Problem 85 - Intermediate Algebra

Question

Find each of the following quotients, and express the answers in the standard form of a complex number. $$ \frac{3 i}{2+4 i} $$

Step-by-Step Solution

Answer

The quotient is \( \frac{3}{5} + \frac{3}{10}i \).

Step 1: Identify the Problem and Recall Complex Number Division

The task is to divide one complex number by another and express the result in standard complex form, which is \( a + bi \) where \( a \) and \( b \) are real numbers. The given expression is \( \frac{3i}{2+4i} \).

Step 2: Multiply by the Conjugate of the Denominator

To simplify the division, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2 + 4i \) is \( 2 - 4i \). Thus, the expression becomes: \[\frac{3i}{2+4i} \times \frac{2-4i}{2-4i}\]

Step 3: Expand the Numerator and Denominator

Expand the numerator and denominator separately.- **Numerator**: \( (3i)(2 - 4i) = 6i - 12i^2 \) Recall that \( i^2 = -1 \), so the expression becomes \( 6i - 12(-1) = 6i + 12 \).- **Denominator**: \( (2+4i)(2-4i) = 4 - 8i + 8i - 16i^2 \) Since \( i^2 = -1 \), it simplifies to \( 4 + 16 = 20 \).

Step 4: Simplify the Quotient

Now put the expanded and simplified numerator and denominator together:\[\frac{6i + 12}{20}\]Separate into real and imaginary components: \[\frac{12}{20} + \frac{6i}{20} = \frac{3}{5} + \frac{3i}{10}\]

Step 5: Express the Result in Standard Form

The quotient \( \frac{3i}{2+4i} \) is expressed in standard complex form as:\[\frac{3}{5} + \frac{3}{10}i\]

Key Concepts

Complex Conjugate

When working with complex numbers, especially in division, the concept of a complex conjugate is crucial. A complex conjugate involves changing the sign of the imaginary part of a complex number. For example, if you have a complex number like \(2 + 4i\), its conjugate will be \(2 - 4i\). This process helps in rationalizing the denominator.
When you multiply a complex number by its conjugate, it results in a real number. Specifically, when you multiply \((2 + 4i)\) by \((2 - 4i)\), the imaginary parts cancel out, leaving you with \(4 + 16 = 20\) since \(i^2 = -1\).

• This method turns the denominator into a real number, making division simpler.
• It helps eliminate the imaginary unit \(i\) from the denominator.

In the context of the problem \(\frac{3i}{2+4i}\), multiplying by the conjugate \(2 - 4i\) is essential.

Standard Form of a Complex Number

The standard form of a complex number is a way of writing it that separates the real and imaginary components. It appears as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary unit \(i\).
To convert a complex fraction like \(\frac{3i}{2+4i}\) into this form, a few key steps are needed. First, multiply by the conjugate to turn the denominator into a real number, then splits the result into real and imaginary parts.
The solution to the problem becomes \(\frac{3}{5} + \frac{3}{10}i\). Here, \(\frac{3}{5}\) is the real part and \(\frac{3}{10}\) is the coefficient of the imaginary part \(i\).

• Always combine real and imaginary results separately.
• This form is essential for simplifying complex number problems.

Ensuring the result is in standard form makes it easier to understand and use.

Imaginary Unit i

The imaginary unit \(i\) is fundamental in dealing with complex numbers. It is defined as \(i = \sqrt{-1}\). This definition implies that \(i^2 = -1\), a property of the imaginary unit that's crucial in calculations. In any operation involving complex numbers, especially multiplication, the existence of \(i^2 = -1\) simplifies expressions.
In our problem, during the expansion of the numerator, you see \(12(-1)\) where \(i^2\) transforms to \(-1\), changing the expression \(6i - 12i^2\) to \(6i + 12\).
This transformation is why the real part of complex numbers often gains or loses values in calculations.

• Understanding \(i\) is critical to comprehending complex arithmetic.
• Always recall \(i^2 = -1\) when simplifying.

Recognizing the role of \(i\) in turning a product of complex numbers into real or imaginary parts is key to mastering complex number operations.