Problem 46 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{4-2 i}{i} $$

Step-by-Step Solution

Answer

The expression in standard form is \(-2 - 4i\).

Step 1: Understand the Expression

The expression provided is \( \frac{4 - 2i}{i} \). Your task is to convert this complex fraction into its standard form, which is \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and the denominator by \( i \) (which is the complex conjugate of \( i \)). The expression becomes \( \frac{(4 - 2i)i}{i^2} \).

Step 3: Simplify the Denominator

Remember that \( i^2 = -1 \). The denominator \( i^2 \) simplifies to \( -1 \). So the modified expression is \( \frac{(4 - 2i)i}{-1} \).

Step 4: Simplify the Numerator

Distribute \( i \) in the numerator: \( (4 - 2i)i = 4i - 2i^2 \).Substitute \( i^2 \) with \( -1 \), resulting in \( 4i - 2(-1) = 4i + 2 \).

Step 5: Combine and Simplify the Expression

Now substituting it back to the expression: \( \frac{4i + 2}{-1} \).Divide each term by \( -1 \): this results in \( -4i - 2 \).

Step 6: Reorder the Expression

Reorder the expression in the form \( a + bi \). Thus, \( -4i - 2 \) becomes \( -2 - 4i \). This is the standard form.

Key Concepts

Standard Form

In complex numbers, the standard form is an essential concept. It allows us to express a complex number as a combination of a real part and an imaginary part.

• The format used is: \(a + bi\).
• Here, \(a\) represents the real part, and \(bi\) represents the imaginary part.

Writing a number in standard form simplifies interpretation and provides a clear way to compare complex numbers. In the original exercise, transforming the expression \( \frac{4 - 2i}{i} \) into the standard form resulted in \(-2 - 4i\). This solution helps us see both components of the complex number distinctly, making it easier to work with in further calculations or comparisons.

Imaginary Unit

The imaginary unit \(i\) is fundamental in dealing with complex numbers. It is defined as the square root of \(-1\).

• This gives us the identity: \(i^2 = -1\).
• The concept of \(i\) allows for the creation of complex numbers of the form \(a + bi\).

In the exercise, \(i\) starts in the denominator, which requires us to multiply by itself to simplify it. By doing so, the complex numbers can be better manipulated and reduced into their more easily understandable form. Recognizing and appropriately manipulating \(i\) is key to solving such problems.

Complex Conjugate

The complex conjugate of a number is another powerful tool, particularly useful in simplifying complex fractions. For any complex number \(a + bi\), the conjugate is \(a - bi\).

• When multiplied together, \(a + bi\) and \(a - bi\) result in a real number \(a^2 + b^2\).
• This technique eliminates the imaginary unit from the denominator.

In the given solution, we multiplied the top and bottom by \(i\) itself, because \(i\) is its own "conjugate" in this context. This multiplication helped eliminate the imaginary unit from the denominator seamlessly, transforming the expression into \(-2 - 4i\), thus successfully reflecting the standard form.