Problem 44 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \frac{-3+5 i}{15 i} $$

Step-by-Step Solution

Answer

The expression evaluates to \( -\frac{1}{3} - \frac{1}{5}i \).

Step 1: Express the Denominator in Standard Form

To evaluate the expression \( \frac{-3+5i}{15i} \), begin by expressing the denominator \( 15i \) in standard form. Note that \( i = \sqrt{-1} \). Therefore, multiply both the numerator and the denominator by the complex conjugate of the denominator, which is \( -i \), to eliminate the imaginary unit \( i \) from the denominator.

Step 2: Multiply Numerator and Denominator by Conjugate

Multiply the numerator and the denominator by \( -i \) (the conjugate of \( 15i \)): \[ \frac{(-3+5i)(-i)}{(15i)(-i)} \] This simplifies the denominator to a real number, since \( i^2 = -1 \).

Step 3: Simplify the Denominator

Simplify the expression in the denominator: \[ (15i)(-i) = 15i^2 = 15(-1) = -15 \] Now the expression becomes: \[ \frac{(-3+5i)(-i)}{-15} \]

Step 4: Distribute in the Numerator

Now simplify the numerator by distributing \( -i \): \[ (-3)(-i) + (5i)(-i) = 3i - 5i^2 \] Recall that \( i^2 = -1 \), so this becomes: \[ 3i + 5 \]

Step 5: Rewrite and Simplify the Expression

Substitute back the simplified numerator: \[ \frac{5 + 3i}{-15} \] Separate into two fractions: \[ \frac{5}{-15} + \frac{3i}{-15} \] This simplifies to: \[ -\frac{1}{3} - \frac{1}{5}i \]

Step 6: Final Answer in Form \( a + bi \)

The result is in the form \( a + bi \), where \( a = -\frac{1}{3} \) and \( b = -\frac{1}{5} \), thus: \[ -\frac{1}{3} - \frac{1}{5}i \].

Key Concepts

Complex Conjugate

A complex conjugate is an essential concept in the world of complex numbers. It is used particularly when simplifying expressions. The complex conjugate of a complex number is formed by changing the sign of the imaginary part. For example, the complex conjugate of a number like \( a + bi \) would be \( a - bi \). This modification helps in removing the imaginary unit from the denominator when dividing complex numbers. In our exercise,

• the denominator obviously contained the imaginary unit, represented by \( 15i \).
• We multiplied the expression by the complex conjugate, which is \(-i\), to simplify the division.

Once the conjugate is employed, the denominator becomes a real number, allowing us to divide without having a complex denominator.

Imaginary Unit

The imaginary unit is denoted by \( i \) and represents the square root of \(-1\). Its importance cannot be overstated as it acts as a building block for complex numbers. Complex numbers include both real and imaginary parts, expressed in the form \( a + bi \). Here, "\( a\)" is the real part, while "\( bi \)" is the imaginary part with \( i \) being the imaginary unit.

• Mathematically, \( i^2 = -1 \) is a fundamental relationship.
• During calculations, substituting \( i^2 \) results in \(-1\) simplifies expressions considerably.

In our problem, multiplying by the conjugate and simplifying led to transformations using \( i^2 = -1 \), which converted complex expressions into an easier-to-manage form.

Standard Form of Complex Numbers

The standard form of complex numbers is \( a + bi \), where \( a \) and \( b \) are real numbers. This form makes it straightforward to perform operations and represent combinations of real and imaginary numbers clearly. Writing complex numbers in this standard form,

• helps in identifying components easily,
• and avails them ready for comparison or further mathematical work.

In our exercise, the goal was to express the final answer in standard form, which means after division and simplification, we achieved \( -\frac{1}{3} - \frac{1}{5}i \). This conforms to the standard form "\( a + bi \)" by separating the real and imaginary parts clearly.

Division of Complex Numbers

The division of complex numbers can seem tricky, but by employing clever strategies, it becomes quite manageable. This process fundamentally involves eliminating any imaginary parts in the denominator. To do this:

• Multiply both the numerator and the denominator by the complex conjugate of the denominator.
• Simplify the outcome to result in a real number in the denominator.

Let's consider dividing \( \frac{-3 + 5i}{15i} \) from our exercise. We multiplied by \(-i\), the conjugate of \( 15i \), which yielded \( (-3+5i)(-i) \) in the numerator and a simplified real number \(-15\) in the denominator. After carrying out the calculations, we could separate the real part from the imaginary part, completing a successful division according to the methods outlined.