Problem 7 Find the indicated sums and diff... [FREE SOLUTION]

Chapter 7: Problem 7

Find the indicated sums and differences of complex numbers. $$(2+3 i)+(-4+5 i)$$

Short Answer

Expert verified

-2 + 8i

Step by step solution

- Identify the Complex Numbers

The given problem involves the addition of two complex numbers: The first complex number is a+bi = 2 + 3iThe second complex number is c+di = -4 + 5i

- Add the Real Parts

Separate the real parts of each complex number and add them together: Real part of the first complex number: 2 Real part of the second complex number: -4 Therefore, the resulting real part is: 2 + (-4) = -2

- Add the Imaginary Parts

Separate the imaginary parts of each complex number and add them together: Imaginary part of the first complex number: 3i Imaginary part of the second complex number: 5i Therefore, the resulting imaginary part is: 3i + 5i = 8i

- Combine the Results

Combine the real part and the imaginary part from the previous steps to obtain the final result: -2 + 8i

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers

Complex numbers might seem confusing at first, but they are quite simple once you break them down. They consist of two parts: a real part and an imaginary part. The general form of a complex number is written as:
a + bi, where a is the real part and bi is the imaginary part.
The imaginary unit denoted as i represents the square root of -1, which is written as:
i^2 = -1.
When solving problems with complex numbers, you treat the real and imaginary parts separately and then combine the results.

Adding Real Parts

The first step in adding complex numbers is to focus on the real parts.
Given two complex numbers, such as 2 + 3i and -4 + 5i, you identify the real parts from each:
For the first complex number, the real part is 2.
For the second complex number, the real part is -4.
You then add these real parts together:
2 + (-4) = -2.
This step can be done quickly with some practice.

Adding Imaginary Parts

Next, you handle the imaginary parts of the complex numbers. Complex numbers are always written in the form a + bi, where bi is the imaginary part.
From our example, the imaginary parts are 3i and 5i.
To add these, you simply add their coefficients (the numbers in front of i):
3i + 5i = (3 + 5)i
= 8i.
This method works the same way for subtraction; just remember to subtract the coefficients instead.

Combining Results

After you have handled the real and imaginary parts separately, combine them to form the final result.
From our steps above, we have a real part of -2 and an imaginary part of 8i.
Put these together to get the final answer:
-2 + 8i.
Make sure to always write your result in the standard form a + bi. Practicing these steps will make working with complex numbers much easier over time.

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91影视

Find the indicated sums and differences of complex numbers. $$(2+3 i)+(-4+5 i)$$

Short Answer

Step by step solution

- Identify the Complex Numbers

- Add the Real Parts

- Add the Imaginary Parts

- Combine the Results

Key Concepts

Complex Numbers

Adding Real Parts

Adding Imaginary Parts

Combining Results

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