To compare and contrast the absolute value of a real number to that of a complex number, let's consider specific examples and detail the steps involved in calculating their absolute values.
### Absolute Value of a Real Number:
The absolute value of a real number is the non-negative value of the number without regard to its sign. In simpler terms, it is the distance of the number from zero on the number line.
#### Example:
Consider the real number [tex]\(-5\)[/tex].
1. The absolute value of [tex]\(-5\)[/tex] is calculated as [tex]\(|-5|\)[/tex].
2. Since the absolute value represents the distance from zero, [tex]\(|-5| = 5\)[/tex].
Thus, the absolute value of [tex]\(-5\)[/tex] is [tex]\(5\)[/tex].
### Absolute Value of a Complex Number:
The absolute value (also known as the magnitude) of a complex number is the distance of the number from the origin in the complex plane. The formula to find the magnitude of a complex number [tex]\(a + bi\)[/tex] is given by [tex]\(\sqrt{a^2 + b^2}\)[/tex], where [tex]\(a\)[/tex] is the real part and [tex]\(b\)[/tex] is the imaginary part.
#### Example:
Consider the complex number [tex]\(3 + 4i\)[/tex].
1. To find the magnitude of [tex]\(3 + 4i\)[/tex], we use the formula [tex]\(\sqrt{a^2 + b^2}\)[/tex].
2. Here, [tex]\(a = 3\)[/tex] and [tex]\(b = 4\)[/tex]. Plugging these values into the formula, we get:
[tex]\[
\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
Thus, the absolute value of [tex]\(3 + 4i\)[/tex] is [tex]\(5\)[/tex].
### Comparison:
1. Both the absolute value of the real number [tex]\(-5\)[/tex] and the magnitude of the complex number [tex]\(3 + 4i\)[/tex] are calculated considering their distances from a specific point (zero on the real line for real numbers, and the origin in the complex plane for complex numbers).
2. The resulting absolute values are the same in this example, both being [tex]\(5\)[/tex]. However, the underlying calculations are different due to the nature of the numbers involved:
- For real numbers, it is a straightforward determination of distance on the real line.
- For complex numbers, it requires the use of the Pythagorean theorem to determine the distance from the origin in the complex plane.
### Conclusion:
In summary, while the methods to calculate the absolute values of real and complex numbers differ due to their mathematical properties, they can sometimes yield the same numerical results. In our examples, both [tex]\(-5\)[/tex] and [tex]\(3 + 4i\)[/tex] have absolute values of [tex]\(5\)[/tex].
