Problem 97 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) $$ (3-i)-(-1+10 i) $$

Step-by-Step Solution

Answer

The result is \(4 - 11i\).

Step 1: Write Down the Expression

The given expression is \((3-i) - (-1+10i)\). We need to simplify this expression and write the result in the form of \(a+b i\).

Step 2: Distribute Negative Sign

Distribute the negative sign across the terms in the second set of parentheses. This changes the expression to \((3-i) + (1-10i)\).

Step 3: Combine Real Parts

Identify and combine the real parts of the expression. In this case, the real parts are \(3\) and \(+1\). Adding these together gives \(3+1 = 4\).

Step 4: Combine Imaginary Parts

Identify and combine the imaginary parts of the expression. The imaginary parts are \(-i\) and \(-10i\). Adding these together gives \(-i - 10i = -11i\).

Step 5: Write the Result

Combine the results of Step 3 and Step 4 to write the final answer in the form \(a+bi\). Thus, the result is \(4 - 11i\).

Key Concepts

Addition and Subtraction of Complex Numbers

When dealing with complex numbers, we often have to perform addition or subtraction.
A complex number takes the form of \(a+bi\), where \(a\) and \(b\) are real numbers, while \(i\) represents the imaginary unit. The imaginary unit \(i\) satisfies the equation \(i^2 = -1\).
To add or subtract complex numbers, follow these simple steps:

• Separate the real parts and the imaginary parts of each complex number.
• Combine the real parts together by either adding or subtracting them.
• Do the same for the imaginary parts.
• Finally, combine these two results to form the new complex number.

Let's take the example \((3-i) - (-1+10i)\), which requires subtraction.
To solve it, we first distribute the negative sign across the second complex number.
This changes the expression to \((3-i) + (1-10i)\). Now, we add the real parts: \(3\) and \(1\) to get \(4\).
Next, the imaginary parts \(-i\) and \(-10i\) are added to yield \(-11i\). Thus, the final result is \(4 - 11i\).
This method of operation makes handling complex numbers simpler and systematic.

Imaginary Numbers

Imaginary numbers can seem a bit puzzling at first. However, once you get the hang of them, they fit naturally into calculations involving complex numbers.
The core of imaginary numbers is the imaginary unit \(i\), which is defined by the property \(i^2 = -1\).
This definition allows us to extend the traditional set of real numbers and include numbers that weren’t possible to represent otherwise.Imagine needing to find a number whose square is \(-1\). In the realm of real numbers, this wasn't possible, but with the introduction of \(i\), it becomes straightforward.
Every imaginary number is simply a real number multiplied by \(i\).
For instance, if we consider \(5i\), it indicates 5 times the imaginary unit \(i\).
In the problem of subtracting complex numbers \((3-i) - (-1+10i)\), the term \(10i\) is an imaginary number. When we combine this with another imaginary number, say \(-i\), as in the solution, our result still follows the rules of real numbers but with the property of \(i^2 = -1\).
Understanding this concept enables clearer navigation through arithmetic operations with complex numbers.

Real and Imaginary Parts

Every complex number is composed of two components: a real part and an imaginary part. Understanding how to identify and work with each part is vital.
The real part of a complex number is the number without the imaginary unit \(i\), while the imaginary part is the coefficient that multiplies \(i\).
For example, in the complex number \(6+4i\), \(6\) is the real part, and \(4\) is the imaginary part.
Identifying these components is essential, especially when performing operations like addition and subtraction, as seen in our exercise.
We initially have the expression \((3-i) - (-1+10i)\).
By rewriting it as \((3-i) + (1-10i)\), we determined the real parts \(3\) and \(1\), adding them to get \(4\).
The imaginary parts are \(-i\) and \(-10i\), summing to \(-11i\).
Recognizing and combining these parts separately ensures the accuracy of the outcome. As you practice more, identifying these parts will become second nature, making operations with complex numbers a breeze.