Problem 16
Question
In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{1}{4}+12 i\right)+\left(7+\frac{i}{10}\right) $$
Step-by-Step Solution
Verified Answer
The sum is \( \frac{29}{4} + \frac{121i}{10} \).
1Step 1: Write Down the Expression
The expression given is \( \left(\frac{1}{4} + 12i\right) + \left(7 + \frac{i}{10}\right) \). We need to find the sum of these two complex numbers.
2Step 2: Separate Real and Imaginary Parts
Identify the real and imaginary parts in each complex number. In \( \frac{1}{4} + 12i \), the real part is \( \frac{1}{4} \) and the imaginary part is \( 12i \). In \( 7 + \frac{i}{10} \), the real part is \( 7 \) and the imaginary part is \( \frac{i}{10} \).
3Step 3: Add Real Parts
Add the real parts of the complex numbers: \( \frac{1}{4} + 7 \). Convert 7 to a fraction with a denominator of 4, which is \( \frac{28}{4} \), resulting in \( \frac{1}{4} + \frac{28}{4} = \frac{29}{4} \).
4Step 4: Add Imaginary Parts
Add the imaginary parts \( 12i \) and \( \frac{i}{10} \). First, express 12 in terms of tenths to align denominators: \( 12 = \frac{120i}{10} \). Then add: \( \frac{120i}{10} + \frac{i}{10} = \frac{121i}{10} \).
5Step 5: Combine Results
Combine the results from Steps 3 and 4. The resulting sum of the complex numbers is \( \frac{29}{4} + \frac{121i}{10} \).
Key Concepts
Real and Imaginary PartsAlgebraic FractionsComplex Number Operations
Real and Imaginary Parts
Complex numbers are composed of two main components: the real part and the imaginary part. The real part is a standard number that you can find on the number line we are used to, while the imaginary part involves the imaginary unit, denoted as \( i \), where \( i^2 = -1 \). For example, in the complex number \( \frac{1}{4} + 12i \), \( \frac{1}{4} \) is the real part and \( 12i \) is the imaginary part.
Understanding these parts is the first step in performing operations on complex numbers. This means that when you're asked to perform addition or subtraction, you focus separately on the real parts and the imaginary parts. For instance, in our given exercise, the first complex number \( \frac{1}{4} + 12i \) has a real part \( \frac{1}{4} \) and an imaginary part \( 12i \), while the second complex number \( 7 + \frac{i}{10} \) has a real part 7 and an imaginary part \( \frac{i}{10} \).
This separation helps us handle complex computations by breaking them down into simpler arithmetic operations. It’s much like organizing different items in a closet; each type of item has its place.
Understanding these parts is the first step in performing operations on complex numbers. This means that when you're asked to perform addition or subtraction, you focus separately on the real parts and the imaginary parts. For instance, in our given exercise, the first complex number \( \frac{1}{4} + 12i \) has a real part \( \frac{1}{4} \) and an imaginary part \( 12i \), while the second complex number \( 7 + \frac{i}{10} \) has a real part 7 and an imaginary part \( \frac{i}{10} \).
This separation helps us handle complex computations by breaking them down into simpler arithmetic operations. It’s much like organizing different items in a closet; each type of item has its place.
Algebraic Fractions
In algebra, we often encounter algebraic fractions, which are similar to standard fractions but involve algebraic expressions in the numerator, the denominator, or both. When dealing with fractions in equations, especially in complex numbers, it's essential to manage them correctly.
To add fractions, they must share a common denominator. For example, in the exercise, to add the real parts \( \frac{1}{4} \) and 7, we convert 7 into a fraction: \( \frac{28}{4} \), because it needs the same denominator. This step ensures that both numbers are compatible for addition: \( \frac{1}{4} + \frac{28}{4} = \frac{29}{4} \).
Similarly, when you add the imaginary parts \( 12i \) and \( \frac{i}{10} \), you first express \( 12i \) as \( \frac{120i}{10} \) to match denominators, allowing the sum: \( \frac{120i}{10} + \frac{i}{10} = \frac{121i}{10} \). Understanding and managing these algebraic fractions correctly is crucial in performing precise and accurate arithmetic with complex numbers.
To add fractions, they must share a common denominator. For example, in the exercise, to add the real parts \( \frac{1}{4} \) and 7, we convert 7 into a fraction: \( \frac{28}{4} \), because it needs the same denominator. This step ensures that both numbers are compatible for addition: \( \frac{1}{4} + \frac{28}{4} = \frac{29}{4} \).
Similarly, when you add the imaginary parts \( 12i \) and \( \frac{i}{10} \), you first express \( 12i \) as \( \frac{120i}{10} \) to match denominators, allowing the sum: \( \frac{120i}{10} + \frac{i}{10} = \frac{121i}{10} \). Understanding and managing these algebraic fractions correctly is crucial in performing precise and accurate arithmetic with complex numbers.
Complex Number Operations
Complex number operations involve manipulating numbers that have both real and imaginary components, such as addition, subtraction, multiplication, and division. In our exercise, we focus on adding complex numbers.
When adding complex numbers, the key is to handle the real and imaginary parts separately. Start by adding the real parts together, and then the imaginary parts:
This way of handling complex number operations transforms what might initially seem like a daunting task into a series of more straightforward arithmetic steps. Mastery of operations like addition helps create a solid foundation for tackling more advanced concepts like multiplication or division of complex numbers in the future.
When adding complex numbers, the key is to handle the real and imaginary parts separately. Start by adding the real parts together, and then the imaginary parts:
- Add the real parts: \( \frac{1}{4} + 7 = \frac{29}{4} \) by adjusting \( 7 \) to \( \frac{28}{4} \).
- Add the imaginary parts: \( 12i + \frac{i}{10} = \frac{121i}{10} \), where \( 12i = \frac{120i}{10} \).
This way of handling complex number operations transforms what might initially seem like a daunting task into a series of more straightforward arithmetic steps. Mastery of operations like addition helps create a solid foundation for tackling more advanced concepts like multiplication or division of complex numbers in the future.
Other exercises in this chapter
Problem 16
Without solving each equation, find the sum and product of the roots. \(x^{2}+2 x-3=0\)
View solution Problem 16
In \(15-23 :\) a. Find the value of the discriminant and determine if the roots of the quadratic equation are \((1)\) rational and unequal, \((2)\) rational and
View solution Problem 16
In \(3-18,\) write each number in terms of \(i\) $$ 1+\sqrt{-3} $$
View solution Problem 16
In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ 4 x^{2}-4 x=11 $$
View solution