Problem 56 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{8+6 i}{2 i} $$

Step-by-Step Solution

Answer

The quotient in \(a + bi\) form is \(3 - 4i\).

Step 1: Write the Expression

The problem asks you to find the expression in the form \(a + bi\) for the complex number quotient \(\frac{8+6i}{2i}\). That means it needs to be rewritten from a fraction into a standard complex number format.

Step 2: Multiply by the Conjugate

To eliminate the imaginary number from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. For \(2i\), the conjugate is \(-2i\). So, multiply \(\frac{8+6i}{2i}\) by \(\frac{-2i}{-2i}\).\[\left( \frac{8+6i}{2i} \right) \cdot \left( \frac{-2i}{-2i} \right) = \frac{(8+6i)(-2i)}{(2i)(-2i)}\]

Step 3: Simplify the Denominator

Calculate the denominator part: \((2i)(-2i) = -4i^2\). Using \(i^2 = -1\), we have:\[-4i^2 = -4(-1) = 4\].Thus, the denominator simplifies to 4.

Step 4: Simplify the Numerator

Expand the numerator using the distributive property: \[(8+6i)(-2i) = 8(-2i) + 6i(-2i)\]Calculate each term: \[8(-2i) = -16i\]\[6i(-2i) = -12i^2 = -12(-1) = 12\]Combine them: \[12 - 16i\].

Step 5: Combine and Simplify

Combine the results from the previous steps and write the expression in the form \(a + bi\):\[\frac{12 - 16i}{4} = \frac{12}{4} - \frac{16i}{4} = 3 - 4i\].

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is denoted by the symbol \(i\). This symbol represents the square root of -1. In other words, \(i^2 = -1\). It's a fundamental component in the realm of complex numbers and allows for the extension of the real number system.

• Understanding \(i^2\): Given that the square of \(i\) is -1, this provides a foundation for many operations with complex numbers.
• Solving equations: Imaginary units enable solutions to equations that have no real solutions, such as subtraction under square roots (e.g., \(\sqrt{-4}\)).

Any number that can be expressed as a multiple of \(i\) is considered to be purely imaginary, such as \(2i\) or 3.5i. By combining the imaginary unit with real numbers, we enter the world of complex numbers, where each number has a real part and an imaginary part.

Complex Conjugate

The complex conjugate of a complex number is a useful concept that can help in simplifying expressions, especially when dividing complex numbers. Consider a complex number of the form \(a + bi\), its conjugate is \(a - bi\).

• Formation: To find the conjugate, simply change the sign of the imaginary part.
• Properties: Multiplying a complex number by its conjugate results in a real number. For instance, \((a + bi)(a - bi) = a^2 + b^2\).

When dividing by a complex number, multiplying the numerator and the denominator by the conjugate of the denominator can remove the imaginary part from the denominator. This was used in your exercise when multiplying by \(-2i\) to simplify the division.

Standard Form of Complex Numbers

The standard form for writing complex numbers is \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is known as the real part, while \(bi\) is the imaginary part.

• Real part: The component \(a\) represents the real number space in a complex number.
• Imaginary part: The component \(bi\) extends the real number into a complex plane.

For example, the solution to your problem was laid out as \(3 - 4i\), which adheres to this standard form. A complex number provides a way to represent numbers that contain both real and imaginary components, expanding the mathematical world significantly.

Distributive Property

The distributive property is a basic algebraic property that plays an important role when working with complex numbers. It states that for all numbers \(a\), \(b\), and \(c\), the expression \(a(b + c) = ab + ac\) holds true.

• Application: This property is key when multiplying a complex number by another number, as it involves distributing the multiplication over addition.
• Expanding expressions: In the exercise, the distributive property was used to handle \((8+6i)(-2i)\) effectively, breaking it into two simpler parts.

By using the distributive property, you can simplify complex number expressions by expanding them in terms of their real and imaginary parts separately. This allows complex numbers to be added, subtracted, multiplied, and divided just like simpler algebraic expressions.