Rational Numbers: Definition & Examples

What are rational numbers

Numbers are core of mathematics. Almost all of mathematics parts consist of numbers. There are some types of numbers. One of them is rational numbers.

Rational numbers were invented by Greek mathematician, Phytagoras.

Rational numbers consist of natural numbers and integers.

Rational numbers symbolized as boldface Q (Q). It is type of numbers that can be expressed by fraction. There is numerator and denominator. All of them are integers.

Even the numbers are written by decimal, if it can be expressed by fraction, then it is rational numbers. But, not all of decimal numbers are rational numbers. If and only of it can be expressed by fraction that is rational numbers.

Another characteristic is if the decimal numbers have repeated numbers after comma (recurring decimal), then it can be expressed to be fraction form.

For example of rational numbers: 1, 2, ½, ¾, 0.123123.., etc. In another words, all of integers are rational numbers. It is because all of integer can be expressed by fraction.

Generally, there is formula or general form of rational numbers.


Where p and q are integer and q is not zero.

Another form of numbers that can not be expressed as fraction is called irrational numbers. For example: Pi value (3.14….), √2, 2/0, etc.

Are Negative Numbers Rational?

What is the answer of the question?

Negative numbers are rational numbers if they can be expressed by fraction.

Rational numbers is not always positive numbers. It is also consist of negative numbers. In another words, it can say that not all of negative numbers are rational numbers.

For example -1.234556789… that can not be written to be fraction form, it is not rational number.

Beside that, negative numbers in fraction that consist of p/q means one of them is negative. If both of them are negative it will be positive rational numbers.

Rational Numbers Example

There are another examples of rational numbers

-0.22/10 = –1/5
0.125125/1000 = 1/8

Rational Number Exercises

1. Which one of this numbers that is not rational number?

  • 1/4
  • 0.123456…..
  • 0.8888..
  • 3/5
  • 6

Answer: 0.123456….. (it is because it can not be fraction form)

2. Determine the length of hypotenuse of the right triangles (and determine whether it rational numbers or not) if the length of right sides are:

  • 3cm and 4cm
  • 2cm and √3 cm
  • √5cm and 1cm

Using Pythagorean theorem, let hypotenuse = h

  1. h = √(32 + 42) = √(9+16) = √25 = 5cm (rational number)
  2. h = √(22 + (√3)2 = √(4+3) = √7 cm (irrational number)
  3. h = √{(√5)2 +212} = √(5+1) = √6 cm (irrational number)

3. Determine the diagonal of a square (and determine whether it rational or irrational numbers) if the length of the side is:

  • 5 cm
  • 10cm
  • 2√2 cm

Diagonal of square is a√2.

  1. a = 5 diagonal is 5√2 cm (irrational)
  2. a = 10 diagonal is 10√2 (irrational)
  3. a = 2√2 diagonal is 2√2.√2 = 4 (Rational)

Learn more

Irrational Numbers

Rational Function

Rationalize the Denominator