Problem 20 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (-4+i)-(2-5 i) $$

Step-by-Step Solution

Answer

The result is -6 + 6i.

Step 1: Distribute the negative sign

When subtracting complex numbers, distribute the negative sign across the second complex number before performing the subtraction. Thus, begin by rewriting \[(-4+i) - (2-5i)\]as:\[(-4+i) + (-2 + 5i)\]

Step 2: Combine real parts

Now, add the real parts of both complex numbers together. From the rewritten expression \[(-4 + i) + (-2 + 5i)\],combine \[-4\] and \[-2\] to get: \[-6\].

Step 3: Combine imaginary parts

Next, add the imaginary parts of both complex numbers together. Using the rewritten expression \[(-4+i) + (-2+5i)\],combine \[i\] and \[5i\] to get: \[6i\].

Step 4: Write the result in the standard form

Having combined both real and imaginary parts, write the result in the standard form \(a + bi\): \[-6 + 6i\]. This is the solution for the expression.

Key Concepts

Real and Imaginary Parts

Complex numbers are comprised of two parts: a real part and an imaginary part. Understanding these components is key to working with complex numbers effectively.

The real part is the portion of the complex number without the imaginary unit, denoted by \(i\). For example, in the complex number \(-4 + i\), \(-4\) is the real part.

On the other hand, the imaginary part is the component that includes the imaginary unit \(i\). In the same complex number \(-4 + i\), the imaginary part is \(1i\), or simply \(i\).

Recognizing and separating these parts allow for operations like addition, subtraction, and multiplication to be performed with ease. Identifying each part is the first step in solving expressions involving complex numbers.

Standard Form

The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form is not only traditional but also essential for clarity when working with complex numbers.

In our example, the expression \(-4 + i\) is already in standard form. After performing operations such as addition or subtraction, the resulting complex number can be rearranged into \(a + bi\) format for convenience and uniformity.

Writing complex numbers in standard form makes it easier to identify the real and imaginary components at a glance. It ensures consistency in mathematical operations and helps in communicating results effectively to others.

Subtraction of Complex Numbers

Subtracting complex numbers is similar to the process used in arithmetic, but it requires careful handling of the imaginary parts. The key step is to distribute the subtraction across both the real and imaginary parts of the numbers involved.

Consider the subtraction expression \((-4+i)-(2-5i)\). First, distribute the negative sign across the second complex number:

• Transform the expression into \((-4+i) + (-2+5i)\), essentially changing it to an addition problem.

Next, combine the real parts and then the imaginary parts:

• Real parts: \(-4\) and \(-2\) combine to make \(-6\).
• Imaginary parts: \(i\) and \(5i\) combine to make \(6i\).

Finally, represent the result in standard form, \(-6 + 6i\). With this approach, the subtraction of complex numbers becomes systematic and manageable.