Problem 18 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$(3-2 i)+\left(-5-\frac{1}{3} i\right)$$

Step-by-Step Solution

Answer

The result is \(-2 - \frac{7}{3}i\).

Step 1: Understand the Expression

We are given the expression \((3-2i) + (-5 -\frac{1}{3}i)\). This is an addition of two complex numbers in the form \(a + bi\). We need to evaluate it and express it in the standard form \(a + bi\).

Step 2: Separate Real and Imaginary Parts

The first complex number is \((3 - 2i)\), where 3 is the real part and \(-2i\) is the imaginary part. The second complex number is \((-5 -\frac{1}{3}i)\), where \(-5\) is the real part and \(-\frac{1}{3}i\) is the imaginary part.

Step 3: Combine Real Parts

Add the real parts of the complex numbers: \[ 3 + (-5) = -2 \]

Step 4: Combine Imaginary Parts

Add the imaginary parts of the complex numbers:\[ -2i + \left(-\frac{1}{3}i\right) = \left(-2 - \frac{1}{3}\right)i = -\frac{7}{3}i \]

Step 5: Write the Expression in Standard Form

Combine the results from the real and imaginary parts to write the expression in the form \(a + bi\):\(-2 - \frac{7}{3}i\).

Key Concepts

Addition of Complex Numbers

Complex numbers are expressed in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When adding complex numbers, our main task is to sum their respective real and imaginary parts separately.
For two complex numbers, say \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\), the addition is carried out as follows:

• Combine the real parts: \(a_1 + a_2\)
• Combine the imaginary parts: \(b_1 + b_2\)

The sum is written as \((a_1 + a_2) + (b_1 + b_2)i\). This method allows us to treat the real and imaginary components independently, ensuring clarity in operations. It makes working with these numbers straightforward, much like handling regular algebraic expressions.

Real and Imaginary Parts

Complex numbers have two components—the real part and the imaginary part. These parts are crucial because they allow us to understand and manipulate the structure of complex numbers.

• The real part of a complex number \(a + bi\) is \(a\).
• The imaginary part is \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit, defined by \(i^2 = -1\).

Understanding which part is which enables us to manage each separately during operations like addition or subtraction. In the expression \((3 - 2i) + (-5 - \frac{1}{3}i)\), the real parts are 3 and -5, while the imaginary parts are \(-2i\) and \(-\frac{1}{3}i\). We process each category—real and imaginary—on their own terms, merging similar types. This separation is essential in ensuring operations maintain mathematical accuracy.

Complex Number Standard Form

Every complex number can be written in the standard form \(a + bi\). This form is essential as it presents a unified way of expressing complex numbers, facilitating simple bookkeeping of the real and imaginary parts.
When we perform operations like adding or subtracting complex numbers, the outcome should also be in this standard format.
For instance, solving the problem \((3 - 2i) + (-5 - \frac{1}{3}i)\) involves combining real and imaginary terms separately and then restructuring the result to fit \(a + bi\). The final evaluation becomes \(-2 - \frac{7}{3}i\). This structured form helps in further calculations, comparisons, and simplifying mathematical expressions.