Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Find the complex conjugate. $$\frac{-11-7 i}{1+2 i}$$

Step-by-Step Solution

Answer

The complex conjugate is \(\frac{3}{5} - 3i\).

Step 1: Identify the Problem

The problem asks us to find the complex conjugate of a given complex number.

Step 2: Recall Definition of Complex Conjugate

The complex conjugate of a complex number \( a + bi \) is \( a - bi \). Thus, for any complex expression, we must switch the sign of the imaginary part to find the conjugate.

Step 3: Simplify the Complex Number

Before finding the conjugate, simplify the given complex expression \(\frac{-11-7i}{1+2i}\). Multiply the numerator and denominator by the conjugate of the denominator to make the denominator a real number: \(\frac{-11-7i}{1+2i} \times \frac{1-2i}{1-2i}\).

Step 4: Multiply Numerator and Denominator

Multiply the numerator: \((-11 - 7i)(1 - 2i)\). Expand to get \(-11 + 22i - 7i + 14i^2\). Simplify using \(i^2 = -1\), which gives \(-11 + 15i + 14\) leading to \(3 + 15i\).Multiplication for the denominator gives \((1 + 2i)(1 - 2i) = 1 - 4i^2 = 1 + 4 = 5\).

Step 5: Simplified Expression

Now we have the simplified form of the complex number: \(\frac{3 + 15i}{5} = \frac{3}{5} + \frac{15}{5}i = \frac{3}{5} + 3i\).

Step 6: Find the Complex Conjugate

Apply the definition of the complex conjugate to the simplified expression \(\frac{3}{5} + 3i\). This gives us the complex conjugate \(\frac{3}{5} - 3i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The term "imaginary" doesn't mean they are fictitious; rather, it is a concept used to extend the traditional realm of real numbers. Complex numbers can be incredibly useful in many areas of mathematics and engineering.

• The real part \( a \) can be any real number such as integers (like 3 or -1.5), decimals, or even algebraic expressions.
• The imaginary part \( bi \) includes the imaginary unit \( i \), which is defined as the square root of -1.

Understanding complex numbers involves visualizing them in a two-dimensional plane, known as the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

Imaginary Unit

The imaginary unit, denoted by \( i \), is fundamental to working with complex numbers. The defining property of \( i \) is that \( i^2 = -1 \). This property is what allows mathematicians and engineers to solve equations that have no real solutions.

• Using \( i \), complex numbers can express numbers that cannot be expressed with real numbers alone.
• In algebra, \( i \) behaves like any other variable: it can be added, subtracted, multiplied, and divided, keeping in mind its special property \( i^2 = -1 \).

For instance, if you encounter an expression like \( 14i^2 \), it simplifies to \( 14(-1) = -14 \), as used in the simplification described in the original step-by-step solution. This simplification plays a crucial role in dealing with complex expressions.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involving complex numbers often requires a blend of arithmetic and algebraic manipulation. This process can involve combining like terms, using properties of \( i \), and rationalizing denominators when necessary.

• To simplify a complex division, like \( \frac{a + bi}{c + di} \), a common technique is to multiply the numerator and denominator by the conjugate of the denominator. This operation removes the imaginary part in the denominator, turning it into a real number.
• For example, in the given problem, the denominator \( 1 + 2i \) was multiplied by its conjugate \( 1 - 2i \). This turned the denominator into 5, a real number.

The end result is a fraction that is easier to interpret and manipulate. Using these techniques ensures the expression is expressed in standard form, \( a + bi \), making it easier to find things like its modulus or conjugate.