Problem 55 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{12+3 i}{3 i} $$

Step-by-Step Solution

Answer

The quotient is \( 1 - 4i \).

Step 1: Understand the problem

We need to express the given quotient \( \frac{12+3i}{3i} \) in the rectangular form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

To simplify the quotient involving complex numbers, multiply both the numerator and the denominator by the conjugate of the imaginary denominator. Here, the conjugate of \( 3i \) is \( -3i \). Thus, we multiply by \( \frac{-3i}{-3i} \):\[\frac{12 + 3i}{3i} \times \frac{-3i}{-3i} = \frac{(12 + 3i)(-3i)}{(3i)(-3i)}\]

Step 3: Simplify the Denominator

Simplify the denominator by using the property \( i^2 = -1 \). We have:\[(3i)(-3i) = -9i^2 = 9\]

Step 4: Expand the Numerator

Expand the numerator:\[(12 + 3i)(-3i) = 12(-3i) + 3i(-3i) = -36i - 9i^2\]

Step 5: Simplify the Numerator

Recall that \( i^2 = -1 \). Substitute to simplify:\[-36i - 9i^2 = -36i + 9 = 9 - 36i\]

Step 6: Write in Standard Form

Divide each term by 9 to write the expression in the standard form:\[\frac{9 - 36i}{9} = 1 - 4i\]Thus, the expression in \( a + bi \) form is \( 1 - 4i \).

Key Concepts

Rectangular Form

When working with complex numbers, it's often helpful to express them in the rectangular form, which is written as \( a + bi \). In this form, \( a \) is the real part and \( b \) is the imaginary part of the complex number, while \( i \) represents the imaginary unit. This form is straightforward because it separates the real and imaginary components, making it easy to visualize and understand complex numbers.

• Real Part (\( a \)): The coefficient of 1.
• Imaginary Part (\( b \)): The coefficient of \( i \).

To express \( \frac{12+3i}{3i} \) in this form, we need to perform operations that separate it into distinct real and imaginary parts, which can be simplified to get \( 1 - 4i \). This makes calculation and further analysis easier.

Conjugate

The concept of conjugates is crucial for simplifying expressions involving complex numbers, especially in division. For any complex number, the conjugate is found by changing the sign of the imaginary part. If you have a complex number \( a + bi \), its conjugate is \( a - bi \).
In our exercise, the denominator is \( 3i \), which is purely imaginary. Its conjugate is \( -3i \).

• Multiplying by the conjugate helps in eliminating the imaginary part in the denominator, simplifying the fraction.
• This is because \( i^2 = -1 \), which transforms the product of conjugates into a purely real number.

Therefore, multiplying both numerator and denominator by \( -3i \) helps simplify the original quotient, allowing it to be expressed more cleanly in rectangular form.

Imaginary Unit i

The imaginary unit, denoted as \( i \), is an essential concept in complex numbers. Defined by the property \( i^2 = -1 \), it allows mathematicians to express and work with numbers that cannot be positioned on the real number line alone.

• The introduction of \( i \) expands the real number system to include complex numbers, extending arithmetic to a two-dimensional plane.
• \( i \) itself represents the square root of \(-1\). Thus, any imaginary number can be seen as a multiple of \( i \).

Understanding \( i \) is vital, as it underlies operations in expressions like \( \frac{12+3i}{3i} \). Knowing how \( i \) behaves under multiplication and exponentiation (e.g., \( i^3 = -i \), \( i^4 = 1 \)) is essential for performing operations involving complex numbers.

Simplification of Complex Fractions

Simplifying complex fractions, like \(\frac{12+3i}{3i}\), involves techniques that help express them in simpler, standard forms. The goal of simplification is to manage arithmetic operations involving complex denominators.
To simplify a complex fraction:

• Identify the conjugate of the denominator.
• Multiply both numerator and denominator by this conjugate. This effectively removes \( i \) from the denominator through reconciliation of \( i^2 \) to \(-1\).
• Apply algebraic manipulation to the resulting real denominator and expanded numerator, using \( i^2 = -1 \) to simplify.
• Distribute terms to segregate real and imaginary parts.

Ultimately, these steps help in rewriting complex expressions in their simplest \( a + bi \) form, making them more usable for analysis and further realizations.