Problem 54 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(7+9 i)+(1-2 i)+(-8-7 i)$$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Identify Real and Imaginary Parts

First, identify the real and imaginary components from each complex number. The given complex numbers are \(7+9i\), \(1-2i\), and \(-8-7i\). The real parts are 7, 1, and -8. The imaginary parts are 9, -2, and -7 (with \(i\) being the imaginary unit).

Step 2: Add Real Parts

Next, add the real parts together. Compute \(7 + 1 + (-8)\): \[ 7 + 1 - 8 = 0 \]

Step 3: Add Imaginary Parts

Then, add the imaginary parts together: Compute \(9 + (-2) + (-7)\): \[ 9 - 2 - 7 = 0 \]

Step 4: Combine Results into Standard Form

Combine the results into a single complex number (in standard form \(a+bi\)). The sum of the real parts is 0, and the sum of the imaginary parts is 0. Therefore, the expression simplifies to \(0 + 0i\), which is simply 0.

Key Concepts

Real and Imaginary Parts

Complex numbers are expressed in the form of \(a + bi\). Here, \(a\) represents the real part, and \(bi\) represents the imaginary part. A complex number combines these two components. The imaginary unit \(i\) is defined such that \(i^2 = -1\). This fundamental characteristic allows us to distinguish imaginary parts from real numbers.

To identify the real and imaginary parts in an expression like \((7 + 9i) + (1 - 2i) + (-8 - 7i)\):

• The real parts are simply the numbers without the \(i\), which are 7, 1, and -8.
• The imaginary parts are those numbers paired with \(i\), which are 9, -2, and -7.

Understanding this separation is crucial because it allows complex arithmetic to be simplified by handling real numbers separately from imaginary ones.

Adding Complex Numbers

When you add or subtract complex numbers, you deal separately with their real and imaginary parts. Each part behaves like a standard arithmetic operation.

Consider the process:

• First, focus on the real parts: add 7, 1, and \(-8\) together to get 0. Handle these as you would in normal numerical addition.
• Next, handle the imaginary parts: add 9, \(-2\), and \(-7\). Again, add them as normal numbers, ensuring that the result retains the \(i\): 9 + (-2) + (-7) = 0.

These steps highlight that adding complex numbers is straightforward once you break down the task into simpler addition of real numbers and separate addition of imaginary numbers.

Standard Form

Once you have computed the sums of the real and imaginary parts separately, combine them back into a complete complex number.

The standard form of a complex number is \(a + bi\). If both the real and imaginary parts add up to zero, as in our example \((0 + 0i)\), the result simplifies to just 0.

• This is key when expressing answers: a result of "0 + 0i" is generally written simply as "0" for simplicity.
• Regardless of the components, any complex number ultimately can be expressed in the form \(a + bi\), clearly indicating both its constituents.

This formal practice helps maintain clarity and consistency when performing complex arithmetic or further mathematical operations with complex numbers.