Problem 67 - Elementary and Intermediate Algebra

Question

Find the product of the given complex number and its conjugate. See Example 7 . $$ -4-7 i $$

Step-by-Step Solution

Answer

The product is 65.

Step 1: Identify the Complex Number

The complex number given is \[-4 - 7i\].

Step 2: Determine the Conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the conjugate of \(-4 - 7i\) is \(-4 + 7i\).

Step 3: Set up the Product

To find the product of the complex number and its conjugate, multiply \(-4 - 7i\) by \(-4 + 7i\). The product formula for a complex number \((a + bi)\) and its conjugate \((a - bi)\) is \a^2 + b^2\.

Step 4: Apply the Formula

For the complex number \(-4 - 7i\), \(a = -4\) and \(b = -7\). Substitute these values into the formula: \ (-4)^2 + (-7)^2\.

Step 5: Calculate the Squares

Calculate each square: \((-4)^2 = 16\) and \((-7)^2 = 49\).

Step 6: Sum of Squares

Add the results from Step 5: \(16 + 49 = 65\).

Step 7: State the Final Result

The product of \(-4 - 7i\) and its conjugate is \(65\).

Key Concepts

Conjugate of a Complex Number

A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The conjugate of a complex number is obtained by changing the sign of the imaginary part. Thus, for a complex number \(a + bi\), its conjugate is \(a - bi\).

Working with conjugates often helps simplify problems, especially when finding the modulus or product of complex numbers. In this context, the conjugate of \(-4 - 7i\) is \(-4 + 7i\).

This operation, although simple, plays an essential role in various computations involving complex numbers, such as simplifying division or finding the squared modulus of a number.

Product of Complex Numbers

Multiplying two complex numbers involves applying the distributive property as if they were algebraic expressions. For any complex numbers \((a + bi)\) and \((c + di)\), their product is \((ac - bd) + (ad + bc)i\). However, when multiplying a complex number by its conjugate, we use a simplified version of this formula.

For a complex number \((a + bi)\) with its conjugate \((a - bi)\), the product is \(a^2 + b^2\). This formula arises because:

• The imaginary parts \(b\) and \(-b\) will cancel each other out, leaving only real components.
• The real part becomes the sum of the squares \(a^2\) and \(b^2\).

This makes calculating products of a number and its conjugate very straightforward, as seen with the complex number \(-4 - 7i\), yielding a product of 65.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. When working with complex numbers, it's often useful to treat them as algebraic expressions to simplify or solve equations.

For example, when multiplying a complex number by its conjugate, we apply algebraic rules just like in polynomials: expand the expression using

• The distributive property, resulting in a combination of real and imaginary terms.
• Combining like terms to simplify the result.

This approach helps in understanding how complex conjugation transforms two-dimensional complex problems into one-dimensional real number problems, which are often easier to solve. Thus, algebraic manipulation is a crucial skill in handling complex numbers efficiently.