Irrational numbers

All decimal numbers that are non terminating (that is, the digits after decimal point keep going on endlessly) and non recurring (that is, the digits after decimal do not show any repetition) are called irrational numbers.

The symbol for irrational numbers is Q.

Irrational numbers can not be written in the form of a/b, where 'a' and 'b' are integers and 'b' is not equal to zero.


Square roots of all numbers that are not perfect squares are irrational numbers. That is, all whole numbers which do not have a whole number square root have irrational square roots. So, √2, √3 and √5 are irrational numbers.

Representing irrational numbers by a number line:

√2 is an irrational number and it can be represented on the number line as follows
Representing √2 on a number line
  Draw a number line with 0, 1 and 2 on it.
  1. Let O be the point marked 0 on the number line.
  2. Let A be the point marked 1 on the number line.
  3. From point A, draw a perpendicular AB to the number line such that this perpendicular is equal to 1 unit length, the same as the length of one unit on the number line (= 0A)
  4. Now join the point B with O to get a right angled triangle OAB, which has a right angle at A.
  5. By Pythagorean theorem, (OA)2 + (AB)2 = (OB)2
  6. Since OA = 1 and AB = 1, therefore 12 + 12 = (OB)2
  7. So 1 + 1 = (OB)2
  8. 2 = (OB)2
  9. Taking square root on both sides, √2 = OB
Thus the length OB represents √2. Take a compass, place its point on O, and extend it to the point B. Draw a curve with radius OB such that it intersects the number line at a point, say C. The length of this point, OC is equal to √2.

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