Problem 15 - Intermediate Algebra

Question

Add or subtract as indicated. $$ (-7+3 i)-(5-2 i) $$

Step-by-Step Solution

Answer

The result of the operation is \(-12 + 5i\).

Step 1: Identify Real and Imaginary Parts in Each Complex Number

Identify the real and imaginary parts of each complex number in the expression. The first complex number is \(-7+3i\), where \(-7\) is the real part and \(3i\) is the imaginary part. The second complex number is \(5-2i\), where \(5\) is the real part and \(-2i\) is the imaginary part.

Step 2: Apply Subtraction to Real Parts

Subtract the real part of the second complex number from the real part of the first complex number: \(-7 - 5 = -12\).

Step 3: Apply Subtraction to Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: \(3i - (-2i) = 3i + 2i = 5i\). Note that subtracting a negative number is equivalent to adding its positive.

Step 4: Combine the Results

Combine the results of the real parts and the imaginary parts. The result is the complex number: \(-12 + 5i\).

Key Concepts

Real Part

In a complex number, the real part is the component that is not associated with the imaginary unit \( i \). Complex numbers are usually expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is known as the real part. It provides the position on the horizontal axis when plotting a complex number on the complex plane.

When identifying the real part in the exercise \((-7 + 3i) - (5 - 2i)\), it is crucial to focus on the parts of the terms that don't involve \( i \):

• First complex number, \(-7\), is the real part.
• Second complex number, \(5\), is the real part.

Recognizing these allows us to handle subtraction operations on complex numbers efficiently by dealing with real and imaginary parts separately.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit \( i \). It is represented by \( bi \) where \( b \) is a real number and \( i \) is the imaginary unit, defined as the square root of \(-1\). In the standard form of a complex number \( a + bi \), \( bi \) constitutes the imaginary part.

In our example, to identify the imaginary parts, observe the terms with \( i \):

• First complex number, \(3i\), is the imaginary part.
• Second complex number, \(-2i\), is the imaginary part.

It's essential to note how operations with imaginary parts work: since both have \( i \), we subtract as we would with real numbers, remembering to keep the \( i \). For example, when faced with \( 3i - (-2i) \), it becomes \( 3i + 2i \), demonstrating that two negatives make a positive.

Complex Number Subtraction

Subtraction of complex numbers involves subtracting their real and imaginary parts separately. The key to performing these operations smoothly is to treat the subtraction like handling two simple arithmetic problems; one with real parts and one with imaginary parts.

For the given exercise \((-7+3i)-(5-2i)\), the steps highlight:

• Subtract the real parts: \(-7 - 5 = -12\).
• Subtract the imaginary parts: \(3i - (-2i) = 3i + 2i = 5i\).

The result, \(-12 + 5i\), succinctly combines the computed real and imaginary results.

Remember, a minus sign in front of a term changes its sign, effectively flipping the operation as seen in \( 3i - (-2i) \), which becomes \( 3i + 2i \). Hence, complex number subtraction is a process of keeping track of signs and ensuring correct operations for both components.