Problem 27 - College Algebra Essentials

Question

In Exercises \(21-28,\) divide and express the result in standard form. $$ \frac{2+3 i}{2+i} $$

Step-by-Step Solution

Answer

The result is \( \frac{7}{5} + \frac{4}{5}i \) in standard form.

Step 1: Identify the Conjugate

The conjugate of a complex number \(a+bi\) is \(a-bi\). Thus the conjugate of \(2+i\) is \(2-i\).

Step 2: Multiply by Conjugate

Multiply the numerator and the denominator by the conjugate of the denominator. Hence, \(\frac{2+3i}{2+i}\) becomes \(\frac{2+3i}{2+i} * \frac{2-i}{2-i}\).

Step 3: Simplify the Numerator

For the numerator, follow the FOIL method (First, Outer, Inner, Last) to get \((2+3i) * (2-i) = 4 - 2i + 6i - 3i^2\). Here, \(i^2 = -1\), so the expression becomes \(4 + 4i + 3 = 7 + 4i\).

Step 4: Simplify the Denominator

In the denominator also follow the FOIL method. The denominator is \((2+i)* (2-i) = 4 - 2i + 2i - i^2\). Here again, \(i^2 = -1\), thus the expression becomes \(4 + 1 = 5\).

Step 5: Express the Result in Standard Form

The result is then \(\frac{7 + 4i}{5}\). This can be written in standard form as \( \frac{7}{5} + \frac{4}{5}i\).

Key Concepts

Understanding Conjugates in Complex Numbers

Conjugates are an important concept when working with complex numbers, especially in division. A complex number is usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The conjugate of this complex number is \(a - bi\).
To form the conjugate, you simply change the sign of the imaginary part. This is crucial for simplifying division involving complex numbers.

• For example, the conjugate of \(2+i\) is \(2-i\).
• Using the conjugate allows us to eliminate the imaginary part in the denominator.

When you multiply a complex number by its conjugate, you end up with a real number. For instance, multiplying \((2+i)(2-i)\) results in a real number, \(4 - i^2\), which simplifies to \(5\) after knowing that \(i^2 = -1\). This property is used to convert the division of complex numbers into a simpler form.

The Standard Form of Complex Numbers

The standard form for complex numbers is important for clearly expressing results. This form consists of a real part and an imaginary part separated by a plus sign, written as \(a + bi\).

• Here, \(a\) is the real component.
• \(b\) is the coefficient of the imaginary part.

In mathematical operations involving complex division, results are often expressed in standard form to maintain consistency and clarity.
For instance, after dividing two complex numbers and simplifying, the result might be \(\frac{7 + 4i}{5}\). This can be further expressed in standard form as \(\frac{7}{5} + \frac{4}{5}i\), where both parts are separated clearly. Using standard form helps in comparing, plotting on the complex plane, and further calculations.

How to Use the FOIL Method with Complex Numbers

The FOIL method is a handy mnemonic for remembering how to multiply binomials and it, surprisingly, works well with complex numbers too. FOIL stands for First, Outer, Inner, and Last. When you multiply two complex numbers in the form \((a+bi)(c+di)\), you apply the FOIL method like this:

• First: Multiply the first terms of each binomial, \(a \cdot c\).
• Outer: Multiply the outer terms, \(a \cdot di\).
• Inner: Multiply the inner terms, \(bi \cdot c\).
• Last: Multiply the last terms, \(bi \cdot di\).

Applying FOIL to \((2+3i)(2-i)\), you get:

• First: \(2 \cdot 2 = 4\)
• Outer: \(2 \cdot (-i) = -2i\)
• Inner: \(3i \cdot 2 = 6i\)
• Last: \(3i \cdot (-i) = -3i^2\)

Remember that \(i^2 = -1\), so the expression \(-3i^2\) becomes \(+3\). Adding all parts together gives \(4 + 4i + 3\), simplifying to \(7 + 4i\). This systematic approach results in fewer errors and a better understanding when dealing with complex multiplications.

Understanding the Imaginary Unit

The imaginary unit \(i\) is a fundamental component of complex numbers. By definition, \(i\) is the square root of \(-1\), which is not possible within the real number system.

• This imaginary unit is denoted as \(i\), where \(i^2 = -1\).
• Complex numbers are constructed using \(i\), combining both real and variable parts.

The imaginary unit is what allows complex numbers to extend beyond the real number line. When you see \(i\) in an expression like \(2 + 3i\), it indicates that \(3i\) is 3 times the imaginary unit. Essential operations like multiplication use the fact about \(i^2 = -1\) to simplify expressions. Thus, in the example \((3i) \cdot (-i) = -3i^2 = 3\), knowing \(i^2 = -1\) enables simplification to a real number. Understanding the imaginary unit and its properties is crucial for anyone working with complex numbers.