Find five rational numbers between 1 and 2

General
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find five rational numbers between 1 and 2

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Find five rational numbers between 1 and 2

Understanding Rational Numbers

To find rational numbers between two given numbers, it’s important to first understand what a rational number is. A rational number is any number that can be expressed as the quotient or fraction \frac{p}{q} of two integers, where the numerator p is an integer, and the denominator q is a non-zero integer. Rational numbers include numbers such as \frac{1}{2}, \frac{3}{4}, and even whole numbers, like 1, which can be expressed as \frac{1}{1}. This implies we can always find an infinite number of rational numbers between any two distinct numbers.

Step-by-Step Process

To find five rational numbers between 1 and 2, we need fractions where each number is greater than 1 and less than 2.

1. Common Denominator Approach

An effective approach is to express the numbers with a common denominator. For instance, let’s express 1 and 2 with a common denominator of 10. So, 1 becomes \frac{10}{10} and 2 becomes \frac{20}{10}. Now, we can easily find rational numbers between these two fractions by choosing numerators between 10 and 20:

  1. \frac{11}{10}
  2. \frac{12}{10}
  3. \frac{13}{10}
  4. \frac{14}{10}
  5. \frac{15}{10}

Each of these fractions, such as \frac{11}{10}, is a rational number between 1 and 2, because 11 is between 10 (which represents 1) and 20 (which represents 2).

2. Simplified and Alternative Fractions

Now let’s express these numbers in their simplest forms, if necessary, though all the initial choices are already simple. However, let’s find a few alternative examples:

  1. \frac{3}{2} = 1.5
  2. \frac{5}{3} \approx 1.67
  3. \frac{7}{5} = 1.4
  4. \frac{9}{8} = 1.125
  5. \frac{17}{12} \approx 1.42

These fractions are also valid because they simplify into numbers (or remain in fractional form) that are greater than 1 and less than 2.

3. Decimals Converted to Fractions

Another approach is to convert decimal numbers between 1 and 2 into fractions. Consider the following examples:

  • 1.1 can be expressed as \frac{11}{10}
  • 1.2 can be expressed as \frac{12}{10} = \frac{6}{5}
  • 1.3 can be expressed as \frac{13}{10}
  • 1.4 can be expressed as \frac{14}{10} = \frac{7}{5}
  • 1.5 can be expressed as \frac{15}{10} = \frac{3}{2}

These fractions all meet the criteria as they are between the stated limits of 1 and 2.

4. Visualizing on a Number Line

A number line can help us visualize how these fractions sit between the integers 1 and 2. Here’s a simple representation:

1    11/10   6/5    13/10   7/5    3/2    2
|-----|------|------|------|------|------|

The fractions are evenly distributed between 1 and 2, illustrating that there are numerous rational numbers populating this interval.

Conclusion Summary

Rational numbers between 1 and 2 are abundant and easily derived through common denominators, simplified fractions, or decimal conversions. By employing varied methods such as direct fraction insertion or decimal conversion, an infinite number of rational numbers can be found between any two given numbers, exemplified by fractions like \frac{11}{10}, \frac{12}{10}, \frac{13}{10}, \frac{7}{5}, and \frac{3}{2}. Each depiction, whether through equations, simplified forms, or number lines, reinforces the flexibility in finding rational numbers within specified intervals, demonstrating their pervasive presence along any segment of the number line.

Feel free to explore more by considering different denominators or using other mathematical techniques to expand this list, @anonymous6!