Problem 35 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{4+6 i}{3 i} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2 - \frac{4}{3}i\).

Step 1: Multiply the Numerator and Denominator by the Conjugate

To eliminate the complex number in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3i\) is \(-3i\). So, our expression becomes: \[\frac{(4+6i)(-3i)}{(3i)(-3i)}\]

Step 2: Simplify the Denominator

Calculate the denominator by multiplying \(3i\) and \(-3i\): \[3i \times (-3i) = -9i^2\]Knowing that \(i^2 = -1\), we have: \[-9(-1) = 9\]So, the denominator simplifies to \(9\).

Step 3: Expand the Numerator and Simplify

Expand the numerator \((4+6i)(-3i)\): \[4(-3i) + 6i(-3i) = -12i - 18i^2\]Substitute \(i^2 = -1\) to simplify:\[-12i - 18(-1) = -12i + 18\] So, the numerator simplifies to \(18 - 12i\).

Step 4: Simplify the Entire Expression

Substitute the simplified numerator and denominator back into the expression: \[\frac{18 - 12i}{9}\]Divide each term in the numerator by \(9\): \[\frac{18}{9} - \frac{12i}{9} = 2 - \frac{4}{3}i\]Thus, the expression is simplified to \(2 - \frac{4}{3}i\) in the form \(a + bi\).

Key Concepts

Conjugate

When working with complex numbers, the conjugate plays a key role in simplifying expressions, especially in division. The conjugate of a complex number is found by changing the sign of its imaginary part. For instance, if we have a complex number in the form \(a + bi\), its conjugate would be \(a - bi\). This concept is crucial in dealing with complex fractions.

• Conjugates are used to rationalize complex denominators by eliminating the imaginary unit under the fraction line.
• Multiplying a complex number by its conjugate results in a real number, which simplifies calculations.
• For instance, in our problem, the conjugate of \(3i\) is \(-3i\).

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This characteristic makes it possible to perform arithmetic operations with negative square roots.

• The imaginary unit helps in representing numbers that are not real, allowing for more complex calculations and solutions.
• Operations involving \(i\) follow the same rules as real numbers, with the additional property \(i^2 = -1\).
• In our original exercise, multiplying \(i\) by itself turns the product into a negative real number, as seen in the denominator's simplification to 9.

Simplification

Simplification is a crucial step in expression evaluation. It involves reducing the complexity of mathematical expressions to make them easier to understand and use. With complex numbers, this often involves distributing terms and substituting \(i^2 = -1\) to handle imaginary parts.

• Simplifying involves combining like terms and using known identities, such as \(i^2 = -1\), to express results in standard form.
• In our exercise, the numerator \((4 + 6i)(-3i)\) was expanded to \(-12i - 18i^2\), which was then simplified by converting \(i^2\) to \(-1\).
• This simplification makes it easy to express the final answer in the standard form \(a + bi\). In this case, it became \(18 - 12i\).

Rationalizing Denominator

Rationalizing the denominator involves transforming a fraction that has an imaginary or complex number in the denominator into an equivalent fraction with a real denominator. This process relies heavily on multiplying by conjugates.

• To rationalize a denominator, multiply both the numerator and the denominator by the conjugate of the denominator's complex part.
• In our example, \(\frac{4+6i}{3i}\) was rationalized by multiplying by the conjugate \(-3i\), converting the denominator to a real number.
• This method ensures that the denominator is no longer complex, ultimately making the expression easier to interpret and use.
  After rationalization, the fraction becomes easier to handle, e.g., \(\frac{18 - 12i}{9} = 2 - \frac{4}{3}i\).