Problem 49 - College Algebra with Modeling and Visualization

Question

Evaluate the expression with a calculator. $$ (23-5.6 i)+(-41.5+93 i) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-18.5 + 87.4i\).

Step 1: Identify the Real Parts

In the expression \((23 - 5.6i) + (-41.5 + 93i)\), identify the real numbers: 23 and -41.5.

Step 2: Identify the Imaginary Parts

Identify the imaginary numbers, which include the imaginary unit \(i\), in the expression: -5.6i and 93i.

Step 3: Add the Real Parts

Add the real parts of the expression: \(23 + (-41.5) = 23 - 41.5 = -18.5\).

Step 4: Add the Imaginary Parts

Add the imaginary coefficients: \(-5.6 + 93 = 87.4\) and keep the imaginary unit \(i\): \(87.4i\).

Step 5: Combine Real and Imaginary Parts

Combine the results from Step 3 and Step 4 to write the final expression: \(-18.5 + 87.4i\).

Key Concepts

Real and Imaginary Parts

Complex numbers are composed of two essential parts: the real part and the imaginary part. Understanding these components is key to working with complex numbers effectively.

• **Real Part**: This is the component of a complex number that acts like an ordinary real number. In the expression, this is any number that does not have the imaginary unit \(i\) attached. For example, in the number \(23 - 5.6i\), \(23\) is the real part.
• **Imaginary Part**: This part includes the coefficient with the imaginary unit \(i\). It represents the imaginary aspect of the complex number, which cannot be found on the real number line. In \(23 - 5.6i\), the imaginary part is \(-5.6i\).

By clearly distinguishing these parts, it becomes straightforward to perform operations such as addition or subtraction on complex numbers. When you realize complex numbers are just simple combinations of real and imaginary parts, tackling problems with complex numbers becomes easier.

Adding Complex Numbers

Adding complex numbers involves combining their real parts and their imaginary parts separately. This systematic approach ensures accurate results every time you add complex numbers. First, target the real parts of the numbers. In our exercise problem, the numbers are \(23\) and \(-41.5\). Add them together: \[ 23 + (-41.5) = -18.5 \] Next, focus on the imaginary parts. These are affected by the imaginary unit \(i\). In our example, note the imaginary coefficients \(-5.6\) and \(93\). Add these coefficients: \[ -5.6 + 93 = 87.4 \] Retain the imaginary unit \(i\) with this result, giving: \[ 87.4i \] Finally, combine your results from both parts to obtain the sum of the complex numbers: \[ -18.5 + 87.4i \] Understanding this process makes it easy to handle any addition operation involving complex numbers.

Imaginary Unit i

The imaginary unit \(i\) is central to understanding complex numbers. It is defined by its unique property: \[ i^2 = -1 \] This definition fundamentally distinguishes it from real numbers, as no real number holds this property. The imaginary unit allows for the creation of numbers that aren’t on the real number line but are crucial in mathematics and many applications, such as engineering and physics. A complex number, hence, can be represented as \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary coefficient, and \(i\) is the imaginary unit. In operations like addition, \(i\) accompanies the imaginary part and maintains its distinct identity. Using \(i\) enables mathematicians and engineers to solve equations that would otherwise have no solution within real numbers alone, broadening the scope of problem-solving in various fields. Understanding and using \(i\) unlocks a deeper appreciation for the power of mathematics in representing real-world phenomena.