Problem 85 - Elementary and Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i\) See Example 10. $$ \frac{5}{3 i} $$

Step-by-Step Solution

Answer

The answer is \( 0 - \frac{5}{3}i \).

Step 1: Identify the Problem

We need to simplify the expression \( \frac{5}{3i} \) and write the answer in the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

To eliminate the imaginary number \( i \) from the denominator, multiply both the numerator and denominator by \( -i \) (the conjugate of \( i \)). This gives: \( \frac{5}{3i} \times \frac{-i}{-i} = \frac{5(-i)}{3i(-i)} \).

Step 3: Simplify the Denominator

Multiply the terms in the denominator: \( 3i \times (-i) = -3i^2 \). Since \( i^2 = -1 \), the denominator becomes \( 3 \).

Step 4: Simplify the Numerator

Multiply the terms in the numerator: \( 5 \times (-i) = -5i \).

Step 5: Combine Results

The expression now becomes \( \frac{-5i}{3} \). Split it into real and imaginary components: \( 0 - \frac{5}{3}i \).

Step 6: Write in Standard Form

Recast the expression in the form \( a + bi \): \( 0 + (-\frac{5}{3})i \), which is simply \( 0 - \frac{5}{3}i \).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept used to extend the notion of real numbers. The defining property of \(i\) is that it satisfies the equation \(i^2 = -1\). This means that \(i\) represents the square root of -1, a number that doesn't have a place on the standard number line of real numbers. - The symbol \(i\) helps us express square roots of negative numbers in a consistent manner.- In complex numbers, \(i\) is used to denote the imaginary part, allowing us to handle equations that have no real solutions. Imaginary numbers open the door to complex numbers, expanding arithmetic and algebra to a broader field. When working with complex numbers, the imaginary unit \(i\) takes center stage as a fundamental building block.

Conjugate Multiplication

Conjugate multiplication is a clever method used when dividing complex numbers, particularly to get rid of imaginary units in the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\). Multiplying a complex number by its conjugate exploits the relation \((a + bi)(a - bi) = a^2 - (bi)^2\).- This multiplication results in \(a^2 + b^2\), a real number. - By using conjugate multiplication, any imaginary component in the denominator can be eliminated, simplifying division problems.In the context of \( \frac{5}{3i} \), we multiply both numerator and denominator by \(-i\), leading to \( \frac{5(-i)}{3i(-i)} \). This action removes \(i\) from the denominator. Recognizing when to apply this technique is a valuable skill when modifying complex expressions.

Standard Form of Complex Numbers

Every complex number can be written in a standard form, \( a + bi \), where \(a\) and \(b\) are real numbers. This form makes calculations involving complex numbers clearer and easier to manage. - \(a\) represents the real component, and \(bi\) is the imaginary component.- When simplified correctly, complex numbers can be effortlessly added, subtracted, multiplied, or divided, maintaining this standard format.For the exercise we're exploring, we aimed to recast \( \frac{5}{3i} \) into this form. Initially, it's transformed into \( 0 - \frac{5}{3}i \). Here, \(a\) is 0 and \(b\) is \(-\frac{5}{3}\), perfectly fitting the standard \(a + bi\) mold. By learning to express complex numbers in standard form, you can compare and perform operations on them with ease.