Compare and contrast the absolute value of a real number to that of a complex number.

The absolute value of a real number and a complex number are both measures of magnitude, but they are defined differently due to the nature of real and complex numbers.

Absolute Value of a Real Number

• Definition: The absolute value of a real number $x$, denoted as $|x|$, is the non-negative value of $x$ without regard to its sign.
• Formula: $|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$
• Example: For $x = -5$, the absolute value is $|-5| = 5$.

Absolute Value of a Complex Number

• Definition: The absolute value (or modulus) of a complex number $z = a + bi$, where $a$ and $b$ are real numbers, is the distance from the origin to the point $(a, b)$ in the complex plane.
• Formula: $|z| = \sqrt{a^2 + b^2}$
• Example: For $z = 3 + 4i$, the absolute value is $|3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

Comparison

• Similarity: Both are measures of magnitude and are always non-negative.
• Difference: The absolute value of a real number is a straightforward measure of distance from zero on the real number line, while the absolute value of a complex number involves calculating the Euclidean distance from the origin in the complex plane.

Final Answer

The absolute value of a real number is its distance from zero on the real line, while the absolute value of a complex number is the Euclidean distance from the origin in the complex plane.