What Is The Difference Of The Complex Numbers Below? { (11-3i)-(4+5i)$}$A. ${ 7-8i\$} B. ${ 15-8i\$} C. ${ 15-2i\$} D. ${ 7-2i\$}
Complex numbers are mathematical expressions that consist of a real number and an imaginary number. They are used to represent points in a two-dimensional plane and are essential in various mathematical and scientific applications. In this article, we will explore the concept of complex numbers and learn how to find the difference between two complex numbers.
What are Complex Numbers?
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.
Example of Complex Numbers
Some examples of complex numbers include:
- 3 + 4i
- 2 - 5i
- 1 + 2i
- 4 - 3i
How to Find the Difference of Complex Numbers
To find the difference of two complex numbers, we need to subtract the real parts and the imaginary parts separately. The difference of two complex numbers a + bi and c + di is given by:
(a - c) + (b - d)i
Step-by-Step Solution
Let's consider the complex numbers (11 - 3i) and (4 + 5i). To find the difference, we need to subtract the real parts and the imaginary parts separately.
- Subtract the real parts: 11 - 4 = 7
- Subtract the imaginary parts: -3i - 5i = -8i
Therefore, the difference of the complex numbers (11 - 3i) and (4 + 5i) is 7 - 8i.
Answer
The correct answer is A. 7 - 8i.
Why is the Answer Correct?
The answer is correct because we have correctly subtracted the real parts and the imaginary parts of the two complex numbers. The real part of the difference is 7, and the imaginary part is -8i.
Conclusion
In conclusion, finding the difference of complex numbers involves subtracting the real parts and the imaginary parts separately. We have learned how to find the difference of two complex numbers and have applied this concept to a specific example. We have also seen why the answer is correct and have understood the importance of complex numbers in mathematics and science.
The difference of two complex numbers a + bi and c + di is given by:
(a - c) + (b - d)i
Example of Complex Numbers
Some examples of complex numbers include:
- 3 + 4i
- 2 - 5i
- 1 + 2i
- 4 - 3i
How to Find the Difference of Complex Numbers
To find the difference of two complex numbers, we need to subtract the real parts and the imaginary parts separately.
Step-by-Step Solution
Let's consider the complex numbers (11 - 3i) and (4 + 5i). To find the difference, we need to subtract the real parts and the imaginary parts separately.
- Subtract the real parts: 11 - 4 = 7
- Subtract the imaginary parts: -3i - 5i = -8i
Therefore, the difference of the complex numbers (11 - 3i) and (4 + 5i) is 7 - 8i.
Answer
The correct answer is A. 7 - 8i.
Why is the Answer Correct?
The answer is correct because we have correctly subtracted the real parts and the imaginary parts of the two complex numbers. The real part of the difference is 7, and the imaginary part is -8i.
Real-World Applications of Complex Numbers
Complex numbers have numerous real-world applications in various fields, including:
- Electrical Engineering: Complex numbers are used to represent AC circuits and to analyze the behavior of electrical systems.
- Signal Processing: Complex numbers are used to represent signals and to analyze the behavior of systems that process signals.
- Navigation: Complex numbers are used to represent the position and velocity of objects in navigation systems.
- Computer Graphics: Complex numbers are used to represent 3D objects and to perform transformations on them.
Conclusion
In conclusion, complex numbers are an essential concept in mathematics and science. They have numerous real-world applications and are used to represent points in a two-dimensional plane. We have learned how to find the difference of complex numbers and have applied this concept to a specific example. We have also seen why the answer is correct and have understood the importance of complex numbers in various fields.
Final Thoughts
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 i?
A: The imaginary unit i is a mathematical concept that is used to extend the real numbers to the complex numbers. It is defined as the square root of -1, denoted by i = √(-1).
Q: How do I add complex numbers?
A: To add complex numbers, we add the real parts and the imaginary parts separately. For example, if we 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, we subtract the real parts and the imaginary parts separately. For example, if we 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, we use the distributive property and the fact that i^2 = -1. For example, if we 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, we multiply the numerator and the denominator by the conjugate of the denominator. For example, if we have two complex numbers a + bi and c + di, their quotient is ((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.
Q: Why do we need complex numbers?
A: Complex numbers are needed to represent points in a two-dimensional plane and to solve equations that involve the square root of a negative number.
Q: What are some real-world applications of complex numbers?
A: Complex numbers have numerous real-world applications in various fields, including electrical engineering, signal processing, navigation, and computer graphics.
Q: Can complex numbers be used to solve real-world problems?
A: Yes, complex numbers can be used to solve real-world problems that involve the representation of points in a two-dimensional plane and the solution of equations that involve the square root of a negative number.
Q: Are complex numbers used in any other fields?
A: Yes, complex numbers are used in various fields, including physics, engineering, and computer science.
