Problem 64 - Elementary and Intermediate Algebra

Question

Find each absolute value. $$ \left|\frac{9}{16}\right| $$

Step-by-Step Solution

Answer

\(\left|\frac{9}{16}\right| = \frac{9}{16}\)

Step 1: Understand Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. Absolute values are always non-negative.

Step 2: Apply Absolute Value to Fraction

For the fraction \(\frac{9}{16}\), the absolute value involves removing any negative sign. However, since \(\frac{9}{16}\) is already positive, the absolute value is simply the fraction itself.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of a numerator and a denominator. The denominator tells us how many equal parts the whole is divided into, while the numerator indicates how many of those parts we are considering. For example, in the fraction \(\frac{9}{16}\), the number 16 is the denominator, which means the whole is divided into 16 parts. The number 9 is the numerator, so we consider 9 out of those 16 parts.

When dealing with fractions in mathematics, it's important to remember:

• Fractions can be less than 1, equal to 1, or greater than 1, depending on the values of the numerator and denominator.
• Fractions can represent rational numbers, which include integers when the denominator is 1.
• You can only add and subtract fractions with the same denominator; otherwise, you need to find a common denominator.

Understanding fractions is a fundamental skill that is used in a wide range of mathematical concepts and real-world applications.

Number Line

The number line is a visual tool that helps in understanding numbers, their order, and their distance from zero. It's a straight line where each point corresponds to a number, placed at equal intervals. Zero is typically in the middle, with positive numbers to the right and negative numbers to the left. On a number line, the distance between any two numbers is determined by subtracting one from the other, and this measure is always positive.

When it comes to absolute value, the number line is especially helpful because:

• It visually represents the distance of a number from zero, regardless of their position on the line (left or right).
• This makes it easier to understand why the absolute value is always non-negative; distances cannot be negative.
• It also helps demonstrate operations like addition and subtraction of numbers.

The number line provides a clear, visual understanding that complements the conceptual knowledge of numbers, positions, and distances.

Non-negative Numbers

Non-negative numbers are numbers that are either positive or zero. This means they include all the positive integers, decimals, and fractions, as well as zero. Non-negative numbers are critical in understanding the absolute value because the absolute value of any real number is a non-negative number.

Here are some key points about non-negative numbers:

• They are found to the right of zero on the number line.
• They include whole numbers and fractions, such as \(\frac{9}{16}\).
• Absolute value ensures that when finding distance or magnitude, the result is always non-negative.

Recognizing non-negative numbers is essential for applications in various fields of science, finance, and everyday life, where it's important to measure and quantify without negatives.