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Density of Rationals

Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers.

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Student-friendly explanation

No two rational numbers are immediate neighbours. If two rational numbers are given, their average is one rational number between them, and the process can be repeated again and again.

How to write this in exams

  1. 1

    Start with the exact idea

    Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers.

  2. 2

    Then show how to use it

    For one number, add the two numbers and divide by 2. For several numbers, convert both numbers to equivalent fractions with a larger common denominator, then choose numerators between them.

  3. 3

    Add one concrete example

    Between 1/3 and 1/2, one rational number is their average: (1/3 + 1/2)/2 = 5/12.

  4. 4

    Avoid this incomplete answer

    Between 3/5 and 4/5, saying no rational exists is wrong because 7/10, 13/20, and many more lie between them.

Definition

Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers.

Example

Between 1/3 and 1/2, one rational number is their average: (1/3 + 1/2)/2 = 5/12.

Rule to remember

If a and b are rational numbers and a < b, then (a + b)/2 is a rational number between them.

Memory hook

There is always room between rationals.

Examples and method

Worked example

Find a rational number between 3/5 and 4/5. Average = (3/5 + 4/5)/2 = (7/5)/2 = 7/10, and 7/10 lies between them.

Method to apply

For one number, add the two numbers and divide by 2. For several numbers, convert both numbers to equivalent fractions with a larger common denominator, then choose numerators between them.

Diagram support

A number-line zoom can show that intervals can always be divided again to locate more rational numbers.

How CBSE asks it

Questions may ask for one, two, or five rational numbers between given rational numbers.

Avoid common mistakes

Common confusion

Students often find only one rational number and think there are no more between the given numbers.

Common wrong answer

Between 3/5 and 4/5, saying no rational exists is wrong because 7/10, 13/20, and many more lie between them.

Exam tip

To find one rational number between two rationals, use the average method; to find many, make denominators same and insert numerators.

Quick check

Find one rational number between 2 and 3.

One rational number between 2 and 3 is 5/2, because 5/2 = 2.5 and it lies greater than 2 but less than 3.

Study the density of rationals diagram carefully

Use the labelled diagram to keep density of rationals clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of density of rationals in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on density of rationals.

Revision cue

Revise density of rationals through the labels before writing the answer.

Answer writing and exam use

1-mark answer

Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers.

2-mark answer

Density of rational numbers means that between any two rational numbers, there are infinitely many rational numbers. If a and b are rational numbers and a < b, then (a + b)/2 is a rational number between them. Between 1/3 and 1/2, one rational number is their average: (1/3 + 1/2)/2 = 5/12.

3-mark answer

No two rational numbers are immediate neighbours. If two rational numbers are given, their average is one rational number between them, and the process can be repeated again and again. If a and b are rational numbers and a < b, then (a + b)/2 is a rational number between them. Find a rational number between 3/5 and 4/5. Average = (3/5 + 4/5)/2 = (7/5)/2 = 7/10, and 7/10 lies between them. Questions may ask for one, two, or five rational numbers between given rational numbers. Between 3/5 and 4/5, saying no rational exists is wrong because 7/10, 13/20, and many more lie between them.
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