Understanding Complex Number Subtraction

Oct 13, 2025 by miodeya.com 126 views

Understanding Complex Number Subtraction

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. When subtracting complex numbers, it's crucial to understand the concept of the difference between two complex numbers.

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 without the imaginary unit, while the imaginary part is the part with the imaginary unit.

Subtracting Complex Numbers

To subtract complex numbers, we need to subtract the real parts and the imaginary parts separately. Let's consider the subtraction expression (a + bi) - (c + di), where a, b, c, and d are real numbers. The difference between the two complex numbers is given by (a - c) + (b - d)i.

Example 1: Subtracting Complex Numbers

Let's consider the subtraction expression (-2 + 6i) - (1 - 2i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: -2 - 1 = -3
Imaginary part: 6 - (-2) = 6 + 2 = 8

Therefore, the difference is -3 + 8i.

Example 2: Subtracting Complex Numbers

Let's consider the subtraction expression (-2 + 6i) - (-1 - 2i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: -2 - (-1) = -2 + 1 = -1
Imaginary part: 6 - (-2) = 6 + 2 = 8

Therefore, the difference is -1 + 8i.

Example 3: Subtracting Complex Numbers

Let's consider the subtraction expression (3 + 5i) - (2 - i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: 3 - 2 = 1
Imaginary part: 5 - (-1) = 5 + 1 = 6

Therefore, the difference is 1 + 6i.

Example 4: Subtracting Complex Numbers

Let's consider the subtraction expression (3 + 5i) - (2 + i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: 3 - 2 = 1
Imaginary part: 5 - 1 = 4

Therefore, the difference is 1 + 4i.

Conclusion

In conclusion, when subtracting complex numbers, we need to subtract the real parts and the imaginary parts separately. The difference between two complex numbers is given by (a - c) + (b - d)i. We have considered four examples of subtracting complex numbers and found the differences in each case.

Which Subtraction Expression Has the Difference 1+4i?

Based on the examples above, we can see that the subtraction expression (3 + 5i) - (2 + i) has the difference 1 + 4i.

Final Answer

The final answer is D. $(3+5i)-(2+i$ ]

Understanding Complex Number Subtraction

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. The real part of a complex number is the part without the imaginary unit, while the imaginary part is the part with the imaginary unit.

Q: How Do I Subtract Complex Numbers?

A: To subtract complex numbers, we need to subtract the real parts and the imaginary parts separately. Let's consider the subtraction expression (a + bi) - (c + di), where a, b, c, and d are real numbers. The difference between the two complex numbers is given by (a - c) + (b - d)i.

Q: What is the Difference Between Two Complex Numbers?

A: The difference between two complex numbers is given by (a - c) + (b - d)i, where a, b, c, and d are real numbers.

Q: Can You Give Me an Example of Subtracting Complex Numbers?

A: Let's consider the subtraction expression (-2 + 6i) - (1 - 2i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: -2 - 1 = -3
Imaginary part: 6 - (-2) = 6 + 2 = 8

Therefore, the difference is -3 + 8i.

Q: How Do I Determine Which Subtraction Expression Has a Certain Difference?

A: To determine which subtraction expression has a certain difference, we need to find the difference between the two complex numbers and compare it with the given difference.

Q: Can You Give Me an Example of Determining Which Subtraction Expression Has a Certain Difference?

A: Let's consider the subtraction expression (3 + 5i) - (2 + i). To find the difference, we need to subtract the real parts and the imaginary parts separately.

Real part: 3 - 2 = 1
Imaginary part: 5 - 1 = 4

Therefore, the difference is 1 + 4i.

Q: Which Subtraction Expression Has the Difference 1+4i?

A: Based on the examples above, we can see that the subtraction expression (3 + 5i) - (2 + i) has the difference 1 + 4i.

Q: What is the Final Answer?

A: The final answer is D. $(3+5i)-(2+i$ ]

Q: Can You Summarize the Key Points of Subtracting Complex Numbers?

A: Here are the key points of subtracting complex numbers:

To subtract complex numbers, we need to subtract the real parts and the imaginary parts separately.
The difference between two complex numbers is given by (a - c) + (b - d)i.
We can determine which subtraction expression has a certain difference by finding the difference between the two complex numbers and comparing it with the given difference.

Q: What is the Importance of Subtracting Complex Numbers?

A: Subtracting complex numbers is an essential concept in mathematics and is used in various applications, such as engineering, physics, and computer science. It is used to represent points in a two-dimensional plane and is used to solve equations and inequalities involving complex numbers.

Q: Can You Give Me a Real-World Example of Subtracting Complex Numbers?

A: Let's consider a real-world example of subtracting complex numbers. Suppose we have two complex numbers, 3 + 5i and 2 + i, and we want to find their difference. We can use the formula for subtracting complex numbers to find the difference.

Real part: 3 - 2 = 1
Imaginary part: 5 - 1 = 4

Therefore, the difference is 1 + 4i.

This example illustrates the importance of subtracting complex numbers in real-world applications.

Q: What is the Final Answer?

A: The final answer is D. $(3+5i)-(2+i$ ]