Problem 6 - Elementary and Intermediate Algebra

Question

Classify each of the following statements as either true or false. The product of a complex number and its conjugate is always a real number.

Step-by-Step Solution

Answer

True

Step 1: Define a Complex Number

Let the complex number be represented as \[ z = a + bi \] where \[ a \] and \[ b \] are real numbers, and \[ i \] is the imaginary unit with \[ i^2 = -1 \].

Step 2: Write the Conjugate of the Complex Number

The conjugate of the complex number \[ z = a + bi \] is given by \[ \overline{z} = a - bi \].

Step 3: Calculate the Product

Multiply the complex number and its conjugate: \[ z \cdot \overline{z} = (a + bi)(a - bi) \]Use the difference of squares formula: \[ z \cdot \overline{z} = a^2 - (bi)^2 \]

Step 4: Simplify the Expression

Since \[ (bi)^2 = b^2i^2 = b^2 (-1) = -b^2 \], the expression simplifies to: \[ z \cdot \overline{z} = a^2 - (-b^2) = a^2 + b^2 \].

Step 5: Determine the Nature of the Result

The result \[ a^2 + b^2 \] is a sum of squares of real numbers, which is always a real number.

Step 6: Conclusion

Since \[ a^2 + b^2 \] is always real, the statement that the product of a complex number and its conjugate is always a real number is true.

Key Concepts

complex number

A complex number is a number that combines a real part and an imaginary part. It is generally written in the form \( z = a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit that satisfies \( i^2 = -1 \). We use complex numbers to solve equations that have no real solutions, like \( x^2 + 1 = 0 \). By introducing the imaginary unit \( i \), we can express the solutions in terms of complex numbers.

• **Real Part**: The part of the complex number that does not have the imaginary unit \( i \), denoted as \( a \) in \( z = a + bi \).
• **Imaginary Part**: The part of the complex number that includes the imaginary unit \( i \), denoted as \( bi \) in \( z = a + bi \).

This combination helps us work with a wider range of mathematical problems. Complex numbers are used in fields such as engineering, physics, and computer science.

conjugate

The conjugate of a complex number is a version of the number in which the sign of the imaginary part is reversed. For a complex number \( z = a + bi \), its conjugate, denoted \( \overline{z} \), is \( a - bi \).

• **Same Real Part**: Both the complex number and its conjugate have the same real part \( a \).
• **Opposite Imaginary Part**: The imaginary part \( b \) of the complex number becomes \( -b \) in its conjugate.

This property is particularly useful for simplifying mathematical operations involving complex numbers. When you multiply a complex number by its conjugate, you eliminate the imaginary part, which results in a real number.

difference of squares

The difference of squares is a useful algebraic formula given by \( a^2 - b^2 = (a + b)(a - b) \). We use this formula in various mathematical problems, including those involving complex numbers. To see its application, let's multiply a complex number by its conjugate:\[ z \cdot \overline{z} = (a + bi)(a - bi) \]Using the difference of squares formula, we can rewrite the expression as:\[ z \cdot \overline{z} = a^2 - (bi)^2 \]Because \( (bi)^2 = b^2i^2 = b^2 (-1) = -b^2 \), the expression simplifies to:\[ z \cdot \overline{z} = a^2 - (-b^2) = a^2 + b^2 \]Since \( a^2 + b^2 \) is a sum of squares of real numbers, it is always a real number. Thus, multiplying a complex number by its conjugate leverages the difference of squares to produce a real number.