Problem 4 - College Algebra

Question

In Exercises \(1-8,\) add or subtract as indicated and write the result in standard form. $$(3+2 i)-(5-7 i)$$

Step-by-Step Solution

Answer

The result of the subtraction of the two complex numbers is \(-2 + 9i\).

Step 1: Real parts subtraction

Subtract the real part of the second complex number from the real part of the first one: \(3 - 5 = -2\).

Step 2: Imaginary parts subtraction

Subtract the imaginary part of the second complex number from the imaginary part of the first one: \(2i - (-7i) = 2i + 7i = 9i\). The '-' sign in front of '-7i' becomes a '+' because the operation is subtraction.

Step 3: Write the result in standard form

The result in standard form is the real part from step 1 plus the imaginary part from step 2: \(-2 + 9i\).

Key Concepts

Standard Form of Complex Numbers

Understanding the standard form of complex numbers is crucial when dealing with any complex number arithmetic. Essentially, complex numbers are composed of two parts: the real part and the imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) represents the real part and \(b\) is the coefficient of the imaginary part, \(i\). It's important to recognize that \(i\) is the imaginary unit and it is defined by the property that \(i^2 = -1\).

When dealing with complex numbers, this standard form allows us to perform addition, subtraction, multiplication, and even division, by treating the real and imaginary parts separately but within the same expression. This form is widely used because it simplifies the handling of complex numbers and provides a clear framework for mathematical operations. In the context of subtraction, which we will explore further, keeping complex numbers in standard form ensures clarity and simplifies the calculation.

Real Parts Subtraction

The process of subtracting the real parts of two complex numbers is similar to subtracting ordinary real numbers. When the problem asks us to subtract one complex number from another, we start by looking at the real components of each.

To illustrate, let's use the real parts from the exercise provided. We have two complex numbers, \(3 + 2i\) and \(5 - 7i\). The real part of the first number is 3, and the real part of the second is 5. We perform the subtraction by calculating \(3 - 5\), which equals \(-2\). This result represents the real part of the resulting complex number after subtraction.

It's helpful to visualize this step as simply taking away or reducing the value you start with by the value you're subtracting. Even though these are part of complex numbers, the real parts behave just like any other real numbers you're used to working with.

Imaginary Parts Subtraction

Subtraction of the imaginary parts follows a similar method to real part subtraction but with mindfulness of the imaginary unit \(i\). Taking the imaginary components from our given complex numbers, \(2i\) and \(-7i\), we proceed with the subtraction just as we would with real numbers; however, we must remember to treat the imaginary unit like a variable that remains constant.

In our exercise, we subtract \(-7i\) from \(2i\). Note that subtracting a negative is the same as adding its positive counterpart. So, this becomes \(2i + 7i = 9i\). This operation yields the imaginary part of our final complex number post-subtraction. It is key to handle the signs correctly, as these can significantly affect the outcome of the calculation. In essence, when subtracting imaginary numbers, focus on the coefficients as if they are regular numbers and keep the \(i\) alongside them.