Problem 17 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ (-12+8 i)-(7+4 i) $$

Step-by-Step Solution

Answer

The result is \(-19 + 4i\).

Step 1: Identify the Expression Format

The expression given is \[ (-12 + 8i) - (7 + 4i) \]which is composed of two complex numbers: \(-12 + 8i\) and \(7 + 4i\). We need to subtract these two complex numbers.

Step 2: Distribute the Negative Sign

Apply the negative sign to the second complex number:\[ (-12 + 8i) - 7 - 4i \]This simplifies the expression into \[-12 + 8i - 7 - 4i\].

Step 3: Combine Real Parts

Combine the real parts of the expression:\(-12 - 7\). This results in\[-19\].

Step 4: Combine Imaginary Parts

Combine the imaginary parts of the expression:\(8i - 4i\). This results in\(4i\).

Step 5: Write the Result

Combine the results from the previous steps to form the final simplified expression:\[-19 + 4i\]. This is the expression in the form \(a + bi\) where \(a = -19\) and \(b = 4\).

Key Concepts

Subtraction of Complex Numbers

Subtraction of complex numbers is quite similar to basic algebraic subtraction. A complex number is composed of a real part and an imaginary part, usually represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). To subtract one complex number from another, you separately subtract the real parts and the imaginary parts.

In our exercise:

• Start with \((-12 + 8i) - (7 + 4i)\).
• First, distribute the negative sign to the second complex number: \(-12 + 8i - 7 - 4i\).
• Then, combine the real parts separately: \(-12 - 7 = -19\).
• Next, combine the imaginary parts: \(8i - 4i = 4i\).

By performing these steps, the subtraction becomes straightforward, concluding with the result \(-19 + 4i\).

Real and Imaginary Parts

Complex numbers are unique because they have two components, the real and imaginary parts. Understanding these parts is crucial when performing operations such as addition and subtraction with complex numbers.

The real part of a complex number \(a + bi\) is the \(a\), and the imaginary part is the \(bi\). Though it has an "imaginary" aspect, the imaginary unit \(i\) helps solve problems that involve square roots of negative numbers.

In subtraction, we simply combine the real parts together and the imaginary parts together separately. This means aligning them as such:

• Real parts: \(-12\) and \(-7\)
• Imaginary parts: \(8i\) and \(-4i\)

Understanding this distinction helps streamline calculations and minimize errors while manipulating complex numbers.

Arithmetic with Complex Numbers

Arithmetic with complex numbers involves a few additional principles compared to dealing with regular numbers, but the operations like addition, subtraction, multiplication, and division remain fundamentally similar in concept.

When conducting arithmetic with complex numbers:

• Always operate on the real and imaginary parts separately.
• The operation results in another complex number in the form \(a + bi\).
• Handle the imaginary unit \(i\) with care, particularly in multiplication where \(i^2\) plays a crucial role.

While it might seem complex initially, arithmetic with complex numbers only needs a bit of practice and familiarity. Realizing that each arithmetic operation will consistently yield a complex number simplifies the process vastly. This consistency eases learning and application in various mathematical and engineering fields.