Problem 98 - Introductory Algebra for College Students

Question

Find the absolute value: \(|-13.4|\)

Step-by-Step Solution

Answer

The absolute value of \(-13.4\) is \(13.4\).

Step 1: Understand Absolute Value

The absolute value of a number is its distance from 0 on the number line, regardless of direction. It is always non-negative: \(|a| \geq 0\).

Step 2: Apply the Definition

We evaluate the absolute value of the given expression. Since the absolute value removes any negative sign, we take the non-negative value.

Step 3: State the Result

The absolute value of \(-13.4\) is \(13.4\).

Key Concepts

Real Numbers

The absolute value topic is deeply rooted in the universe of real numbers. Real numbers include all the numbers you can think of, such as whole numbers, fractions, and decimals. They encompass all rational and irrational numbers, essentially any number that can exist on a continuous number line. Absolute values apply to real numbers because they help us measure the distance of a number from zero.

• Whole numbers: Include numbers like 0, 1, 2...
• Fractions: Numbers like \(\frac{1}{2}, \frac{3}{4} \), etc.
• Decimals: Such as 3.5, 0.125, etc.
• Negative numbers: Like -2, -3.5...
• Positive numbers: Like 2, 15.8...

Since real numbers can be either positive or negative, absolute value gives us a consistent way to measure their size without considering their sign.

Positive Magnitude

Positive magnitude refers to the size or extent of a real number, ignoring whether it's positive or negative in nature. The concept of positive magnitude helps to simplify our understanding of numbers by focusing on their absolute value. Magnitude effectively strips away any negative signs, offering a straightforward, non-negative perspective of the number involved.
For example, when we consider the absolute value of \(-13.4\), we are only concerned with its size, resulting in a positive magnitude of 13.4. This is because absolute value always yields a non-negative result, ensuring we only concentrate on the real number's scale or size.

• Transforms any negative into a positive
• Leaves positive numbers unchanged

This method is particularly useful in various mathematical and practical calculations, where positive values simplify the interpretation of numerical data.

Modulus

The term modulus is another way of referring to the absolute value in mathematics. Modulus is symbolized by vertical bars \(|x|\) placed around a number. The modulus essentially denotes the non-negative magnitude of that number. It effectively *modifies* the number to remove any negative sign, presenting only its raw magnitude.
Using the concept of modulus makes it easier to perform consistent mathematical calculations that require focus on size rather than size and direction (positive or negative nature).
For instance, the modulus of \(-13.4\) is 13.4, as the bars indicate we must take the non-negative measure of the number. This simple change aids greatly in real-world applications such as engineering, computer science, and even finance.

Non-Negative

A non-negative number is any number that is zero or positive. In the context of absolute value, the principle of non-negativity ensures that distances and magnitudes are straightforward to interpret and use.
When considering the absolute value of a given number, we expect the result to always be non-negative. This aligns with distance measurement on a number line, where distances are naturally non-negative. For the value \(-13.4\), its absolute value produces a non-negative result of 13.4, aligning the number with this principle.

• Numbers like 0, 5, 7.9, etc., are non-negative
• Absolute value of any real number is always non-negative

Non-negative results are essential for error-free calculations in mathematics and applied sciences, reinforcing clarity and unambiguity.