Problem 53 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(-6+5 i)+(4-4 i)+(2-i)$$

Step-by-Step Solution

Answer

The sum in standard form is 0.

Step 1: Identify Like Terms

Begin by grouping the real and imaginary parts of each complex number. We have the expression \((-6 + 5i) + (4 - 4i) + (2 - i)\). The real terms are \(-6, 4,\) and \(2\). The imaginary terms are \(5i, -4i,\) and \(-i\).

Step 2: Add Real Parts

Add the real parts together. This involves calculating \(-6 + 4 + 2\). Performing this addition gives \((-6 + 4 + 2 = 0)\).

Step 3: Add Imaginary Parts

Add the imaginary parts together. This means we need to calculate \(5i - 4i - i\). Performing this addition results in \((5i - 4i - i = 0i)\).

Step 4: Combine Results

Combine the results from Step 2 and Step 3 to form the sum in standard form, which is \(0+0i\). In this case, since both the real and imaginary parts are zero, the final simplified form is just \(0\).

Key Concepts

Standard Form

Complex numbers can often be intimidating, but understanding their standard form provides clarity. The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) represents the square root of \(-1\). All complex numbers can be broken into these two components which makes operations like addition and subtraction easier to manage. Remembering this form helps keep the real and imaginary parts distinct and organized.

Real and Imaginary Parts

Understanding the real and imaginary parts of complex numbers is critical for manipulation and operations. In any complex number expressed as \(a + bi\):

• The real part is \(a\).
• The imaginary part is \(bi\).

These parts dictate how we manage complex number operations. In our example, the numbers \(-6, 4,\) and \(2\) are real parts, while \(5i, -4i,\) and \(-i\) are imaginary parts. Each part is treated separately when adding or subtracting, but ultimately combined into the standard form. Separating these components helps to see the structure clearly and simplifies calculations.

Addition and Subtraction of Complex Numbers

Adding or subtracting complex numbers is similar to combining like terms in algebra. You separately add or subtract the real parts and the imaginary parts.

• First, identify and group the real numbers together. In our exercise, we have \(-6, 4,\) and \(2\). Performing \(-6 + 4 + 2\) equals \(0\).
• Next, group the imaginary terms. For our example, that's \(5i, -4i,\) and \(-i\). Calculating \(5i - 4i - i\) also results in \(0i\).

When both sums lead to zero, we get \(0+0i\), reducing just to \(0\). This addition and subtraction process ensures we correctly handle each part of complex numbers individually, before combining them for a final answer.