Problem 17 - Algebra for College Students

Question

Add or subtract as indicated. $$(-3-10 i)+(2-13 i)$$

Step-by-Step Solution

Answer

The sum is \(-1 - 23i\).

Step 1: Identify Real and Imaginary Parts

The expression given is \((-3 - 10i) + (2 - 13i)\). Here the real parts are \(-3\) and \(2\), and the imaginary parts are \(-10i\) and \(-13i\).

Step 2: Add the Real Parts

Add the real parts of the complex numbers together. This involves the numbers:\(-3\) and \(2\).Calculate:\[-3 + 2 = -1\]

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers. This involves: \(-10i\) and \(-13i\).Calculate:\[-10i + (-13i) = -23i\]

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to get the sum of the complex numbers.The resulting complex number is: \(-1 - 23i\).

Key Concepts

Real and Imaginary Parts

Complex numbers are numbers that have two components: a real part and an imaginary part. Understanding these components is crucial in handling complex number operations. In any complex number expressed in the standard form, usually written as \(a + bi\), the "real part" is represented by \(a\) and the "imaginary part" is represented by \(bi\).

• The real part is simply a standard real number without any imaginary "\(i\)" component. For instance, in the expression \(2 - 3i\), the number 2 is the real part.
• The imaginary part involves the imaginary unit \(i\). In \(2 - 3i\), the number \(-3i\) is the imaginary part, where "\(i\)" denotes the imaginary unit.

This division is fundamental in performing any addition, subtraction, or any other operations involving complex numbers.

Addition of Complex Numbers

Adding complex numbers might sound complicated, but just like real numbers, it follows a straightforward rule: combine like terms. This means real parts are added together, and imaginary parts are added together. Let's examine the example:
Consider \((-3 - 10i) + (2 - 13i)\).

• Add the real parts: Take \(-3\) from the first complex number and \(2\) from the second. Compute the sum: \[-3 + 2 = -1\]
• Add the imaginary parts: Combine \(-10i\) and \(-13i\) accordingly. The sum is \[-10i + (-13i) = -23i\]

Thus, the result of adding these complex numbers is a new complex number, \(-1 - 23i\). Each part—real and imaginary—is processed separately and then combined to form the final result.

Algebraic Operations

When dealing with algebraic operations on complex numbers, focus on treating each part carefully, adhering to the basic algebraic rules. In our context of addition, it emphasizes recognizing that each term must be simplified within its "like" group, i.e., real with real and imaginary with imaginary.

• Combining Terms: This is similar to how you combine like terms in algebra involving real numbers.
• Subtraction: If we were tasked with subtracting instead, the same principle applies. Real parts are handled separately from imaginary parts but instead subtracted from one another.

Each of these operations relies heavily on a solid understanding of arithmetic—specifically how addition and subtraction work within and between these categories. Proper execution of these operations can simplify complex expressions effectively and accurately.