Problem 51 - Algebra 2 and Trigonometry

Question

In \(46-60,\) write each quotient in \(a+b i\) form. $$ \frac{10-5 i}{2+6 i} $$

Step-by-Step Solution

Answer

The quotient \( \frac{10-5i}{2+6i} \) in \(a + bi\) form is \( -\frac{1}{4} - \frac{7}{4}i \).

Step 1: Identify the Conjugate of the Denominator

To simplify the expression \( \frac{10-5i}{2+6i} \), we need to multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2+6i \) is \( 2-6i \).

Step 2: Multiply the Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate \( 2-6i \), resulting in: \[ \frac{(10-5i)(2-6i)}{(2+6i)(2-6i)} \]

Step 3: Simplify the Denominator

Use the formula \( (a+bi)(a-bi) = a^2 + b^2 \). For the denominator, \( (2+6i)(2-6i) = 2^2 + 6^2 = 4 + 36 = 40 \). Thus, the denominator becomes 40.

Step 4: Expand the Numerator

Expand \((10-5i)(2-6i)\) using the distributive property (FOIL method):\[10 \cdot 2 + 10 \cdot (-6i) + (-5i) \cdot 2 + (-5i) \cdot (-6i)\]Calculating each term gives:\[20 - 60i - 10i + 30i^2\]

Step 5: Simplify the Numerator

Remember that \( i^2 = -1 \). Therefore, simplify the last term:\( 30i^2 = 30(-1) = -30 \).So, the expression becomes:\( 20 - 60i - 10i - 30 \).Combine like terms:\(20-30 = -10 \) (real part) and \(-60i - 10i = -70i \) (imaginary part).The simplified numerator is \(-10 - 70i\).

Step 6: Combine and Simplify the Expression

Combine the simplified numerator and denominator to get the expression \(-\frac{10 + 70i}{40}\). Divide each term by 40:\[ -\frac{10}{40} - \frac{70i}{40} = -\frac{1}{4} - \frac{7}{4}i\]This places the quotient in \(a + bi\) form.

Key Concepts

Conjugate of a Complex Number

Complex numbers have both a real part and an imaginary part and are often expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the coefficient of the imaginary part \( i \). When dealing with complex numbers, especially in division, the concept of a 'conjugate' comes into play. The conjugate of a complex number is obtained by changing the sign of its imaginary part. This means the conjugate of \( a + bi \) is \( a - bi \). Utilizing the conjugate helps in simplifying expressions and rationalizing denominators.

When you multiply a complex number by its conjugate, you eliminate the imaginary part because the result is a real number. In mathematical terms, for a complex number \( a+bi \), its multiplication with its conjugate \( a-bi \) is given by:
\[ (a+bi)(a-bi) = a^2 + b^2 \].
This result is key, as it turns the denominator of a fraction into a real number, making further simplification possible without imaginary components in the denominator.

Imaginary Numbers

Imaginary numbers root back to the need to solve equations that are impossible to solve with real numbers alone. They are based on \( i \), the imaginary unit, which is defined by its remarkable property \( i^2 = -1 \).

Imaginary numbers are crucial, not just for theoretical aspects of mathematics, but for practical applications as well, such as engineering and physics. They often describe phenomena like electrical currents and wave behavior.
If we consider a number like \( 6i \), it's purely imaginary because it has no real component. In expressions such as \( 10 + 6i \), we have both real and imaginary parts, forming what we know as a complex number. Understanding this concept helps in operations involving these numbers, as they behave differently from real numbers.

Complex Number Division

Dividing complex numbers is a process that often involves the use of conjugates. This ensures that the division results in a number that's simpler and devoid of imaginary components in the denominator. With the examplar expression \( \frac{10-5i}{2+6i} \), the first step is to eliminate the imaginary unit from the denominator.

To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. For the denominator \( 2+6i \), its conjugate would be \( 2-6i \).

• Multiplying the numerator by the conjugate involves expanding the expression through distribution. It's akin to how you expand brackets using the FOIL method, focusing on combining like terms.
• The denominator, when multiplied by its conjugate, always results in a real number, \( a^2 + b^2 \).

After the multiplication and simplification steps, each component of the complex number is divided separately. This leads to expressing the division result in what's known as \( a + bi \) form, offering a neat and interpretable expression of the division result.