Subtract { (3 + 2i)$}$ From { (-9 - 8i)$}$.A. { -17 - 5i$}$ B. { -6 - 6i$}$ C. { -12 - 10i$}$ D. ${ 12 + 10i\$}
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Introduction
Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are used to represent points in a two-dimensional plane and are essential in various fields, including algebra, geometry, and engineering. In this article, we will focus on subtracting complex numbers, which is a fundamental operation in complex number arithmetic.
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, while the imaginary part is the part that is multiplied by i.
Subtracting Complex Numbers
Subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. The formula for subtracting complex numbers is:
(a + bi) - (c + di) = (a - c) + (b - d)i
where a, b, c, and d are real numbers.
Example: Subtracting Complex Numbers
Let's consider the example of subtracting (3 + 2i) from (-9 - 8i).
Step 1: Identify the Real and Imaginary Parts
The real part of the first complex number is 3, and the imaginary part is 2i. The real part of the second complex number is -9, and the imaginary part is -8i.
Step 2: Subtract the Real Parts
Subtract the real parts of the two complex numbers: 3 - (-9) = 3 + 9 = 12.
Step 3: Subtract the Imaginary Parts
Subtract the imaginary parts of the two complex numbers: 2i - (-8i) = 2i + 8i = 10i.
Step 4: Combine the Results
Combine the results of the real and imaginary parts: 12 + 10i.
Conclusion
In conclusion, subtracting complex numbers involves subtracting the real parts and the imaginary parts separately. By following the formula (a + bi) - (c + di) = (a - c) + (b - d)i, we can perform complex number subtraction with ease.
Answer
The correct answer is C. [12 + 10i].
Discussion
- What is the formula for subtracting complex numbers?
- How do you subtract the real and imaginary parts of complex numbers?
- Can you provide an example of subtracting complex numbers?
Related Topics
- Adding Complex Numbers
- Multiplying Complex Numbers
- Dividing Complex Numbers
- Complex Number Arithmetic
References
Further Reading
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Introduction
Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are used to represent points in a two-dimensional plane and are essential in various fields, including algebra, geometry, and engineering. In this article, we will focus on frequently asked questions about subtracting complex numbers.
Q&A
Q1: What is the formula for subtracting complex numbers?
A1: The formula for subtracting complex numbers is (a + bi) - (c + di) = (a - c) + (b - d)i, where a, b, c, and d are real numbers.
Q2: How do you subtract the real and imaginary parts of complex numbers?
A2: To subtract the real and imaginary parts of complex numbers, you simply subtract the real parts and the imaginary parts separately.
Q3: Can you provide an example of subtracting complex numbers?
A3: Let's consider the example of subtracting (3 + 2i) from (-9 - 8i). The real part of the first complex number is 3, and the imaginary part is 2i. The real part of the second complex number is -9, and the imaginary part is -8i. Subtracting the real parts gives 3 - (-9) = 12, and subtracting the imaginary parts gives 2i - (-8i) = 10i. Therefore, the result of subtracting (3 + 2i) from (-9 - 8i) is 12 + 10i.
Q4: What is the difference between subtracting complex numbers and subtracting real numbers?
A4: The main difference between subtracting complex numbers and subtracting real numbers is that complex numbers have both real and imaginary parts. When subtracting complex numbers, you must subtract both the real and imaginary parts separately.
Q5: Can you subtract complex numbers with different magnitudes?
A5: Yes, you can subtract complex numbers with different magnitudes. The formula for subtracting complex numbers is (a + bi) - (c + di) = (a - c) + (b - d)i, where a, b, c, and d are real numbers.
Q6: Can you subtract complex numbers with different angles?
A6: Yes, you can subtract complex numbers with different angles. The formula for subtracting complex numbers is (a + bi) - (c + di) = (a - c) + (b - d)i, where a, b, c, and d are real numbers.
Q7: How do you handle complex numbers with zero imaginary part?
A7: When a complex number has a zero imaginary part, it is simply a real number. In this case, you can subtract the real parts as usual.
Q8: Can you provide a real-world example of subtracting complex numbers?
A8: Let's consider a real-world example of subtracting complex numbers. Suppose we have two complex numbers representing the positions of two objects in a two-dimensional plane. The first complex number is (3 + 2i), and the second complex number is (-9 - 8i). To find the position of the second object relative to the first object, we subtract the first complex number from the second complex number. Using the formula for subtracting complex numbers, we get (-9 - 8i) - (3 + 2i) = (-9 - 3) + (-8 - 2)i = -12 - 10i.
Conclusion
In conclusion, subtracting complex numbers is a fundamental operation in complex number arithmetic. By following the formula (a + bi) - (c + di) = (a - c) + (b - d)i, we can perform complex number subtraction with ease.
Related Topics
- Adding Complex Numbers
- Multiplying Complex Numbers
- Dividing Complex Numbers
- Complex Number Arithmetic