Problem 21 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (6+5 i)-(8-i) $$

Step-by-Step Solution

Answer

The result is \(-2 + 6i\).

Step 1: Distribute the Negative Sign

Rewrite the subtraction as adding the opposite by distributing the negative sign through the second parentheses. This changes the expression from \((6 + 5i) - (8 - i)\) to \((6 + 5i) + (-8 + i)\).

Step 2: Combine Real Parts

Identify and combine the real parts of the expression. The real parts are 6 and -8. Calculate: \(6 + (-8) = -2\).

Step 3: Combine Imaginary Parts

Identify and combine the imaginary parts of the expression. The imaginary parts are \(5i\) and \(i\). Calculate: \(5i + i = 6i\).

Step 4: Form the Result

Combine the results from Step 2 and Step 3 to obtain the final answer. The solution in the form \(a + bi\) is \(-2 + 6i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers involves combining both their real and imaginary parts. Think of a complex number as having two separate components: the real part and the imaginary part. Each component operates independently during addition. Let's break it down step-by-step for clarity:

• Identify the Real Parts: These are the non-"i" containing parts of the numbers. When adding complex numbers, sum up their respective real parts.


• Identify the Imaginary Parts: These parts contain the "i" (which represents the square root of -1). Similarly, sum the imaginary components separately.


• Combine: The total result of adding complex numbers will have both a real and an imaginary component, written as \(a + bi\).

By treating each part separately, you can easily manage the addition of complex numbers, ensuring you maintain a clear distinction between real and imaginary components. This method also helps in visualizing operations like vector addition in two-dimensional planes, where each part represents a different axis.

Subtraction of Complex Numbers

Subtracting complex numbers is very similar to addition, but with an important twist: subtraction can be thought of as adding a negative. To manage this operation simply, follow these steps:

• Distribute the Negative: When subtracting, change the subtraction of a complex number to the addition of its negatives. For example, \((a + bi) - (c + di)\) becomes \((a + bi) + (-c - di)\).


• Identify and Combine Real Parts: After distributing the negative, group and calculate the real parts.


• Identify and Combine Imaginary Parts: Similarly, do the same for the imaginary parts, ensuring you manage the signs accordingly.


• Write the Result: Combine the results into the standard form \(a + bi\).

Handling subtraction through distribution helps prevent errors that can occur with sign management, especially within the imaginary components. With practice, these operations become organized and predictable.

Imaginary Numbers

Imaginary numbers are a fascinating concept in mathematics, extending beyond the regular number line. They are defined as numbers that, when squared, give a negative result. The basic imaginary unit is \(i\), where \(i = \sqrt{-1}\).

Imaginary numbers often arise in real-world applications ranging from electrical engineering to signal processing, offering solutions to equations that do not have real-number answers. Here are some key points about imaginary numbers:

• The square of \(i\) is -1, i.e., \(i^2 = -1\).


• Imaginary numbers help represent complex numbers which combine real numbers with multiples of \(i\).


• They allow for the description of phenomena such as oscillations, complex waveforms, and even more abstract theoretical constructs in physics and other sciences.

Understanding imaginary numbers deepens the comprehension of logarithmic and exponential functions, often appearing in more advanced topics of calculus and algebra.