Problem 21 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(12+5 i)-(7-i)$$

Step-by-Step Solution

Answer

The result is \(5 + 6i\).

Step 1: Understand the complex numbers structure

Complex numbers are in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Here we have two complex numbers and we need to subtract one from the other, which means we should subtract the real part from the real part and the imaginary part from the imaginary part.

Step 2: Perform the subtraction for the real parts

We subtract \(7\) (the real part of the second number) from \(12\) (the real part of the first number). This results in \(12 - 7 = 5\).

Step 3: Perform the subtraction for the imaginary parts

We subtract \(-1\) (the imaginary part of the second number, don't forget about the minus in front of it) from \(5\) (the imaginary part of the first number). This results in \(5 -(-1) = 5 + 1 = 6\). After the subtraction, we multiply the result by \(i\) (the imaginary unit) to retain the imaginary part.

Step 4: Write out the answer in standard form

The result of the subtraction of the two complex numbers is \(5 + 6i\), which is the final result in standard form for a complex number.

Key Concepts

Standard Form of a Complex Number

A complex number is a mathematical entity comprising both a real number and an imaginary number. It's expressed in the standard form as:
\( a + bi \),
where \( a \) represents the real part and \( bi \) represents the imaginary part, with \( i \) being the imaginary unit, satisfying \( i^2 = -1 \). The beauty of the standard form lies in its simplicity, allowing us to perform arithmetic operations much like we do with real numbers, while keeping the distinct components organized.
For instance, the complex numbers in our exercise, \( 12 + 5i \) and \( 7 - i \), are displayed in this clear, standard form, making it straightforward to identify and manipulate their respective real and imaginary parts.

Complex Numbers Operations

Operations with complex numbers follow the same fundamental principles as with real numbers but with added attention to both the real and imaginary components. When adding or subtracting complex numbers, we combine like terms; that is, real numbers with real numbers, and imaginary numbers with imaginary numbers.

Subtraction of Complex Numbers

In subtraction, such as in our example, we individually subtract the real part of the second number from the real part of the first, and do the same with the imaginary parts. The subtraction is performed as:

• Real Parts: \( 12 - 7 = 5 \)
• Imaginary Parts: \( 5i - (-i) = 5i + i = 6i \)

This approached ensures we handle the real and imaginary portions appropriately, resulting in precise and correct calculations in the complex plane.

Imaginary Numbers Arithmetic

Imaginary number arithmetic involves performing operations with the imaginary unit \( i \), which can seem daunting at first. However, treating \( i \) as a variable—with the unique property that \( i^2 = -1 \)—simplifies the process. We can then apply the standard arithmetic rules as we would with algebraic expressions.
In the subtraction task from our exercise, one key step is managing the sign in front of the imaginary part, ensuring that \( -(-i) \) becomes \( +i \), due to the rule of signs in standard arithmetic. This results in an intuitive addition:

• \( 5i + i = 6i \)

Understanding this arithmetic is essential in both simplifying complex expressions and in ensuring seamless transition from complex number operations to more advanced topics involving complex analysis.