Problem 26 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(1.6+3.2 i)+(-5.8+4.3 i)$$

Step-by-Step Solution

Answer

-4.2 + 7.5i

Step 1: Identify the Real and Imaginary Parts

The real parts of the complex numbers are 1.6 and -5.8, while the imaginary parts (without considering the 'i') are 3.2 and 4.3.

Step 2: Add the Real Parts

Add the real parts of the complex numbers: \(1.6 + (-5.8) = -4.2\)

Step 3: Add the Imaginary Parts

Add the imaginary parts of the complex numbers: \(3.2 + 4.3 = 7.5\)

Step 4: Write the answer in standard form

Combine the results from step 2 and 3 to write the result in standard form, which is \(a + bi\). Here, a=-4.2 and b=7.5, so the result is \(-4.2 + 7.5i\).

Key Concepts

Real Numbers

Real numbers are numbers that we commonly use in everyday life. They include all the possible numbers that can sit on the number line, such as positive numbers, negative numbers, and zero. When we talk about complex numbers, the real part is the number without any imaginary component.

Here are a few characteristics of real numbers:

• They can be whole numbers like 7 or -3.
• They can be fractions like \( \frac{1}{2} \).
• They can also be decimals such as 1.6 or -5.8.

In our example, the real parts are 1.6 from the first complex number, and -5.8 from the second complex number. When combined, these give us the real part of the final sum. Understanding the role of real numbers helps in splitting and properly recombining any complex number.

Imaginary Numbers

Imaginary numbers might sound a bit unusual, but they make perfect sense in mathematics. An imaginary number is essentially a real number multiplied by the imaginary unit \(i\), where \(i^2 = -1\). These numbers are crucial for working with anything that involves quadratic equations with negative roots, among other things.

Consider a few points about imaginary numbers:

• The simplest form is just \(i\), or one times the imaginary unit.
• They can be scaled, like with 3.2 \(i\) or 4.3 \(i\), exactly as in our example.
• Combining them follows similar arithmetic rules as with real numbers, just respect \(i^2 = -1\).

In the given exercise, 3.2 and 4.3 are the coefficients of \(i\) for the corresponding terms. These are the parts that make up the imaginary component of our complex number when summed up.

Standard Form

The standard form for writing complex numbers is \(a + bi\), where \(a\) represents the real part and \(b\) denotes the imaginary part. This form is extremely useful because it neatly segregates the different elements of a complex number, making it straightforward to understand and operate on.

To identify and write numbers in standard form:

• Determine the real part, \(a\).
• Identify the imaginary part, represented as \(b \cdot i\).
• Combine them as \(a + bi\).

In our exercise, after completing all operations, the combined form of \(-4.2\) (real part) and \(7.5i\) (imaginary part) is written as \(-4.2 + 7.5i\). This process of combining ensures clarity and precision, especially when working within mathematical problems involving complex numbers.