Problem 10 - Algebra for College Students

Question

Add or subtract as indicated. $$(5+2 i)+(7+10 i)$$

Step-by-Step Solution

Answer

The sum is \(12+12i\).

Step 1: Understand the Complex Numbers

The given expression is \[ (5+2i) + (7+10i) \].In this expression, each complex number has a real part and an imaginary part. The real parts are \(5\) and \(7\), and the imaginary parts are \(2i\) and \(10i\). Your task is to add these parts separately.

Step 2: Add the Real Parts

Add the real parts of the complex numbers together:\[ 5 + 7 = 12 \].The result, \(12\), is the real part of the sum.

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers:\[ 2i + 10i = 12i \].The result, \(12i\), is the imaginary part of the sum.

Step 4: Combine the Results

Combine the results of the real parts and the imaginary parts:\[ 12 + 12i \].This is the sum of the two complex numbers.

Key Concepts

Real Numbers

Real numbers are those numbers that do not include any imaginary components. They can be positive, negative, or zero. Real numbers are used in everyday life and include familiar types of numbers, such as whole numbers, fractions, and decimals. For example:

• Whole numbers: 3, -1, 0
• Fractions: \( \frac{1}{2}, -\frac{5}{4} \)
• Decimals: 4.5, -3.3

In the context of complex numbers, the real part is the component without the imaginary unit \( i \). For a complex number \( a + bi \), \( a \) represents the real part. Understanding real numbers is essential when working with complex numbers as they form one-half of the pair in any complex number.

Imaginary Numbers

Imaginary numbers involve the unit \( i \), defined as the square root of \(-1\). Imaginary numbers are crucial when numbers are extended beyond the real line. They allow us to solve equations that would be impossible otherwise. The component inline with \( i \) in a complex number is its imaginary part. For instance, in \( a + bi \), \( bi \) is the imaginary part, where \( b \) is a real number while \( i \) turns it into the imaginary component.

Imaginary numbers help in solving equations like \( x^2 + 1 = 0 \), where the solution is \( x = \pm i \). These numbers are fundamental in complex numbers, where the imaginary part can determine the direction, completeness, and functionality of the numbers within equations or systems.

Addition of Complex Numbers

Adding complex numbers is straightforward when you separate their real and imaginary parts. A complex number is usually expressed in the form \( a + bi \), with \( a \) as the real part and \( bi \) as the imaginary part. When you are adding two complex numbers, like \( (a+bi) \) and \( (c+di) \), you add their real parts together and their imaginary parts together independently. The formula looks like this:

• Real parts: \( a + c \)
• Imaginary parts: \( b + d \)

For example, in the original exercise, we had \( (5+2i) + (7+10i) \). You add the real parts \( 5 + 7 = 12 \) and the imaginary parts \( 2i + 10i = 12i \). Therefore, the sum is \( 12 + 12i \).

Being able to efficiently add complex numbers enables solving more complicated mathematical problems and is an essential technique in fields such as engineering and physics as it simplifies circular and wave equations.