Find three rational numbers between

General
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find three rational numbers between

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Find three rational numbers between two given rational numbers

Understanding Rational Numbers

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. Examples of rational numbers include whole numbers like 3 (which can be expressed as \frac{3}{1}), fractions like \frac{3}{4}, and even decimals that terminate (like 0.75) or repeat (like 0.333…).

Finding Rational Numbers Between Two Rational Numbers

Suppose we are tasked with finding three rational numbers between two given rational numbers, say a = \frac{1}{4} and b = \frac{3}{4}.

  1. Convert to Equivalent Fractions:
    Start by making both fractions have a common denominator if they do not already. This simplifies comparison and calculation. In this case, both fractions already have the same denominator (4), so they are \frac{1}{4} and \frac{3}{4} respectively.

  2. Find the Midpoints:
    To find rational numbers between \frac{1}{4} and \frac{3}{4}, it’s helpful to consider midpoints or averages of these fractions. One straightforward way is to find the average of two fractions:

    $$m_1 = \frac{\frac{1}{4} + \frac{3}{4}}{2} = \frac{\frac{4}{4}}{2} = \frac{1}{2}$$

    So, \frac{1}{2} is a rational number between \frac{1}{4} and \frac{3}{4}.

  3. Find Additional Rational Numbers:
    Next, find more rational numbers within the segments \frac{1}{4} to \frac{1}{2} and \frac{1}{2} to \frac{3}{4}.

    Between \frac{1}{4} and \frac{1}{2}, calculate the midpoint:

    $$m_2 = \frac{\frac{1}{4} + \frac{1}{2}}{2} = \frac{\frac{1}{4} + \frac{2}{4}}{2} = \frac{\frac{3}{4}}{2} = \frac{3}{8}$$

    Between \frac{1/2} and \frac{3/4}, calculate another midpoint:

    $$m_3 = \frac{\frac{1}{2} + \frac{3}{4}}{2} = \frac{\frac{2}{4} + \frac{3}{4}}{2} = \frac{\frac{5}{4}}{2} = \frac{5}{8}$$

  4. Listing the Rational Numbers:
    With the calculations above, we now have three rational numbers between \frac{1}{4} and \frac{3}{4}:

    • \frac{3}{8}
    • \frac{1}{2}
    • \frac{5}{8}

Alternative Method: Using Arbitrary Denominator Expansion

Instead of using midpoints, expand each fraction’s denominator to a common larger number that makes it easier to find numbers in between. For instance:

  • Convert \frac{1}{4} to \frac{25}{100} and \frac{3}{4} to \frac{75}{100}, which uses a common denominator of 100.

In this expanded range:

  • Rational numbers between \frac{25}{100} and \frac{75}{100} can be selected easily, such as \frac{30}{100}, \frac{50}{100}, and \frac{60}{100} which reduce to \frac{3}{10}, \frac{1}{2}, and \frac{3}{5} respectively.

Generalizing the Approach

This method is versatile and can apply to any two rational numbers, regardless of their specific values. If the fractions don’t initially share a common denominator, convert both to equivalent fractions with a common denominator first. Then, either use the interval splitting (midpoint) method or expand both fractions into a larger and common base to identify intermediate rational numbers.

Summary

Finding rational numbers between two rational numbers involves converting to common denominators, dividing intervals for various points, and expanding fractions to obtain direct intermediate points. These approaches reliably provide solutions for any two rational numbers. Remember, the set of rational numbers is dense, meaning between any two rational numbers there are infinitely many other rational numbers. This ensures that finding more rational numbers between any two initial choices is always possible.

Feel free to ask further if you’d like examples with other numbers or details on particular steps. @anonymous6