Problem 21 - Algebra 2

Question

Simplify. $$ \frac{3+i}{1+4 i} $$

Step-by-Step Solution

Answer

\(\frac{7}{17} - \frac{11}{17}i\)

Step 1: Identify the Objective

The goal is to simplify the expression \( \frac{3+i}{1+4i} \) by removing the imaginary number from the denominator.

Step 2: Multiply by the Conjugate

To remove the imaginary unit \(i\) from the denominator, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(1 + 4i\) is \(1 - 4i\). Thus, multiply the entire fraction by \(\frac{1 - 4i}{1 - 4i}\).

Step 3: Apply Multiplication for Numerator and Denominator

Multiply the numerator: \((3+i)(1-4i) = 3(1) + 3(-4i) + i(1) + i(-4i)\). Calculate each part: \(3 \times 1 = 3\), \(3 \times -4i = -12i\), \(i \times 1 = i\), \(i \times -4i = -4i^2\). Since \(i^2 = -1\), it becomes \(-4(-1) = 4\), so the numerator becomes \(3 - 12i + i + 4 = 7 - 11i\).

Step 4: Simplify the Denominator

Multiply the denominator: \((1+4i)(1-4i) = 1^2 - (4i)^2\). Apply \(a^2 - b^2 = (a+b)(a-b)\), where \(a = 1\) and \(b = 4i\). Compute it as:\(1 - (4i)^2 = 1 - 16(-1) = 1 + 16 = 17\). The denominator becomes 17.

Step 5: Write the Simplified Expression

Now, the simplified expression is \(\frac{7 - 11i}{17}\). Separate the real and imaginary parts: \(\frac{7}{17} - \frac{11}{17}i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is the core of complex numbers. It is defined by the property that \( i^2 = -1 \). This fundamental aspect allows us to extend the real number system to include solutions to equations like \( x^2 + 1 = 0 \), which have no real solutions.

In complex numbers, the imaginary unit is used to express numbers in the form \( a + bi \), where \( a \) and \( b \) are real numbers. The "\( a \)" is the real part, and "\( bi \)" is the imaginary part. Here, \( i \) is essential as it provides a way to deal with square roots of negative numbers.

Imaginary numbers, therefore, seem to be abstract, but they are very useful in applied mathematics and engineering. They help model and solve problems in areas such as electrical engineering, quantum physics, and control theory, where real numbers alone are insufficient.

Conjugate

The conjugate of a complex number is a crucial tool for simplifying expressions. For a given complex number \( a + bi \), its conjugate is \( a - bi \). This operation involves changing the sign of the imaginary part while keeping the real part unchanged.

Conjugates are especially handy when it comes to division of complex numbers. Multiplying a complex number by its conjugate results in a real number. This is because

• The formula, \( (a + bi)(a - bi) = a^2 - (bi)^2 \), simplifies to \( a^2 + b^2 \) since \( (bi)^2 = -b^2 \).
• The resulting expression, \( a^2 + b^2 \), is a real number because it comprises no imaginary part.

In the exercise, the conjugate of the denominator \( 1 + 4i \) was \( 1 - 4i \). This conversion facilitates the reduction of the fraction into a simpler form that has no imaginary unit in the denominator.

Simplification

Simplifying expressions with complex numbers often involves removing imaginary units from the denominator through multiplication by the conjugate. This process stabilizes the fraction into a form that's simpler and real.

In our exercise, we began with the fraction \( \frac{3+i}{1+4i} \). To simplify it:

• We multiplied both the numerator and the denominator by the conjugate of the denominator, \( \frac{1-4i}{1-4i} \).
• This led to a new numerator of \( 7 - 11i \) after expanding and combining like terms.
• The denominator became \( 17 \), a real number, through the formula \( a^2 + b^2 \).

The final simplified expression was \( \frac{7}{17} - \frac{11}{17}i \), separating the real and imaginary parts. Simplifying like this is essential in both theoretical math and practical applications, ensuring that operations involving complex numbers remain manageable and clear.