Problem 21 - Elementary Algebra

Question

Perform the operations. $$ (114+32 i)-(47-34 i) $$

Step-by-Step Solution

Answer

The result is \(67 + 66i\).

Step 1: Identify Real and Imaginary Parts

In a complex number, the real part and imaginary part need to be identified first. The expression given is \((114 + 32i) - (47 - 34i)\). Here, the real parts are 114 and 47, and the imaginary parts are \(32i\) and \(-34i\).

Step 2: Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number. That is, compute: \(114 - 47\). This results in: \(67\).

Step 3: Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number. This involves \(32i - (-34i)\). This simplifies to \(32i + 34i\) because subtracting a negative is equivalent to adding, resulting in \(66i\).

Step 4: Combine the Results

Combine the results of Step 2 and Step 3 to express the final complex number. The real part is 67 and the imaginary part is \(66i\), combining them gives: \(67 + 66i\).

Key Concepts

Real and Imaginary Parts

In complex numbers, each number is expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers. The "real part" is the \(a\) component, and it represents the number without any imaginary unit. The "imaginary part" is \(bi\), which incorporates the imaginary unit \(i\), defined by \(i^2 = -1\).
Identifying these parts properly is the first step when working with complex numbers. It is key for operations such as addition, subtraction, multiplication, and division. For example, given the complex numbers \(114 + 32i\) and \(47 - 34i\), the real parts are 114 and 47, while the imaginary parts are \(32i\) and \(-34i\). Correctly distinguishing these components allows for the seamless execution of mathematical operations.

Complex Number Subtraction

Subtracting complex numbers requires you to focus separately on the real and imaginary parts. Here's the process in simple terms:

• Subtract the real part of the second complex number from the real part of the first.
• Subtract the imaginary part of the second complex number from the imaginary part of the first.

In our exercise, when we subtract \((47 - 34i)\) from \((114 + 32i)\), the calculation begins with two distinct steps:

• Real part: \(114 - 47 = 67\)
• Imaginary part: \(32i - (-34i) = 32i + 34i = 66i\)

This approach of individual subtraction ensures clarity and accuracy in the results!

Combining Complex Numbers

After performing separate operations on the real and imaginary parts, it's time to bring them back together.
The combined form of a complex number is simply the addition of these two parts in their respective places. From our subtraction step, we derived the real part as 67 and the imaginary part as \(66i\). Now, by combining them, the resultant complex number becomes \(67 + 66i\).
This synthesis of the components is what gives a complex number its final form, allowing further utilization in other mathematical contexts.

Mathematical Operations with Complex Numbers

Complex numbers are fundamental to various fields, requiring operations beyond just subtraction. Here are some key mathematical operations you can perform:

• Addition: Similar to subtraction, add the real parts and imaginary parts separately.
• Multiplication: Use distribution, remembering that \(i^2 = -1\).
• Division: Involve the complex conjugate for rationalizing the denominator.

Complex numbers allow for a rich set of operations, contributing to applications in engineering, physics, and more. Practicing these various calculations will deepen your understanding and enable you to handle complex mathematics with ease.