Problem 68 - Intermediate Algebra

Question

Find the product of the given complex number and its conjugate. $$ -10-9 i $$

Step-by-Step Solution

Answer

The product is 181.

Step 1: Identify the Complex Number

The given complex number is \(-10 - 9i\). Recall that a complex number is generally in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Step 2: Find the Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the conjugate of \(-10 - 9i\) is \(-10 + 9i\).

Step 3: Apply the Product Formula

The product of a complex number and its conjugate is given by:\((a + bi)(a - bi) = a^2 + b^2\). This formula simplifies the multiplication process using the properties of imaginary numbers.

Step 4: Calculate the Product

Using the formula, calculate the product of \(-10 - 9i\) and its conjugate \(-10 + 9i\):- Obtain \(a^2\): \((-10)^2 = 100\),- Obtain \(b^2\): \(9^2 = 81\).Thus, the product is \(100 + 81 = 181\).

Key Concepts

Conjugate of a Complex Number

The conjugate of a complex number is an essential concept to understand when working with complex numbers. In mathematics, the conjugate of a complex number is found by changing the sign of the imaginary part. For a complex number in the form of \(a + bi\), its conjugate is \(a - bi\). This means that you take the same real part \(a\), but you flip the sign of the imaginary part \(bi\).

Finding the conjugate is particularly useful because when a complex number is multiplied by its conjugate, the result is a real number. This is the basis for simplifying complex expressions and finding products of complex numbers by eliminating the imaginary part.

In our example, the complex number \(-10 - 9i\) has the conjugate \(-10 + 9i\). This forms the pair necessary for our product formula, leading to an easier calculation with real results.

Product of Complex Numbers

Multiplying complex numbers involves applying the distributive property and simplifying terms. However, when multiplying a complex number by its conjugate, a specific formula can be used to simplify the process.

The formula for the product of a complex number \(a + bi\) and its conjugate \(a - bi\) is straightforward: \((a + bi)(a - bi) = a^2 + b^2\).

This formula works as follows:

• The cross terms \(abi - abi\) cancel each other because they are additive inverses.
• The imaginary unit \(i^2\) is replaced with \(-1\), which is why the combined imaginary terms disappear, giving a real number \(a^2 + b^2\).

Using this relationship, you simplify the calculation substantially. In the provided example, the product of \(-10 - 9i\) and its conjugate \(-10 + 9i\) is simply calculated by evaluating \(a^2\) and \(b^2\): \((-10)^2 + (9)^2\), resulting in \(100 + 81 = 181\).

Imaginary Unit

The imaginary unit \(i\) is a fundamental concept in complex numbers, enabling the representation of square roots of negative numbers. Defined as \(i = \sqrt{-1}\), the imaginary unit allows calculations beyond real numbers and opens up analysis in the complex plane.

Here are some key points about the imaginary unit:

• \(i^2 = -1\): This property is used in computations involving complex numbers to convert expressions with \(i^2\) into real numbers.
• Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(bi\) represents the imaginary part.

In practical terms, the use of \(i\) means you can extend algebraic operations to numbers that are otherwise impossible with just real numbers.

When you multiply complex numbers, the imaginary unit is pivotal in understanding how complex conjugates produce real number products by effectively canceling out the imaginary components.