Problem 19 Perform the operation(s) and wri... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 19

Perform the operation(s) and write the result in standard form. $$(4-3 i)+(6+7 i)$$

Short Answer

Expert verified

The result of the operation is the complex number $10 + 4i$.

Step by step solution

Identify the Real Parts

The real parts of the complex numbers $4-3i$ and $6+7i$ are $4$ and $6$ respectively. These can be identified as the numbers not involving the imaginary unit $i$.

Add the Real Parts

Add together the real parts identified in Step 1.\n So, $4 + 6 = 10$. This will form the real part of the resulting complex number.

Identify the Imaginary Parts

The imaginary parts of the complex numbers $4-3i$ and $6+7i$ are $-3$ and $7$ respectively. These can be identified as the numbers prior to the imaginary unit $i$.

Add the Imaginary Parts

Add together the imaginary parts identified in Step 3.\n So, $-3 + 7 = 4$. This will form the imaginary part of the resulting complex number.

Write the Final Result

Combine together the real and imaginary parts to form the final complex number.\n So, our final result is $10 + 4i$. This is in the standard form $a+bi$ for a complex number.

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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 is similar to adding polynomials or like terms. Breaking it down makes it simple:

Identify the real components of each complex number.
Sum up these real parts.
Do the same for the imaginary components.

To illustrate, let's take the complex numbers: $ (4 - 3i) + (6 + 7i) $. Here's the breakdown:

The real parts are 4 and 6. When you add them, you get 10.
The imaginary parts are $-3i$ and $7i$. Combining them gives $4i$.

Join your results: The final product, 10 for the real part and 4i for the imaginary part, makes the resulting complex number: $10 + 4i$. It's that easy!

Imaginary Unit

The concept of the imaginary unit is fascinating and fundamental to complex numbers. The imaginary unit, symbolized as $i$, is defined as the square root of $-1$. This:

Allows us to extend the real number system.
Provides a way to solve equations like $x^2 = -1$ which are impossible in the realm of real numbers.

For instance, when dealing with the imaginary components of our example: in $4 - 3i$, the $-3i$ involves the imaginary unit. The beauty of $i$ is in its cyclical patterns, where:

$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

Recognizing these patterns helps in simplifying expressions and understanding the nature of complex numbers.

Standard Form of Complex Numbers

Complex numbers come in a specific format that blends real and imaginary components. The standard form, $a + bi$, consists of two parts:

$a$ is the real part.
$bi$ is the imaginary part, where $b$ is a real number multiplied by the imaginary unit $i$.

In our exercise, the final result $10 + 4i$ is neatly arranged in this standard form.Keeping complex numbers in this structure has benefits:

It facilitates easy comparison with other complex numbers.
Makes operations like addition or multiplication straightforward.

Imagine solving more intricate problems or modeling phenomena in engineering and physics using complex numbers. The standard form gives you a clean, accessible handle on them all.

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91影视

Perform the operation(s) and write the result in standard form. $$(4-3 i)+(6+7 i)$$

Short Answer

Step by step solution

Identify the Real Parts

Add the Real Parts

Identify the Imaginary Parts

Add the Imaginary Parts

Write the Final Result

Key Concepts

Addition of Complex Numbers

Imaginary Unit

Standard Form of Complex Numbers

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