medium difficulty5 min

Density of Rationals

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

In one line: There is always room between rationals.

Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1Find a rational number between 3/5 and 4/5.
2Average = (3/5 + 4/5)/2 = (7/5)/2 = 7/10, and 7/10 lies between them.

Method you can reuse

1For one number, add the two numbers and divide by 2.
2For several numbers, convert both numbers to equivalent fractions with a larger common denominator, then choose numerators between them.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find one rational number between 2 and 3.

Think first, then show answer

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.

Common wrong answer

Between 3/5 and 4/5, saying no rational exists.

Why it loses marks: 7/10, 13/20, and many more lie between them.

Don't mix this up

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

Exam tip

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