Problem 97 - Intermediate Algebra

Question

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{-2+7 i}{-1+i} $$

Step-by-Step Solution

Answer

\(-\frac{5}{2} - \frac{9}{2}i\)

Step 1: Identify the Complex Numbers

We are given two complex numbers: the numerator \(-2 + 7i\) and the denominator \(-1 + i\). Our goal is to divide these two complex numbers.

Step 2: Multiply by the Conjugate

To simplify the division, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(-1 + i\) is \(-1 - i\). So, we multiply:\[\frac{-2 + 7i}{-1 + i} \times \frac{-1 - i}{-1 - i}\]

Step 3: Perform the Multiplications

Calculate the multiplication for both the numerator and the denominator:- Numerator: \((-2 + 7i)(-1 - i) = -2(-1) - 2i + 7i + 7i^2\) Simplify using \(i^2 = -1\): \(2 - 2i - 7i - 7 = -5 - 9i\)- Denominator: \((-1 + i)(-1 - i) = (-1)(-1) - i^2 = 1 + 1 = 2\).

Step 4: Write the Result as a Complex Number

The multiplication results give us a new fraction:\[\frac{-5 - 9i}{2}\]Divide each term by 2 to express this in standard form, \(a + bi\):\(-\frac{5}{2} - \frac{9}{2}i\).

Key Concepts

Quotients

When dealing with complex numbers, finding quotients can be a little different than with regular numbers. A quotient in mathematics refers to the result of dividing one number by another. For complex numbers, the process is similar, but it involves an extra step to handle the imaginary parts. Let's break it down:

• Start by identifying the complex numbers involved. Typically, you'll have one in the numerator and one in the denominator — much like a regular fraction.
• Division involves creating a "fraction" with the complex numbers. However, because of the presence of imaginary numbers, you can't simply "divide" as with regular numbers.
• To clear the imaginary parts from the denominator, you multiply both the numerator and the denominator by the conjugate of the denominator. This is the crucial step that allows you to manage complex division effectively.

This method ensures that your result is still a valid complex number but without an imaginary unit in the denominator.

Complex Conjugate

The concept of a complex conjugate is fundamental when simplifying complex quotients. Imagine a complex number expressed as \(a + bi\). Its conjugate is simply \(a - bi\). So, you change the sign of the imaginary part. Why do we use it?

• When you multiply a complex number by its conjugate, the result is a real number. Specifically, you get \(a^2 + b^2\), removing the imaginary part.
• This makes it incredibly useful for division because it clears out the imaginary part in the denominator.
• In our problem, we used the conjugate \(-1 - i\) to multiply both parts of the fraction \(\frac{-2+7i}{-1+i}\).

Using the conjugate ensures the denominator becomes a real number, making it easier to express the quotient in standard form, \(a + bi\).

Standard Form

Expressing a complex number in standard form is an essential final step in completing the division of complex numbers. The standard form \(a + bi\) consists of a real part \(a\) and an imaginary part \(b\), where \(i\) represents the square root of \(-1\). This tidy form helps in clearly distinguishing between the real and imaginary components. How to achieve it:

• After completing the multiplication steps when dividing, you obtain a new expression, which often looks complex at first glance.
• To transition that result into standard form, divide each term in the resulting fraction individually — the real part and the imaginary part — by the denominator.
• In our example, after multiplying by the conjugate and simplifying, we had \(-\frac{5}{2} - \frac{9}{2}i\), which is already a good representation of \(a + bi\).

Getting the complex number into this neat format allows for easy interpretation and further mathematical operations.