Problem 13
Question
In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{2}{3}-\frac{1}{6} i\right)+(2-i) $$
Step-by-Step Solution
Verified Answer
\( \frac{8}{3} - \frac{7}{6} i \)
1Step 1: Identify the Real and Imaginary Parts
Separate the real and imaginary parts for each complex number:\( \frac{2}{3} - \frac{1}{6} i \) has a real part of \( \frac{2}{3} \) and an imaginary part of \( -\frac{1}{6} \).\( 2 - i \) has a real part of \( 2 \) and an imaginary part of \( -1 \).
2Step 2: Add the Real Parts Together
Add the real parts of the two complex numbers together:\[ \text{Real part sum} = \frac{2}{3} + 2 \]
3Step 3: Convert Mixed Numbers to Common Denominator
Convert the whole number \(2\) into a fraction with a common denominator of 3:\[ 2 = \frac{6}{3} \]
4Step 4: Compute the Real Sum
Now add the fractions:\[ \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \]
5Step 5: Add the Imaginary Parts Together
Add the imaginary parts of the two complex numbers together:\[ \text{Imaginary part sum} = -\frac{1}{6} + (-1) \]
6Step 6: Convert Imaginary Whole Number to Common Denominator
Convert the whole number \(-1\) into a fraction with a common denominator of 6:\[ -1 = -\frac{6}{6} \]
7Step 7: Compute the Imaginary Sum
Add the fractions:\[ -\frac{1}{6} + (-\frac{6}{6}) = -\frac{7}{6} \]
8Step 8: Combine Real and Imaginary Sums
Combine the real and imaginary sums to write the final complex number in \(a + bi\) form:\[ \frac{8}{3} - \frac{7}{6} i \]
Key Concepts
Real and Imaginary PartsAddition of Complex NumbersFractions with Common Denominators
Real and Imaginary Parts
Complex numbers consist of two parts: the real part and the imaginary part. Understanding these components is key to working with complex numbers.
Real numbers are numbers that you are already familiar with, such as 2, -5, or \( \frac{3}{4} \). Imaginary numbers are a bit different; they involve the imaginary unit \( i \), which is defined as \( i^2 = -1 \).
In complex numbers like \( a + bi \), \( a \) is the real part and \( b \) is the imaginary part, with \( i \) indicating the imaginary component. For instance, in the complex number \( \frac{2}{3} - \frac{1}{6} i \), the real part is \( \frac{2}{3} \), and the imaginary part is \( -\frac{1}{6} \). The real and imaginary parts are distinct, yet they combine to form the whole complex number, making it unique and versatile.
Identifying these parts is crucial because it allows us to handle operations like addition and subtraction smoothly.
Real numbers are numbers that you are already familiar with, such as 2, -5, or \( \frac{3}{4} \). Imaginary numbers are a bit different; they involve the imaginary unit \( i \), which is defined as \( i^2 = -1 \).
In complex numbers like \( a + bi \), \( a \) is the real part and \( b \) is the imaginary part, with \( i \) indicating the imaginary component. For instance, in the complex number \( \frac{2}{3} - \frac{1}{6} i \), the real part is \( \frac{2}{3} \), and the imaginary part is \( -\frac{1}{6} \). The real and imaginary parts are distinct, yet they combine to form the whole complex number, making it unique and versatile.
Identifying these parts is crucial because it allows us to handle operations like addition and subtraction smoothly.
Addition of Complex Numbers
Adding complex numbers is very similar to adding two polynomials. The main rule is to add the like terms: add the real parts with the real parts and the imaginary parts with the imaginary parts.
Let's look at the example: \( \left(\frac{2}{3}-\frac{1}{6} i\right) + (2-i) \). First, we separate the two components: the real numbers are \( \frac{2}{3} \) and 2, while the imaginary parts are \( -\frac{1}{6} \) and \(-1\).
First, add the real parts together: \( \frac{2}{3} + 2 \). To make the addition easier, convert 2 to a fraction with a common denominator of 3, which becomes \( \frac{6}{3} \). Then, add \( \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \).
Next, add the imaginary parts: \( -\frac{1}{6} + (-1) \). We again look for a common denominator, which is 6, converting \(-1\) to \(-\frac{6}{6}\). Add these to get \(-\frac{1}{6} + (-\frac{6}{6}) = -\frac{7}{6} \).
This combination of separate operations results in the overall sum: \( \frac{8}{3} - \frac{7}{6} i \). This process shows how complex numbers can seamlessly combine by treating their real and imaginary parts separately.
Let's look at the example: \( \left(\frac{2}{3}-\frac{1}{6} i\right) + (2-i) \). First, we separate the two components: the real numbers are \( \frac{2}{3} \) and 2, while the imaginary parts are \( -\frac{1}{6} \) and \(-1\).
First, add the real parts together: \( \frac{2}{3} + 2 \). To make the addition easier, convert 2 to a fraction with a common denominator of 3, which becomes \( \frac{6}{3} \). Then, add \( \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \).
Next, add the imaginary parts: \( -\frac{1}{6} + (-1) \). We again look for a common denominator, which is 6, converting \(-1\) to \(-\frac{6}{6}\). Add these to get \(-\frac{1}{6} + (-\frac{6}{6}) = -\frac{7}{6} \).
This combination of separate operations results in the overall sum: \( \frac{8}{3} - \frac{7}{6} i \). This process shows how complex numbers can seamlessly combine by treating their real and imaginary parts separately.
Fractions with Common Denominators
Handling fractions is an essential skill, especially when performing operations with complex numbers. A common denominator allows you to add or subtract fractions efficiently and smoothly.
Suppose you have the fractions \( \frac{2}{3} \) and \( 2 \) in the real part addition. To combine them properly, convert any whole numbers to fractions that share common denominators. For \( 2 \), we convert it as \( \frac{6}{3} \).
Similarly, when dealing with the imaginary parts, \( -\frac{1}{6} \) and \(-1\), we convert the whole number \(-1\) to \(-\frac{6}{6}\), sharing the same denominator of 6. It is easier to work with fractions like this because you only add or subtract the top numbers, leaving the denominator as it is.
In practice, once fractions have a common denominator, you can directly perform the simplification: \( \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \) for the real parts and \(-\frac{1}{6} + (-\frac{6}{6}) = -\frac{7}{6} \) for the imaginary parts.
This method ensures accurate and organized computations, making it much easier to handle various mathematical operations seamlessly.
Suppose you have the fractions \( \frac{2}{3} \) and \( 2 \) in the real part addition. To combine them properly, convert any whole numbers to fractions that share common denominators. For \( 2 \), we convert it as \( \frac{6}{3} \).
Similarly, when dealing with the imaginary parts, \( -\frac{1}{6} \) and \(-1\), we convert the whole number \(-1\) to \(-\frac{6}{6}\), sharing the same denominator of 6. It is easier to work with fractions like this because you only add or subtract the top numbers, leaving the denominator as it is.
In practice, once fractions have a common denominator, you can directly perform the simplification: \( \frac{2}{3} + \frac{6}{3} = \frac{8}{3} \) for the real parts and \(-\frac{1}{6} + (-\frac{6}{6}) = -\frac{7}{6} \) for the imaginary parts.
This method ensures accurate and organized computations, making it much easier to handle various mathematical operations seamlessly.
Other exercises in this chapter
Problem 13
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