Problem 45 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{3}{-i} $$

Step-by-Step Solution

Answer

The expression in standard form is \( 3i \).

Step 1: Recognize the Complex Denominator

The denominator in the fraction \( \frac{3}{-i} \) is a purely imaginary number \(-i\). To convert this into a standard form, which means the form of \( a + bi \), we need to eliminate the imaginary unit \( i \) from the denominator.

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator of the fraction \( \frac{3}{-i} \) by \( i \), the conjugate of \(-i\). This gives: \[ \frac{3 \cdot i}{-i \cdot i} \] which simplifies the denominator.

Step 3: Simplify the Denominator

Calculate \(-i \cdot i = (-1)(i^2) = -(-1) = 1\) since \(i^2 = -1\). So, the denominator becomes \(1\). The fraction now looks like \( \frac{3i}{1} \).

Step 4: Simplify the Fraction

Now, \( \frac{3i}{1} \) simplifies to \( 3i \). This is in the form \( 0 + 3i \), which is the standard form a complex number.

Key Concepts

Imaginary Numbers

Imaginary numbers are numbers that can be expressed in the form of a real number multiplied by the imaginary unit, which is denoted by the letter \(i\). The imaginary unit is a mathematical concept used to represent the square root of \(-1\). This means that \(i^2 = -1\), which is a key property of imaginary numbers.
For instance, what makes \(-i\) in our original exercise a purely imaginary number is its form. It is essentially \(0 + (-1)i\), where the real part is zero, and the imaginary part is "\(-1i\)".

Standard Form of Complex Numbers

A complex number is typically expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This way of writing a complex number is known as its standard form.
In our context, put simply, the goal is to write a number in the form \(a + bi\), even if either \(a\) or \(b\) is zero. For example:

• If \(a = 0\), then the complex number is purely imaginary.
• If \(b = 0\), then the complex number is purely real.

In the problem at hand, the result \(3i\) can be seen as \(0 + 3i\), which perfectly adheres to this standard form.

Complex Conjugate

The complex conjugate of a complex number is found by changing the sign of its imaginary part. For a number \(a + bi\), the complex conjugate is \(a - bi\). Conjugates have a unique utility: when multiplied by each other, they yield a real number.
In the exercise given,

• The conjugate of \(-i\) (which is \(0 - i\)) is \(i\) (or \(0 + i\)).
• This means wherever we see \(-i\) in the denominator, multiplying by its conjugate \(i\) will help eliminate the imaginary unit.

By multiplying the denominator and the numerator by \(i\), we managed to transform the fractional expression into a standard form where the imaginary part is in the numerator.

Simplifying Fractions with Imaginary Denominators

Ensuring fractions with imaginary denominators are simplified into a recognizable form is essential. The method for doing so is to eliminate the imaginary number from the denominator by multiplication.
Using the complex conjugate is a systematic way to achieve this simplification. In the exercise example, the fraction \(\frac{3}{-i}\) was transformed by multiplying both the numerator and the denominator by \(i\). This yielded:

• The denominator simplified from \(-i \cdot i\) to \(1\), as \(-i^2 = 1\).
• The numerator became \(3i\), resulting in a fraction simplification to \(3i\), or \(0 + 3i\) in standard form.

In any fraction involving imaginary numbers as denominators, this technique of using the complex conjugate is vital in achieving the desired standard form.