Problem 85 - Intermediate Algebra

Question

Divide. Write all answers in the form \(a+b i.\) $$ \frac{5}{3 i} $$

Step-by-Step Solution

Answer

The result is \(0 + (-\frac{5}{3})i\) or simply \(-\frac{5}{3}i\).

Step 1: Multiply Numerator and Denominator by the Conjugate

To remove the imaginary unit from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of any imaginary number of the form \(bi\) is \(-bi\). Thus, we multiply by \(-i\):\[\frac{5}{3i} imes \frac{-i}{-i} = \frac{5(-i)}{3i(-i)}\]

Step 2: Simplify the Denominator

Now, simplify the denominator. Using the identity \(i^2 = -1\), we have:\[3i(-i) = 3(-i^2) = 3(-(-1)) = 3(1) = 3\]

Step 3: Simplify the Numerator

Simplify the numerator:\[5(-i) = -5i\]

Step 4: Write the Result in Standard Form \(a + bi\)

Now, put together the simplified numerator and denominator:\[\frac{-5i}{3} = 0 - \frac{5}{3}i\] In standard form, this means:\(0 + \left(-\frac{5}{3}\right)i\)

Step 5: Conclusion

The division results in the complex number \(0 - \frac{5}{3}i\) in the form of \(a + bi\). The real part is \(0\) and the imaginary part is \(-\frac{5}{3}\). This is the final answer.

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers. It is denoted as \(i\) and is defined as the square root of \(-1\). This characteristic gives the imaginary unit its unique property: \(i^2 = -1\). Understanding the imaginary unit is crucial when dealing with complex numbers, as it serves as the building block for all imaginary and thus complex numbers.

In simple terms:

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)

These properties of \(i\) allow us to work with complex numbers and perform operations such as addition, subtraction, multiplication, and division. Remember that when you see \(i\), think of it as a way to incorporate the concept of the square root of a negative number into everyday mathematics. Without it, complex numbers could not exist, as they rely on the imaginary unit to reflect the presence of an imaginary component.

Conjugate

The conjugate of a complex number is another concept that is essential for various operations with complex numbers, especially for division. If you have a complex number of the form \(a + bi\), its conjugate is \(a - bi\).

Here's why this is important:

• Conjugates help "cancel out" the imaginary unit when dividing complex numbers.
• When a complex number is multiplied by its conjugate, the result is a real number.

For instance, in the exercise, the denominator is \(3i\), an imaginary number. Its conjugate would be \(-3i\). By multiplying the numerator and denominator by \(-i\), we are, in effect, using the conjugate method to simplify the expression.

Multiplying \(3i\) by \(-i\) results in a real number, simplifying the division process.

Standard Form

Complex numbers are expressed in a standardized way for easier manipulation and understanding. This is known as the standard form and is typically written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

For clarity:

• The real part \(a\) can be any real number, including zero.
• The imaginary part \(bi\) involves the imaginary unit \(i\).

In our exercise, the result of the division needs to be put into standard form, even if one part (the real or imaginary) is zero. The final result \(0 + (-\frac{5}{3})i\) is a demonstration of presenting a complex number properly by showing both the real and imaginary parts, even if the real part is zero.

Division of Complex Numbers

Dividing complex numbers can initially seem complicated due to the presence of \(i\) in the denominator. However, the use of conjugates simplifies this process significantly. The main steps involved include multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary unit from the denominator.

Here's how this works:

• Identify the complex number to be divided, such as \(\frac{5}{3i}\).
• Find the conjugate of the denominator, which in this case is \(-i\).
• Multiply both numerator and denominator by \(-i\) to simplify.
• Simplify the resulting expression to get a real denominator.

This process ensures that the division results in a complex number expressed in standard form \(a + bi\). Through this method, the complexities of the imaginary components are handled efficiently, making complex arithmetic accessible and understandable.