solution to find the additive inverse of the complex number $9 - 4i$ .

What is the Additive Inverse of a Complex Number?

The additive inverse of a complex number is another complex number that, when added to the original complex number, results in a real number equal to zero. In other words, if we have a complex number $a + bi$ , its additive inverse is $-a - bi$ . This concept is similar to the additive inverse in the real number system, where the additive inverse of a number $a$ is $-a$ .

Properties of the Additive Inverse of Complex Numbers

The additive inverse of a complex number has several important properties:

The additive inverse of a complex number is unique.
The additive inverse of a complex number is commutative, meaning that the order of the complex numbers does not change the result.
The additive inverse of a complex number is associative, meaning that the order in which we add the complex numbers does not change the result.

Finding the Additive Inverse of a Complex Number

To find the additive inverse of a complex number, we need to follow these steps:

Identify the real and imaginary parts of the complex number.
Change the sign of both the real and imaginary parts.
Write the resulting complex number as the additive inverse.

Step-by-Step Solution to Find the Additive Inverse of $9 - 4i$

Now, let's apply the steps above to find the additive inverse of the complex number $9 - 4i$ .

Step 1: Identify the Real and Imaginary Parts

The complex number $9 - 4i$ has a real part of 9 and an imaginary part of -4.

Step 2: Change the Sign of Both the Real and Imaginary Parts

To find the additive inverse, we need to change the sign of both the real and imaginary parts. The real part becomes -9, and the imaginary part becomes 4.

Step 3: Write the Resulting Complex Number as the Additive Inverse

The resulting complex number is $-9 + 4i$ . Therefore, the additive inverse of the complex number $9 - 4i$ is $-9 + 4i$ .

Conclusion

In conclusion, the additive inverse of a complex number is a crucial concept in understanding the properties of complex numbers. By following the steps outlined above, we can find the additive inverse of any complex number. In this article, we found the additive inverse of the complex number $9 - 4i$ to be $-9 + 4i$ . This concept is essential in various mathematical applications, including algebra, geometry, and calculus.

Answer

The correct answer is B. $-9 + 4i$ .

Additional Resources

For more information on complex numbers and their properties, we recommend the following resources:

Khan Academy: Complex Numbers
MIT OpenCourseWare: Complex Variables
Wolfram MathWorld: Complex Numbers

Frequently Asked Questions

Q: What is the additive inverse of a complex number?

A: The additive inverse of a complex number is another complex number that, when added to the original complex number, results in a real number equal to zero.

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

A: To find the additive inverse of a complex number, you need to follow these steps:

Identify the real and imaginary parts of the complex number.
Change the sign of both the real and imaginary parts.
Write the resulting complex number as the additive inverse.

Q: What is the additive inverse of the complex number $9 - 4i$ ?

Introduction

Complex numbers are a fundamental concept in mathematics that extends the real number system to include numbers with both real and imaginary parts. In our previous article, we discussed the concept of the additive inverse of complex numbers and provided a step-by-step solution to find the additive inverse of the complex number $9 - 4i$ . In this article, we will continue to explore the world of complex numbers by answering some frequently asked questions.

Q&A Session

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 difference between a real number and a complex number?

A: A real number is a number that can be expressed in the form $a$ , where $a$ is a real number. A complex number, on the other hand, 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.

Q: What is the imaginary unit $i$ ?

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

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 have two complex numbers $a + bi$ and $c + di$ , their sum is $(a + c) + (b + d)i$ .

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 have two complex numbers $a + bi$ and $c + di$ , their difference is $(a - c) + (b - d)i$ .

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to use the distributive property and the fact that $i^2 = -1$ . For example, if you have two complex numbers $a + bi$ and $c + di$ , their product is $(ac - bd) + (ad + bc)i$ .

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers $a + bi$ and $c + di$ , their quotient is $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$ .

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$ . It is used to simplify complex expressions and to divide complex numbers.

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$ . It is used to measure the distance of the complex number from the origin.

Q: What is the argument of a complex number?

A: The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the complex number. It is used to represent complex numbers in polar form.

Conclusion

In conclusion, complex numbers are a fundamental concept in mathematics that extends the real number system to include numbers with both real and imaginary parts. By understanding the properties and operations of complex numbers, we can solve a wide range of mathematical problems. In this article, we answered some frequently asked questions about complex numbers and provided a comprehensive overview of the subject.

Additional Resources

For more information on complex numbers and their properties, we recommend the following resources:

Khan Academy: Complex Numbers
MIT OpenCourseWare: Complex Variables
Wolfram MathWorld: Complex Numbers

Frequently Asked Questions (FAQs)

Q: What is the additive inverse of a complex number?

A: The additive inverse of a complex number is another complex number that, when added to the original complex number, results in a real number equal to zero.

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

A: To find the additive inverse of a complex number, you need to follow these steps:

Identify the real and imaginary parts of the complex number.
Change the sign of both the real and imaginary parts.
Write the resulting complex number as the additive inverse.

Q: What is the additive inverse of the complex number $9 - 4i$ ?

A: The additive inverse of the complex number $9 - 4i$ is $-9 + 4i$ .

Answer Key

What is a complex number?
- A number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit.
What is the difference between a real number and a complex number?
- A real number is a number that can be expressed in the form $a$ , where $a$ is a real number. A complex number, on the other hand, 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.
What is the imaginary unit $i$ $i$ ?
- A mathematical constant that satisfies the equation $i^2 = -1$ .
How do I add complex numbers?
- To add complex numbers, you need to add the real parts and the imaginary parts separately.
How do I subtract complex numbers?
- To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately.
How do I multiply complex numbers?
- To multiply complex numbers, you need to use the distributive property and the fact that $i^2 = -1$ .
How do I divide complex numbers?
- To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator.
What is the conjugate of a complex number?
- The conjugate of a complex number $a + bi$ is $a - bi$ .
What is the modulus of a complex number?
- The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$ .
What is the argument of a complex number?
- The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the complex number.