Verified Answer :

  • Identify that the distance of a number \(x\) from \(0\) is given by its absolute value, \(|x|\).
  • Convert the mixed number \(5 \frac{2}{3}\) into an improper fraction: \(5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{17}{3}\).
  • Conclude that the distance is equal to the value of the number itself because it is positive.
  • State the final result: \(\boxed{5 \frac{2}{3}}\).

Explanation

  1. Analysis of the Problem

The problem asks for the distance of the number \(5 \frac{2}{3}\) from \(0\) on the number line. The distance of any real number \(x\) from \(0\) is defined as its absolute value, denoted by \(|x|\). Since \(5 \frac{2}{3}\) is a positive number, its distance from \(0\) is simply the number itself.

  1. Calculating the Distance

A mixed number \(a \frac{b}{c}\) can be converted into an improper fraction using the expression \(\frac{a \times c + b}{c}\). Applying this to \(5 \frac{2}{3}\):
\[5 \frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}.\]
Thus, the distance of \(5 \frac{2}{3}\) from \(0\) is \(\left| \frac{17}{3} \right| = \frac{17}{3}\). In mixed number notation, this remains \(5 \frac{2}{3}\).

  1. Final Answer

The distance of \(5 \frac{2}{3}\) from \(0\) is \(5 \frac{2}{3}\) units. In improper fraction form, this is \(\frac{17}{3}\) units. Therefore, the distance is \(\boxed{5 \frac{2}{3}}\).

Examples

In real-world applications, measuring distance from a zero point is fundamental. For example, if you are hiking and a sign indicates that a scenic viewpoint is \(5 \frac{2}{3}\) miles from your current location (the 'zero' point), understanding that distance represents the absolute displacement regardless of the direction you are walking is essential for planning your trip and estimating travel time. Knowing how to convert mixed numbers to improper fractions helps in these calculations, such as dividing the total distance by your average speed to find the time needed.

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