Problem 106 - A Graphical Approach to Precalculus with Limits

Question

Find the complex conjugate. \(\frac{-19-9 i}{i}\)

Step-by-Step Solution

Answer

The complex conjugate is \(-9 - 19i\).

Step 1: Multiply top and bottom by the conjugate of the denominator

First, identify the conjugate of the denominator, which is \(-i\). Multiply both the numerator and the denominator by this conjugate to rationalize the denominator:\[\frac{-19 - 9i}{i} \times \frac{-i}{-i} = \frac{(-19 - 9i)(-i)}{i(-i)}\]

Step 2: Simplify the denominator

The denominator now becomes \(i \times (-i)\). Calculate it as follows:\[i \times (-i) = -i^2 = -(-1) = 1\]Thus, the denominator simplifies to \(1\).

Step 3: Distribute in the numerator

Now distribute \(-i\) across the terms in the numerator:\[(-19 - 9i)(-i) = (-19)(-i) + (-9i)(-i)\]This simplifies to:\[19i + 9i^2\]

Step 4: Simplify the numerator further

Recall that \(i^2 = -1\). Substitute \(-1\) for \(i^2\) in the expression:\[19i + 9(-1) = 19i - 9\]

Step 5: Write the simplified expression

After simplification of both the numerator and the denominator, the expression is:\[\frac{19i - 9}{1} = 19i - 9\]Thus, the result is \(-9 + 19i\).

Step 6: Find the complex conjugate of the expression

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, for the expression \(-9 + 19i\), the complex conjugate is:\[-9 - 19i\]

Key Concepts

Complex Conjugate

A complex conjugate is a fundamental concept when working with complex numbers, which are numbers in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate simply flips the sign of the imaginary part, turning \(a + bi\) into \(a - bi\). This operation is crucial in many areas of mathematics, including simplifying complex fractions and calculating magnitudes.

Utilizing complex conjugates can help simplify expressions, especially when rationalizing denominators. They are helpful because multiplying complex conjugates results in a real number. In this operation, the imaginary parts cancel each other out, which aids in simplifying complex fractions. So, when you're given a complex fraction, taking the complex conjugate of the denominator can make handling the fraction much easier.

Imaginary Unit

The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\). It is used to create complex numbers and plays a key role in operations involving these numbers. In mathematics, the imaginary unit extends the real numbers to complex numbers by enabling roots of negative numbers.

For example, the square root of \(-1\) is \(i\), as it satisfies the equation \(i^2 = -1\). This property is crucial when simplifying expressions involving powers of \(i\). When simplifying a multiplication like \(i \times (-i)\), it translates to \(-i^2\), which further simplifies to \(1\) since \(-1\times -1 = 1\). This cycle of powers of \(i\) (i.e., \(i, -1, -i, 1\)) repeats every four steps, which is a useful pattern in calculations.

Rationalizing Denominator

Rationalizing the denominator is the process of eliminating the imaginary unit from the denominator of a fraction. This technique simplifies the expression and makes interpreting the results more straightforward. To rationalize a denominator containing \(i\), you multiply both the numerator and the denominator by the conjugate of the denominator.

For example, if the original fraction is \(\frac{-19 - 9i}{i}\), you would multiply by \(\frac{-i}{-i}\). This use of the conjugate helps because, as seen, multiplying a complex number by its conjugate results in a real number. The product of the denominator, \(i \times (-i)\), gives \(1\), effectively eliminating any imaginary component from the denominator of the fraction. This principle is not only used in simplifying single complex fractions but is a foundational tool in higher-level math problems involving complex numbers.