Determine whether each statement is true or false. If false, explain why. The product of a complex number and its conjugate is always a real number.

Verified step by step guidance

1. Recall the definition of a complex number: it is of the form \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with \(i^2 = -1\).
2. The conjugate of a complex number \(z = a + bi\) is given by \(\overline{z} = a - bi\).
3. To find the product of a complex number and its conjugate, multiply \(z\) and \(\overline{z}\): \(z \times \overline{z} = (a + bi)(a - bi)\).
4. Use the distributive property (FOIL) to expand the product: \((a + bi)(a - bi) = a^2 - abi + abi - b^2 i^2\).
5. Simplify the expression by combining like terms and using \(i^2 = -1\): the middle terms cancel out, and \(-b^2 i^2\) becomes \(+b^2\), so the product is \(a^2 + b^2\), which is a real number.

Key Concepts

Complex Numbers

A complex number is a number in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Complex numbers extend the real number system and are used to represent quantities with both real and imaginary parts.

Complex Conjugate

The complex conjugate of a complex number a + bi is a - bi. It reflects the number across the real axis in the complex plane, changing the sign of the imaginary part while keeping the real part the same.

Product of a Complex Number and Its Conjugate

Multiplying a complex number by its conjugate results in a real number equal to a² + b², which is the sum of the squares of the real and imaginary parts. This product is always non-negative and has no imaginary component.