Problem 29 Add or subtract as indicated. Wr... [FREE SOLUTION]

Chapter 10: Problem 29

Add or subtract as indicated. Write your answers in the form $a+b i$ $$ (3+2 i)+(-4+5 i) $$

Short Answer

Expert verified

-1 + 7i

Step by step solution

Identify the real and imaginary parts

First, rewrite the complex numbers by separately identifying the real and imaginary parts. For the complex numbers (3+2i) and (-4+5i), notice that 3 and -4 are the real parts, whereas 2i and 5i are the imaginary parts.

Add the real parts

Add the real parts of the complex numbers: 3 + (-4) = -1.

Add the imaginary parts

Add the imaginary parts of the complex numbers: 2i + 5i = 7i.

Form the resulting complex number

Combine the results from Steps 2 and 3 to form the resulting complex number: (-1 + 7i).

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Key Concepts

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

real_parts

To understand complex numbers addition, you need to first identify the real parts of the complex numbers involved. Complex numbers are written in the form of $a + bi$, where $a$ is the real part and $bi$ (\b\bi\bi) is the imaginary part. Identifying the real parts clarifies the first step in complex number addition. For instance, in the given problem, we have two complex numbers: $3 + 2i$ and $-4 + 5i$. Here, $3$ and $-4$ are the real parts. When adding these, focus only on the numbers without the imaginary unit $i$. So, $3 + (-4) = -1$. This is how we gather the real parts together in the addition process.

imaginary_parts

Alongside the real parts, complex numbers have imaginary parts, which include the imaginary unit $i$. Imaginary parts are numbers paired with the unit $i$, representing $\sqrt{-1}$. In our problem, the imaginary parts are $2i$ and $5i$. Similar to real parts, when adding the imaginary parts, you treat the coefficients of $i$ as regular numbers. So, $2i + 5i = 7i$. Always remember: it's important to add imaginary parts to imaginary parts only. This keeps the integral structure of the complex number.

resulting_complex_number

Once you've added both the real and imaginary parts separately, you combine the results to form the final, resulting complex number. From our exercise, we obtained the sums: $-1$ from the real parts and $7i$ from the imaginary parts. Combining these gives us the resulting complex number: $-1 + 7i$. This is your final answer. By doing so, you conform to the standard complex number format: $a + bi$. It helps in consistently recognizing and working with complex numbers in further calculations.

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91影视

Add or subtract as indicated. Write your answers in the form \(a+b i\) $$ (3+2 i)+(-4+5 i) $$

Short Answer

Step by step solution

Identify the real and imaginary parts

Add the real parts

Add the imaginary parts

Form the resulting complex number

Key Concepts

real_parts

imaginary_parts

resulting_complex_number

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