Q: Can complex numbers be used to represent points in a three-dimensional space?
A: Yes, complex numbers can be used to represent points in a three-dimensional space by using quaternions, which are a type of complex number that has four components.
Q: What is the difference between a complex number and a quaternion?
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 quaternion is a number that has four components and is used to represent points in a three-dimensional space.
Q: Can complex numbers be used to solve equations that involve the square root of a negative number?
A: Yes, complex numbers can be used to solve equations that involve the square root of a negative number by using the imaginary unit i.
Q: Are complex numbers used in any other mathematical operations?
A: Yes, complex numbers are used in various mathematical operations, including addition, subtraction, multiplication, and division.
Q: Can complex numbers be used to represent points in a higher-dimensional space?
A: Yes, complex numbers can be used to represent points in a higher-dimensional space by using higher-dimensional quaternions.
Q: What are some of the limitations of complex numbers?
A: Some of the limitations of complex numbers include the fact that they are not as intuitive as real numbers and that they can be difficult to work with in certain situations.
Q: Can complex numbers be used to solve real-world problems that involve the representation of points in a two-dimensional plane?
A: Yes, complex numbers can be used to solve real-world problems that involve the representation of points in a two-dimensional plane.
Q: Are complex numbers used in any other fields besides mathematics?
A: Yes, complex numbers are used in various fields, including physics, engineering, and computer science.
Q: Can complex numbers be used to represent points in a three-dimensional space?
A: Yes, complex numbers can be used to represent points in a three-dimensional space by using quaternions.
Q: What is the difference between a complex number and a quaternion?
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 quaternion is a number that has four components and is used to represent points in a three-dimensional space.
Q: Can complex numbers be used to solve equations that involve the square root of a negative number?
A: Yes, complex numbers can be used to solve equations that involve the square root of a negative number by using the imaginary unit i.
Q: Are complex numbers used in any other mathematical operations?
A: Yes, complex numbers are used in various mathematical operations, including addition, subtraction, multiplication, and division.
Q: Can complex numbers be used to represent points in a higher-dimensional space?
A: Yes, complex numbers can be used to represent points in a higher-dimensional space by using higher-dimensional quaternions.
Q: What are some of the limitations of complex numbers?
A: Some of the limitations of complex numbers include the fact that they are not as intuitive as real numbers and that they can be difficult to work with in certain situations.
Q: Can complex numbers be used to solve real-world problems that involve the representation of points in a two-dimensional plane?
A: Yes, complex numbers can be used to solve real-world problems that involve the representation of points in a two-dimensional plane.
Q: Are complex numbers used in any other fields besides mathematics?
A: Yes, complex numbers are used in various fields, including physics, engineering, and computer science.
Q: Can complex numbers be used to represent points in a three-dimensional space?
A: Yes, complex numbers can be used to represent points in a three-dimensional space by using quaternions.
Q: What is the difference between a complex number and a quaternion?
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 quaternion is a number that has four components and is used to represent points in a three-dimensional space.
Q: Can complex numbers be used to solve equations that involve the square root of a negative number?
A: Yes, complex numbers can be used to solve equations that involve the square root of a negative number by using the imaginary unit i.
Q: Are complex numbers used in any other mathematical operations?
A: Yes, complex numbers are used in various mathematical operations, including addition, subtraction, multiplication, and division.
Q: Can complex numbers be used to represent points in a higher-dimensional space?
A: Yes, complex numbers can be used to represent points in a higher-dimensional space by using higher-dimensional quaternions.
Q: What are some of the limitations of complex numbers?
A: Some of the limitations of complex numbers include the fact that they are not as intuitive as real numbers and that they can be difficult to work with in certain situations.
Q: Can complex numbers be used to solve real-world problems that involve the representation of points in a two-dimensional plane?
A: Yes, complex numbers can be used to solve real-world problems that involve the representation of points in a two-dimensional plane.
Q: Are complex numbers used in any other fields besides mathematics?
A: Yes, complex numbers are used in various fields, including physics, engineering, and computer science.
Q: Can complex numbers be used to represent points in a three-dimensional space?
A: Yes, complex numbers can be used to represent points in a three-dimensional space by using quaternions.
Q: What is the difference between a complex number and a quaternion?
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 quaternion is a number that has four components and is used to represent points in a three-dimensional space.
Q: Can complex numbers be used to solve equations that involve the square root of a negative number?
A: Yes, complex numbers can be used to solve equations that involve the square root of a negative number by using the imaginary unit i.
Q: Are complex numbers used in any other mathematical operations?
A: Yes, complex numbers are used in various mathematical operations, including addition, subtraction, multiplication, and division.
Q: Can complex numbers be used to represent points in a higher-dimensional space?
A: Yes, complex numbers can be used to represent points in a higher-dimensional space by using higher-dimensional