Problem 20 Perform the operation(s) and wri... [FREE SOLUTION]

91影视

Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 20

Perform the operation(s) and write the result in standard form. $$(-10+2 i)+(4-7 i)$$

Short Answer

Expert verified

The result of the operation is $ -6 - 5i $.

Step by step solution

Identify the real and imaginary parts.

In the first complex number, -10 is the real part and 2i is the imaginary part. In the second complex number, 4 is the real part and -7i is the imaginary part.

Add the real parts.

We add -10 and 4, which gives us -6.

Add the imaginary parts.

Then we add 2i and -7i, which gives us -5i.

Write the final solution in the form of a complex number.

Combine the results from Step 2 and Step 3 to form a complex number. The complex number is obtained as -6 - 5i.

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Key Concepts

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

Addition of Complex Numbers

Adding complex numbers might sound a bit challenging at first, but it's simpler than you think. To add two complex numbers, you treat the real and imaginary parts separately. Let's take the expression \[(-10+2i) + (4-7i)\]as an example. Here's how addition works:

Identify the Real Parts: For the first number $-10 + 2i$, the real part is $-10$. For $4 - 7i$, the real part is $4$.
Identify the Imaginary Parts: In $-10 + 2i$, the imaginary part is $2i$, and in $4 - 7i$, it is $-7i$.
Add Separately:
- Real parts: $-10 + 4 = -6$
- Imaginary parts: $2i - 7i = -5i$

Adding these independently simplifies the entire process of adding complex numbers, leading directly to the result in standard form.

Real and Imaginary Parts

In any complex number, understanding the structure of its real and imaginary parts is crucial. A complex number usually appears in the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. Here's a closer look at finding these parts:

Real Part $a$: This is the normal number without $i$. In $-10 + 2i$, $-10$ is real, and in $4 - 7i$, $4$ is real.
Imaginary Part $bi$: This part contains the imaginary unit $i$. For $-10 + 2i$, the imaginary part is $2i$, and for $4 - 7i$, it is $-7i$.

Each complex number contains both parts, which are handled separately in operations such as addition or subtraction.

Standard Form of Complex Numbers

Complex numbers often need to be expressed in standard form, allowing for ease of interpretation and calculation. Standard form refers to writing the number as$a + bi$, where $a$ and $b$ are real numbers. For our exercise, after performing addition, the result $-6 - 5i$ is already in standard form:

Real Part: $-6$ represents this part.
Imaginary Part: $-5i$ completes the complex number.

Arranging complex numbers in this form helps in visualizing and solving problems involving complex number operations. It's a consistent way to ensure that both the real and imaginary components are right where you expect them to be.

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91影视

Perform the operation(s) and write the result in standard form. $$(-10+2 i)+(4-7 i)$$

Short Answer

Step by step solution

Identify the real and imaginary parts.

Add the real parts.

Add the imaginary parts.

Write the final solution in the form of a complex number.

Key Concepts

Addition of Complex Numbers

Real and Imaginary Parts

Standard Form of Complex Numbers

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